Question

If $ \sec \theta = 4 $ , how do you use the reciprocal identity to find $ \cos \theta $ ?

Answer 1

Hint: In this question we need to determine the value of $ \cos \theta $ by using the reciprocal identity. Hence, we will use the reciprocal property, $ \sec \theta = \dfrac{1}{{\cos \theta }} $ . And substitute the value of $ \sec \theta = 4 $ , then determine the value of $ \cos \theta $

Complete step-by-step answer:
The cosine function is the reciprocal of secant and secant function is also a reciprocal of cosine function.
We know that from reciprocal identity, $ \sec \theta = \dfrac{1}{{\cos \theta }} $
Then, we can say $ \cos \theta = \dfrac{1}{{\sec \theta }} $ $ \to \left( 1 \right) $
Therefore, here it is given that $ \sec \theta = 4 $ .
By substituting the value in the equation $ \left( 1 \right) $ , we have,
 $ \cos \theta = \dfrac{1}{4} $
Hence, by using the reciprocal identity $ \cos \theta = \dfrac{1}{4} $ .
So, the correct answer is “ $ \cos \theta = \dfrac{1}{4} $ ”.

Note: The reciprocal relation of a trigonometric function with another trigonometric function is called reciprocal identity. Every trigonometric function has a reciprocal relation with one another trigonometric function. The sine function is a reciprocal function of cosecant function and cosecant is also a reciprocal of sine. The cosine function is the reciprocal of secant and secant function is also a reciprocal of cosine function. Tangent function is a reciprocal of cotangent and cotangent function is also reciprocal of tangent function. The reciprocal identities are $ \sin \theta = \dfrac{1}{{\csc \theta }} $ , $ \cos \theta = \dfrac{1}{{\sec \theta }} $ , $ \tan \theta = \dfrac{1}{{\cot \theta }} $ , $ \csc \theta = \dfrac{1}{{\sin \theta }} $ , $ \sec \theta = \dfrac{1}{{\cos \theta }} $ and $ \cot \theta = \dfrac{1}{{\tan \theta }} $ .