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Inverse Trigonometric Functions

Given an angle θ, we now know how to calculate the trig functions sin θ, cos θ and tan θ using a scientific calculator. Often, however, we need to be able to find the angle θ given the values of one of the trig functions sin θ, cos θ or tan θ.

To do this we need the inverse trigonometric functions:


If sin θ = x, then θ = sin-1x.


If cos θ = x, then θ= cos-1x.


if tan θ = x, then θ = tan-1x.

Warning: The exponent -1 on the inverse trig functions does not mean the reciprocal. For example,

Instead it is the inverse function of the sine. That is sin-1(sinθ) = θ and sin(sin-1θ) = θ. To avoid confusion, we also write  arcsin x,  or simply  asin x,   for  sin-1x (read as arc sine of x).

Similar notation is used for other trig functions:


If sin θ = x, then θ = sin-1x = arcsin x.


If cos θ = x, then θ = cos-1x = arccos x.


if tan θ = x, then θ = tan-1x = arctan x.

For example, in the following triangle sin θ = 4/5.



Check that this is equal to and using your scientific calculator.

Exercise 3

Given the value of a trig function at θ, use a scientific calculator to find the angle θ in degrees.

(Round your answer to 2 decimal places)


θ =
           θ = 2 d.p.


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