Misplaced Pages

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78-416: The Sinclair Scientific was a 12-function, pocket-sized scientific calculator introduced in 1974, dramatically undercutting in price other calculators available at the time. The Sinclair Scientific Programmable , released a year later, was advertised as the first budget programmable calculator. Significant modifications to the algorithms used meant that a chipset intended for a four-function calculator

156-590: A calculator to compete with the HP-35 using this series of chips. Despite scepticism about the feasibility of the project from Texas Instruments engineers, Nigel Searle was able to design algorithms that sacrificed some speed and accuracy in order to implement scientific functions on the TMS0805 variation. The Sinclair Scientific first appeared in a case derived from that of the Sinclair Cambridge , but it

234-463: A carry case. The build time was advertised as being around three hours, and required a soldering iron and a pair of cutters. In January 1975, the kit was available for US$ 49.95 , half the price at the time of introduction a year earlier, and in December 1975 it was available for £9.95 , less than a quarter of the introductory price. The Sinclair Scientific Programmable was introduced in 1975, with

312-489: A cost of a million dollars, leading the Texas Instruments engineers to think that Sinclair's aim to build a scientific calculator around the TMS0805 chip, which could barely handle four-function arithmetic, was impossible. However, by sacrificing some speed and accuracy, Sinclair used clever algorithms to run scientific operations on a chip with room for just 320 instructions. Constants , rather than being stored in

390-744: A fragment of a sine table: Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real). A useful form that follows directly from the table above is It is obtained by recognizing that cos ⁡ ( arctan ⁡ ( x ) ) = 1 1 + x 2 = cos ⁡ ( arccos ⁡ ( 1 1 + x 2 ) ) {\displaystyle \cos \left(\arctan \left(x\right)\right)={\sqrt {\frac {1}{1+x^{2}}}}=\cos \left(\arccos \left({\sqrt {\frac {1}{1+x^{2}}}}\right)\right)} . From

468-1044: A modern scientific calculator include: In addition, high-end scientific calculators generally include some or all of the following: While most scientific calculators have traditionally used a single-line display similar to traditional pocket calculators , many of them have more digits (10 to 12), sometimes with extra digits for the floating-point exponent. A few have multi-line displays, with some models from Hewlett-Packard , Texas Instruments (both US manufacturers), Casio , Sharp , and Canon (all three Japanese makers) using dot matrix displays similar to those found on graphing calculators . Scientific calculators are used widely in situations that require quick access to certain mathematical functions, especially those that were once looked up in mathematical tables , such as trigonometric functions or logarithms . They are also used for calculations of very large or very small numbers, as in some aspects of astronomy , physics , and chemistry . They are very often required for math classes from

546-662: A result of 1.2500 01 (representing 12.5 which is equal to 25 sin(30°) ). Sine is selected with the combination of the ▲ key followed by the + key. The ▼ (down) and ▲ (up) arrow keys are function select keys. The four operation keys ( − , + , ÷ and × ) all have two other functions, activated by using one of the arrow keys. The functions available are sine, arcsine, cosine, arccosine, tangent, arctangent, logarithm and antilogarithm. Scientific calculator In some areas of study and professions scientific calculators have been replaced by graphing calculators and financial calculators which have

624-498: A single solution θ {\displaystyle \theta } to an elementary trigonometric equation ( sin ⁡ θ = y {\displaystyle \sin \theta =y} is such an equation, for instance, and because sin ⁡ ( arcsin ⁡ y ) = y {\displaystyle \sin(\arcsin y)=y} always holds, θ := arcsin ⁡ y {\displaystyle \theta :=\arcsin y}

702-477: A slightly altered Reverse Polish Notation method; lacking an enter key, the operation keys enter a number into the appropriate register and the calculation is performed. For example, (1+2) × 3 could be calculated as: C 1 + 2 + 3 × to give the result of 9.0000 00 ( 9.0000 × 10 , or 9). The C key performs a clear; pressing it sets the calculator to a state with zero in the internal registers. Pressing "C" followed by number keys then + effectively adds

780-449: A very large following. The HP-35 , introduced on February 1, 1972, was Hewlett-Packard's first pocket calculator and the world's first handheld scientific calculator. Like some of HP's desktop calculators it used RPN. Introduced at US$ 395, the HP-35 was available from 1972 to 1975. Texas Instruments (TI), after the production of several units with scientific notation , introduced a handheld scientific calculator on January 15, 1974, in

858-454: Is a shorthand way of saying that (at least) one of the following statement is true: As mentioned above, if arccos ⁡ x = π {\displaystyle \,\arccos x=\pi \,} (which by definition only happens when x = cos ⁡ π = − 1 {\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for

SECTION 10

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936-446: Is allowed to be a complex number , then the range of y {\displaystyle y} applies only to its real part. The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians . The symbol R = ( − ∞ , ∞ ) {\displaystyle \mathbb {R} =(-\infty ,\infty )} denotes

1014-5368: Is always a solution), the set of all solutions to it are: The equations above can be transformed by using the reflection and shift identities: These formulas imply, in particular, that the following hold: sin ⁡ θ = − sin ⁡ ( − θ ) = − sin ⁡ ( π + θ ) = − sin ⁡ ( π − θ ) = − cos ⁡ ( π 2 + θ ) = − cos ⁡ ( π 2 − θ ) = − cos ⁡ ( − π 2 − θ ) = − cos ⁡ ( − π 2 + θ ) = − cos ⁡ ( 3 π 2 − θ ) = − cos ⁡ ( − 3 π 2 + θ ) cos ⁡ θ = − cos ⁡ ( − θ ) = − cos ⁡ ( π + θ ) = − cos ⁡ ( π − θ ) = − sin ⁡ ( π 2 + θ ) = − sin ⁡ ( π 2 − θ ) = − sin ⁡ ( − π 2 − θ ) = − sin ⁡ ( − π 2 + θ ) = − sin ⁡ ( 3 π 2 − θ ) = − sin ⁡ ( − 3 π 2 + θ ) tan ⁡ θ = − tan ⁡ ( − θ ) = − tan ⁡ ( π + θ ) = − tan ⁡ ( π − θ ) = − cot ⁡ ( π 2 + θ ) = − cot ⁡ ( π 2 − θ ) = − cot ⁡ ( − π 2 − θ ) = − cot ⁡ ( − π 2 + θ ) = − cot ⁡ ( 3 π 2 − θ ) = − cot ⁡ ( − 3 π 2 + θ ) {\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&=-\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}} where swapping sin ↔ csc , {\displaystyle \sin \leftrightarrow \csc ,} swapping cos ↔ sec , {\displaystyle \cos \leftrightarrow \sec ,} and swapping tan ↔ cot {\displaystyle \tan \leftrightarrow \cot } gives

1092-582: Is ambiguous. Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “ −1 ” superscript: Sin ( x ) , Cos ( x ) , Tan ( x ) , etc. Although it is intended to avoid confusion with the reciprocal , which should be represented by sin ( x ) , cos ( x ) , etc., or, better, by sin x , cos x , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA ) use those very same capitalised representations for

1170-608: Is an angle θ {\displaystyle \theta } in some interval that satisfies cos ⁡ θ = x . {\displaystyle \cos \theta =x.} We want to find this θ . {\displaystyle \theta .} The table above indicates that the solution is θ = ± arccos ⁡ x + 2 π k  for some  k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which

1248-564: Is below the x {\displaystyle x} -axis (in which case θ = − π / 2 {\displaystyle \theta =-\pi /2} ). The table below shows how two angles θ {\displaystyle \theta } and φ {\displaystyle \varphi } must be related if their values under a given trigonometric function are equal or negatives of each other. The vertical double arrow ⇕ {\displaystyle \Updownarrow } in

1326-905: Is defined so that sin ⁡ ( y ) = x . {\displaystyle \sin(y)=x.} For a given real number x , {\displaystyle x,} with − 1 ≤ x ≤ 1 , {\displaystyle -1\leq x\leq 1,} there are multiple (in fact, countably infinitely many) numbers y {\displaystyle y} such that sin ⁡ ( y ) = x {\displaystyle \sin(y)=x} ; for example, sin ⁡ ( 0 ) = 0 , {\displaystyle \sin(0)=0,} but also sin ⁡ ( π ) = 0 , {\displaystyle \sin(\pi )=0,} sin ⁡ ( 2 π ) = 0 , {\displaystyle \sin(2\pi )=0,} etc. When only one value

1404-873: Is denoted by π Z   :=   { π n : n ∈ Z }   =   { … , − 2 π , − π , 0 , π , 2 π , … } . {\displaystyle \pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.} The symbol ∖ {\displaystyle \,\setminus \,} denotes set subtraction so that, for instance, R ∖ ( − 1 , 1 ) = ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) {\displaystyle \mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )}

1482-439: Is desired, the function may be restricted to its principal branch . With this restriction, for each x {\displaystyle x} in the domain, the expression arcsin ⁡ ( x ) {\displaystyle \arcsin(x)} will evaluate only to a single value, called its principal value . These properties apply to all the inverse trigonometric functions. The principal inverses are listed in

1560-419: Is equal to 0 {\displaystyle 0} ) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases arccos ⁡ x = 0 {\displaystyle \,\arccos x=0\,} and arccos ⁡ x = π , {\displaystyle \,\arccos x=\pi ,\,} we now focus on

1638-596: Is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos ⁡ θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol: And similarly for the secant function, where π h + π Parity ⁡ ( h ) {\displaystyle \pi h+\pi \operatorname {Parity} (h)} equals π h {\displaystyle \pi h} when

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1716-453: Is just another way of saying "for some integer k . {\displaystyle k.} " The symbol ⟺ {\displaystyle \,\iff \,} is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote for more details and an example illustrating this concept). where

1794-1186: Is known). Then arccos ⁡ x = arccos ⁡ 0 = π 2 {\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}} and moreover, in this particular case k = 0 {\displaystyle k=0} (for both the + {\displaystyle \,+\,} case and the − {\displaystyle \,-\,} case) and so consequently, θ   =   ± arccos ⁡ x + 2 π k   =   ± ( π 2 ) + 2 π ( 0 )   =   ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyle \,-\pi /2.} Without additional information it

1872-727: Is nonnegative on 0 ≤ y < π 2 , {\textstyle 0\leq y<{\frac {\pi }{2}},} but nonpositive on π 2 < y ≤ π . {\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).} If x {\displaystyle x}

1950-497: Is not possible to determine which of these values θ {\displaystyle \theta } has. An example of some additional information that could determine the value of θ {\displaystyle \theta } would be knowing that the angle is above the x {\displaystyle x} -axis (in which case θ = π / 2 {\displaystyle \theta =\pi /2} ) or alternatively, knowing that it

2028-416: Is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccos ⁡ x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and 0 < arccos ⁡ x < π , {\displaystyle \,0<\arccos x<\pi ,\,} statements (1) and (2) are different and furthermore, exactly one of

2106-404: Is the radius of the circle. Thus in the unit circle , the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is x " is the same as "the angle whose cosine is x ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages,

2184-643: Is the set of points in R {\displaystyle \mathbb {R} } (that is, real numbers) that are not in the interval ( − 1 , 1 ) . {\displaystyle (-1,1).} The Minkowski sum notation π Z + ( 0 , π ) {\textstyle \pi \mathbb {Z} +(0,\pi )} and π Z + ( − π 2 , π 2 ) {\displaystyle \pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}} that

2262-494: Is used above to concisely write the domains of cot , csc , tan ,  and  sec {\displaystyle \cot ,\csc ,\tan ,{\text{ and }}\sec } is now explained. Domain of cotangent cot {\displaystyle \cot } and cosecant csc {\displaystyle \csc } : The domains of cot {\displaystyle \,\cot \,} and csc {\displaystyle \,\csc \,} are

2340-443: Is useful (for example) to define the multivalued version of each inverse trigonometric function: tan − 1 ⁡ ( x ) = { arctan ⁡ ( x ) + π k ∣ k ∈ Z }   . {\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.} However, this might appear to conflict logically with

2418-566: Is written on the page using simple formatting tools. The first scientific calculator that included all of the basic ideas above was the programmable Hewlett-Packard HP-9100A , released in 1968, though the Wang LOCI-2 and the Mathatronics Mathatron had some features later identified with scientific calculator designs. The HP-9100 series was built entirely from discrete transistor logic with no integrated circuits , and

Sinclair Scientific - Misplaced Pages Continue

2496-458: The HP-35 was designed to complete calculations in under a second. Accuracy in scientific functions was also limited to around three digits at best, and there were a number of bugs and limitations. Ken Shirriff, an employee of Google , reverse engineered a Sinclair Scientific in 2013 and built a simulator using the original algorithms. The assembly kit consisted of eight groups of components, plus

2574-508: The Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x {\displaystyle x} is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation. Complementary angles: Negative arguments: Reciprocal arguments: The identities above can be used with (and derived from)

2652-554: The inverse trigonometric functions (occasionally also called antitrigonometric , cyclometric , or arcus functions ) are the inverse functions of the trigonometric functions , under suitably restricted domains . Specifically, they are the inverses of the sine , cosine , tangent , cotangent , secant , and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering , navigation , physics , and geometry . Several notations for

2730-424: The junior high school level through college , and are generally either permitted or required on many standardized tests covering math and science subjects; as a result, many are sold into educational markets to cover this demand, and some high-end models include features making it easier to translate a problem on a textbook page into calculator input, e.g. by providing a method to enter an entire problem in as it

2808-599: The "plus or minus" symbol ± , {\displaystyle \,\pm ,\,} whose meaning is now clarified. Only the solution to cos ⁡ θ = x {\displaystyle \cos \theta =x} will be discussed since the discussion for sec ⁡ θ = x {\displaystyle \sec \theta =x} is the same. We are given x {\displaystyle x} between − 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1} and we know that there

2886-431: The analogous equations for csc , sec ,  and  cot , {\displaystyle \csc ,\sec ,{\text{ and }}\cot ,} respectively. So for example, by using the equality sin ⁡ ( π 2 − θ ) = cos ⁡ θ , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,}

2964-423: The calculator, were printed on the case below the screen. It displays only in scientific notation, with a five digit mantissa and a two digit exponent , although a sixth digit of the mantissa was stored internally. Because of the way the processor was designed, it uses Reverse Polish notation (RPN) to input calculations. RPN meant that the difficult implementation of brackets, and the associated recursive logic,

3042-528: The capabilities of a scientific calculator along with the capability to graph input data and functions , as well as by numerical computing , computer algebra , statistical, and spreadsheet software packages running on personal computers . Both desktop and mobile software calculators can also emulate many functions of a physical scientific calculator. Standalone scientific calculators remain popular in secondary and tertiary education because computers and smartphones are often prohibited during exams to reduce

3120-680: The case where arccos ⁡ x ≠ 0 {\displaystyle \,\arccos x\neq 0\,} and arccos ⁡ x ≠ π , {\displaystyle \,\arccos x\neq \pi ,\,} So assume this from now on. The solution to cos ⁡ θ = x {\displaystyle \cos \theta =x} is still θ = ± arccos ⁡ x + 2 π k  for some  k ∈ Z {\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} } which as before

3198-546: The common semantics for expressions such as sin ( x ) (although only sin x , without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal ( multiplicative inverse ) and inverse function . The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos( x )) = sec( x ) . Nevertheless, certain authors advise against using it, since it

Sinclair Scientific - Misplaced Pages Continue

3276-549: The decimal point in the right place, some space was freed up. Trigonometric functions were implemented in about 40 instructions, and inverse trigonometric functions took almost 30 more instructions. Logarithms are about 40 instructions, with anti-log taking about 20 more. The code to normalize and display the computed values is roughly the same in the TI and Sinclair programs. The design of the algorithms meant that some calculations, such as arccos 0.2, could take up to 15 seconds, whereas

3354-432: The domains of the original functions. For example, using function in the sense of multivalued functions , just as the square root function y = x {\displaystyle y={\sqrt {x}}} could be defined from y 2 = x , {\displaystyle y^{2}=x,} the function y = arcsin ⁡ ( x ) {\displaystyle y=\arcsin(x)}

3432-1347: The equation cos ⁡ θ = x {\displaystyle \cos \theta =x} can be transformed into sin ⁡ ( π 2 − θ ) = x , {\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,} which allows for the solution to the equation sin ⁡ φ = x {\displaystyle \;\sin \varphi =x\;} (where φ := π 2 − θ {\textstyle \varphi :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: φ = ( − 1 ) k arcsin ⁡ ( x ) + π k  for some  k ∈ Z , {\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ   =   ( − 1 ) k arcsin ⁡ ( x ) + π k  for some  k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using

3510-1004: The fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k {\displaystyle h:=-k} proves that another solution to cos ⁡ θ = x {\displaystyle \;\cos \theta =x\;} is: θ   =   ( − 1 ) h + 1 arcsin ⁡ ( x ) + π h + π 2  for some  h ∈ Z . {\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .} The substitution arcsin ⁡ x = π 2 − arccos ⁡ x {\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;} may be used express

3588-523: The fact that sin {\displaystyle \sin } and csc {\displaystyle \csc } are reciprocals (i.e. csc = 1 sin {\displaystyle \csc ={\tfrac {1}{\sin }}} ), as are cos {\displaystyle \cos } and sec , {\displaystyle \sec ,} and tan {\displaystyle \tan } and cot . {\displaystyle \cot .} Useful identities if one only has

3666-1533: The first four solutions can be written in expanded form as: For example, if cos ⁡ θ = − 1 {\displaystyle \cos \theta =-1} then θ = π + 2 π k = − π + 2 π ( 1 + k ) {\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)} for some k ∈ Z . {\displaystyle k\in \mathbb {Z} .} While if sin ⁡ θ = ± 1 {\displaystyle \sin \theta =\pm 1} then θ = π 2 + π k = − π 2 + π ( k + 1 ) {\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)} for some k ∈ Z , {\displaystyle k\in \mathbb {Z} ,} where k {\displaystyle k} will be even if sin ⁡ θ = 1 {\displaystyle \sin \theta =1} and it will be odd if sin ⁡ θ = − 1. {\displaystyle \sin \theta =-1.} The equations sec ⁡ θ = − 1 {\displaystyle \sec \theta =-1} and csc ⁡ θ = ± 1 {\displaystyle \csc \theta =\pm 1} have

3744-717: The following table. Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π 2  or  π ≤ y < 3 π 2 ) {\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\pi \leq y<{\frac {3\pi }{2}})} , because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan ⁡ ( arcsec ⁡ ( x ) ) = x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},} whereas with

3822-522: The form of the SR-50 . TI's long-running TI-30 series being one of the most widely used scientific calculators in classrooms. Casio , Canon , and Sharp , produced their graphing calculators, with Casio's FX series (beginning with the Casio FX-1 in 1972 ). Casio was the first company to produce a Graphing calculator ( Casio fx-7000G ). Inverse trigonometric function In mathematics ,

3900-403: The integer h {\displaystyle h} is even, and equals π h + π {\displaystyle \pi h+\pi } when it's odd. The solutions to cos ⁡ θ = x {\displaystyle \cos \theta =x} and sec ⁡ θ = x {\displaystyle \sec \theta =x} involve

3978-636: The integer k {\displaystyle k} in the solution's formula is uniquely determined by θ {\displaystyle \theta } (for fixed r , s , x , {\displaystyle r,s,x,} and y {\displaystyle y} ). With the help of integer parity Parity ⁡ ( h ) = { 0 if  h  is even  1 if  h  is odd  {\displaystyle \operatorname {Parity} (h)={\begin{cases}0&{\text{if }}h{\text{

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4056-900: The integer k {\displaystyle k} is unique and completely determined by θ . {\displaystyle \theta .} If arccos ⁡ x = 0 {\displaystyle \,\arccos x=0\,} (which by definition only happens when x = cos ⁡ 0 = 1 {\displaystyle x=\cos 0=1} ) then ± arccos ⁡ x = 0 {\displaystyle \,\pm \arccos x=0\,} (because + arccos ⁡ x = + 0 = 0 {\displaystyle \,+\arccos x=+0=0\,} and − arccos ⁡ x = − 0 = 0 {\displaystyle \,-\arccos x=-0=0\,} so in both cases ± arccos ⁡ x {\displaystyle \,\pm \arccos x\,}

4134-691: The integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement (2) is K + 1 {\displaystyle K+1} (because θ = − π + 2 π ( 1 + K ) {\displaystyle \theta =-\pi +2\pi (1+K)} ). However, if x ≠ − 1 {\displaystyle x\neq -1} then

4212-477: The inverse trigonometric functions are often called by the abbreviated forms asin , acos , atan . The notations sin ( x ) , cos ( x ) , tan ( x ) , etc., as introduced by John Herschel in 1813, are often used as well in English-language sources, much more than the also established sin ( x ) , cos ( x ) , tan ( x ) – conventions consistent with the notation of an inverse function , that

4290-413: The inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin( x ) , arccos( x ) , arctan( x ) , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ , where r

4368-605: The last row indicates that θ {\displaystyle \theta } and φ {\displaystyle \varphi } satisfy | sin ⁡ θ | = | sin ⁡ φ | {\displaystyle \left|\sin \theta \right|=\left|\sin \varphi \right|} if and only if they satisfy | cos ⁡ θ | = | cos ⁡ φ | . {\displaystyle \left|\cos \theta \right|=\left|\cos \varphi \right|.} Thus given

4446-417: The likelihood of cheating. When electronic calculators were originally marketed they normally had only four or five capabilities ( addition , subtraction , multiplication , division and square root ). Modern scientific calculators generally have many more capabilities than the original four- or five-function calculator, and the capabilities differ between manufacturers and models. The capabilities of

4524-473: The natural logarithm of 10 (2.30259) and e (2.71828) are printed on the case. π (3.14159) and 57.2958 (180 / π ) are also on the case for trigonometry calculations. There was not enough internal memory to store these constants internally. Angles are computed using radians; degree values must be converted to radians by dividing by 57.2958. As an example, to calculate 25 sin (600×0.05°) one would enter C 6 E 2 + 0 0 5 × 5 7 2 9 5 8 E 1 ÷ ▲ + 2 5 E 1 × to get

4602-426: The number entered to the zero and stores it internally to be worked on in subsequent calculations. If the − key is pressed instead, the number is subtracted from zero, effectively entering a negative number. All numbers are entered in scientific notation. After entering the mantissa part of the number, the "E" exponent key is pressed prior to entering the integer exponent of the number. The task of ordering

4680-477: The operations is placed on the user, and there are no bracket keys. The display shows only five digits, but six digits can be entered. As an example 12.3×(−123.4+123.456) could be entered as C 1 2 3 4 E 2 - 1 2 3 4 5 6 E 2 + 1 2 3 E 1 × for a displayed result of 6.8880 -01 (representing 6.8880 × 10 , or 0.68880). Four constants are printed on the calculator case for easy reference. For converting to and from base 10 logarithms and natural logarithms,

4758-421: The programming was wasteful, with start and end quotes needed to use a constant in a program. However, included with the calculator was a library of over 120 programs that performed common operations in mathematics, geometry, statistics, finance, physics, electronics, engineering, as well as fluid mechanics and materials science. There were over 400 programs in the full Sinclair Program Library. The Sinclair used

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4836-541: The range ( 0 ≤ y < π 2  or  π 2 < y ≤ π ) {\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi )} , we would have to write tan ⁡ ( arcsec ⁡ ( x ) ) = ± x 2 − 1 , {\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},} since tangent

4914-526: The right hand side of the above formula in terms of arccos ⁡ x {\displaystyle \;\arccos x\;} instead of arcsin ⁡ x . {\displaystyle \;\arcsin x.\;} Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length x , {\displaystyle x,} then applying

4992-601: The same case as the Sinclair Oxford . It was larger than the Scientific, at 73 by 155 by 34 millimetres (2.9 in × 6.1 in × 1.3 in), and used a larger PP3 battery , but could also be powered by mains electricity . It had 24-step programming abilities, which meant it was highly limited for many purposes. It also lacked functions for the natural logarithm and exponential function . Constants used in programs were required to be integers , and

5070-678: The same solutions as cos ⁡ θ = − 1 {\displaystyle \cos \theta =-1} and sin ⁡ θ = ± 1 , {\displaystyle \sin \theta =\pm 1,} respectively. In all equations above except for those just solved (i.e. except for sin {\displaystyle \sin } / csc ⁡ θ = ± 1 {\displaystyle \csc \theta =\pm 1} and cos {\displaystyle \cos } / sec ⁡ θ = − 1 {\displaystyle \sec \theta =-1} ),

5148-1358: The same. They are the set of all angles θ {\displaystyle \theta } at which cos ⁡ θ ≠ 0 , {\displaystyle \cos \theta \neq 0,} π Z + ( − π 2 , π 2 ) = ⋯ ∪ ( − 3 π 2 , − π 2 ) ∪ ( − π 2 , π 2 ) ∪ ( π 2 , 3 π 2 ) ∪ ⋯ = R ∖ ( π 2 + π Z ) {\displaystyle {\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}} Each of

5226-1359: The same. They are the set of all angles θ {\displaystyle \theta } at which sin ⁡ θ ≠ 0 , {\displaystyle \sin \theta \neq 0,} i.e. all real numbers that are not of the form π n {\displaystyle \pi n} for some integer n , {\displaystyle n,} π Z + ( 0 , π ) = ⋯ ∪ ( − 2 π , − π ) ∪ ( − π , 0 ) ∪ ( 0 , π ) ∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}} Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } : The domains of tan {\displaystyle \,\tan \,} and sec {\displaystyle \,\sec \,} are

5304-392: The set of all real numbers and Z = { … , − 2 , − 1 , 0 , 1 , 2 , … } {\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}} denotes the set of all integers . The set of all integer multiples of π {\displaystyle \pi }

5382-549: The six standard trigonometric functions. It is assumed that the given values θ , {\displaystyle \theta ,} r , {\displaystyle r,} s , {\displaystyle s,} x , {\displaystyle x,} and y {\displaystyle y} all lie within appropriate ranges so that the relevant expressions below are well-defined . Note that "for some k ∈ Z {\displaystyle k\in \mathbb {Z} } "

5460-495: The standard trig functions, whereas others ( Python , SymPy , NumPy , Matlab , MAPLE , etc.) use lower-case. Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions. Since none of the six trigonometric functions are one-to-one , they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of

5538-431: The trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2 π : {\displaystyle 2\pi :} This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer. The following table shows how inverse trigonometric functions may be used to solve equalities involving

5616-524: The two equalities holds (not both). Additional information about θ {\displaystyle \theta } is needed to determine which one holds. For example, suppose that x = 0 {\displaystyle x=0} and that all that is known about θ {\displaystyle \theta } is that − π ≤ θ ≤ π {\displaystyle \,-\pi \leq \theta \leq \pi \,} (and nothing more

5694-516: Was able to process scientific functions , but at the cost of reduced speed and accuracy. Compared to contemporary scientific calculators, some functions were slow to execute, and others had limited accuracy or gave the wrong answer, but the cost of the Sinclair was a fraction of the cost of competing calculators. In 1972, Hewlett-Packard launched the HP-35 , the world's first handheld scientific calculator. Despite market research suggesting that it

5772-712: Was advertised as "the first ... calculator to offer a ... programming facility ... at a price within the reach of the general public," but was limited by having only 24 program steps. Both the Sinclair Scientific and the Sinclair Scientific Programmable were manufactured in England, like all other Sinclair calculators except the Sinclair President . The HP-35 used five chips, and had been developed by twenty engineers at

5850-485: Was not necessary to implement in the hardware, but the effort was instead offloaded to the user. Instead of an "equals" button, the + or − keys are used to enter the initial value of a calculation, followed by subsequent operand(s) each followed by their appropriate operator(s). To fit the program into the 320 words available on the chip, some significant modification was used. By not using ordinary floating point numbers, which require many instructions to keep

5928-631: Was not part of the same range. The initial retail price was £49.95 in the UK (equivalent to £478 in 2016), and in the US for US$ 99.95 as a kit or US$ 139.95 fully assembled. By July 1976, however, it was possible to purchase one for £7 (equivalent to £46 in 2016). The Sinclair Scientific Programmable was introduced in August 1975, and was larger than the Sinclair Scientific, at 73 by 155 by 34 millimetres (2.9 in × 6.1 in × 1.3 in). It

6006-586: Was one of the first uses of the CORDIC algorithm for trigonometric computation in a personal computing device, as well as the first calculator based on reverse Polish notation (RPN) entry. HP became closely identified with RPN calculators from then on, and even today some of their high-end calculators (particularly the long-lived HP-12C financial calculator and the HP-48 series of graphing calculators ) still offer RPN as their default input mode due to having garnered

6084-487: Was too expensive for there to be any real demand, production went ahead. It cost US$ 395 (about £165 ), but despite the price, over 300,000 were sold in the three and a half years for which it was produced. From 1971, Texas Instruments had been making available the building block for a simple calculator on a single chip and the TMS0803 chipset appeared in a number of Sinclair calculators. Clive Sinclair wanted to design

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