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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
$x = sin θ$	$θ = sin^{- 1} x$
$x = cos θ$	$θ = cos^{- 1} x$
$x = tan θ$	$θ = tan^{- 1} x$
$x = cot θ$	$θ = cot^{- 1} x$
$x = sec θ$	$θ = sec^{- 1} x$
$x = csc θ$	$θ = csc^{- 1} x$

Equivalent notations for the inverse trigonometric ratios include

arcsin x

and

A r c s i n 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 $ENTER .$

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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.

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

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Digital Tools