# Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 218: 27

$\csc(-56^\circ42')$ is the cofunction needed to be found.

#### Work Step by Step

$$\sec146^\circ42'$$ Cosecant and secant are cofunctions. So to write $\sec146^\circ42'$ in terms of cofunction, cosecant would be included. Now the job is to figure out the value of $\theta$, which must satisfy $$\csc\theta=\sec146^\circ42'\hspace{1cm}(1)$$ According to Cofunction Identity: $\csc\theta=\sec(90^\circ-\theta)$ Apply this to the equation $(1)$: $$\sec(90^\circ-\theta)=\sec146^\circ42'$$ $$90^\circ-\theta=146^\circ42'$$ $$\theta=90^\circ-146^\circ42'=-56^\circ42'$$ $\csc(-56^\circ42')$ is the answer to this exercise.

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