Video: Using Trigonometric Ratios to Find the Exact Value of Trigonometric Functions

Find the exact value of cos (sin⁻¹(5/13)).

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Video Transcript

Find the exact value of cos of sin inverse of five over 13.

As we’re asked for the exact value, we might not be able to just put this expression into our calculator. And even if our calculator does give us the exact value of cos of sin inverse of five over 13, it’s probably a good idea to work out why this expression has such a nice exact value.

Let’s start by thinking about sin inverse of five over 13. What does this mean? In the right triangle shown the sine of 𝜃 is the ratio of the length of the side opposite 𝜃 to the length of the hypotenuse. In some ways, this is the definition of the sine function.

Applying the inverse sine function on both sides of this equation, we see that 𝜃 is the inverse sine of 𝑜 over ℎ. If we compare this to sine inverse of five over 13, we can see that 𝜃 is sine inverse five over 13 when the length of the opposite side 𝑜 is five and the length of the hypotenuse ℎ is 13. Changing the values of 𝑜 and ℎ on the diagram to five and 13, respectively, sin 𝜃 is now five over 13 and so 𝜃 is sin inverse of five over 13. Of course, we’re looking for cos of sine inverse of five over 13, which as 𝜃 is sine inverse of five over 13 is cos 𝜃.

In a right triangle, the cosine of an angle is the length of the adjacent side of that angle divided by the length of the hypotenuse of the right triangle. And so looking at our diagram, we can see that cos 𝜃 is 𝑎 — the length of the adjacent side — over 13 — the length of the hypotenuse. So the exact value that we’re looking for is just 𝑎 over 13, where 𝑎 is the length of the adjacent side.

We just have to find the value of 𝑎. How do we find this value? Well, we know that by the Pythagorean theorem that in a right triangle the square of the length of the hypotenuse is equal to the sum of squares of the lengths of the other two sides. And so applying this to our triangle, 𝑎 squared plus five squared is 13 squared. And we can solve this equation for 𝑎, finding that 𝑎 squared is 144. And so as 𝑎 is positive, 𝑎 must be 12. Using this value of 𝑎, we find that the exact value of cos of sine inverse of five over 13 is 12 over 13.

Whenever you want to evaluate the composition of a trigonometric function with the inverse of another trigonometric function, so for example, cos of sine inverse as we have here or also we could have cos inverse of sine or cot inverse of tan, it’s a good idea to draw a right triangle like we have or alternatively use the unit circle.

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