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Concept

Inverse Trigonometric Ratios

If the value of a trigonometric ratio for a specific angle θ is known, it is possible to calculate the measure of the angle using inverse trigonometric ratios. The inverse ratio of sine is called inverse sine and is written as sin^(- 1).
A right triangle with the acute angle of 30 degrees, opposite side 3 units long and the hypotenuse 6 units long; sin(30)=3/6; 30=sin^(-1)(3/6)

The rest of the inverse trigonometric ratios are defined in a similar way.

Trigonometric Ratio Inverse Ratio
x=sinθ θ = sin^(-1)x
x=cosθ θ=cos^(-1)x
x=tanθ θ=tan^(-1)x
x=cotθ θ=cot^(-1)x
x=secθ θ=sec^(-1)x
x=cscθ θ=csc^(-1)x
Equivalent notations for the inverse trigonometric ratios include arcsinx and Arcsin x. This is similar for the rest of the inverse ratios.
Alternative names of inverse trigonometric ratios

As long as the appropriate sides are being used, the same angle can be found by using different inverse trigonometric ratios.

The right triangle; 53\Deg=sin^(-1)(4/5); 53\Deg=cos^(-1)(3/5); 53\Deg=tan^(-1)(4/3)

Digital Tools

A graphing calculator can be used to find the values of the main inverse trigonometric functions. This can be done by pressing 2ND and the desired trigonometric function. Enter the desired value, close the parentheses, and press ENTER.

Since angles can be measured in degrees or radians, this must be specified in the calculator. This can be done by pressing MODE and selecting the desired output in the third row. The default option is usually Radian.

If Degree is selected, the output will be shown in degrees.

Extra

 Difference in Notations

There are three different notations for each inverse trigonometric ratios. For example, inverse trigonometric sine of x can be written as follows. sin^(- 1)x arcsinx Arcsin x In the first notation, it is important to not mistake - 1 for the negative power of 1. For that reason, writing arcsinx or Arcsin x is sometimes preferred as it leaves no space for misinterpretation. Note that there is a slight difference between these two notations.

arcsinx Arcsin x
Algebraic Definition { θ ∈ R: sin θ =x } { θ ∈ [- π2, π2]: sin θ =x }
Meaning All real angles θ that satisfy sin θ =x. One or more unique angles θ in the interval [- π2, π2] that satisfy sin θ =x.

In other words, arcsinx represents all the angles whose sine equals x, while Arcsin x represents only the principal angle in the main interval [- π2, π2]. However, this notation is not always followed. In practice, in most cases the lower case notation is used to denote the principal value, and the upper case notation can be not used at all.

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