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 $\cot \theta = \dfrac{1}{{\tan \theta }}$ .