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Video: Using Reciprocal Identities to Complete Trigonometric Expressions

Alex Cutbill

Complete the following: cos 𝜃 = 1/_. [A] sin 𝜃 [𝐵] sec 𝜃 [C] cot 𝜃 [𝐷] csc 𝜃


Video Transcript

Complete the following: cos 𝜃 is equal to one over something. And we have four options: option A is sin 𝜃, 𝐵 is sec 𝜃, C is cot 𝜃, and 𝐷 is csc 𝜃.

Okay. So let’s give that unknown something a name; let’s call it 𝑥. So now we have cos 𝜃 is equal to one over 𝑥. And we want to find what 𝑥 is which means rearranging this equation for 𝑥.

So first we multiply both sides by 𝑥 to get 𝑥 cos 𝜃 is equal to one. Dividing both sides by cos 𝜃, we get that 𝑥 is equal to one over cos 𝜃. But unfortunately that isn’t one of our options. We have either A sin 𝜃, B sec 𝜃, C cot 𝜃, or D csc 𝜃 to choose from.

So at this point, we need to remember that the reciprocals of the trig functions sin, cos, and tan have other names. For example, csc 𝜃 is defined to be one over sin 𝜃. Cot 𝜃 is one over 10 𝜃. And the one that’s useful to us is sec 𝜃 is equal to one over cos 𝜃.

So 𝑥, which is one over cos 𝜃, can also be written as sec 𝜃. And if we look through our options, we’ll see that B is our answer.