Chapter 8.6, Problem 11CE

To determine

To explain: the relationship between cosine of $\angle A$ and the sine of its complement $\angle B$ .

Answer to Problem 11CE

  $\cos A=\sin B$.

Explanation of Solution

Given information: Two angles $\angle A\text{ and }\angle \text{B}$ are given, forming a complementary pair, that is $\angle A\text{ +}\angle \text{B=90}°$ .

From figure,

  

  $\cos A=\frac{AC}{AB}$, and $\sin B=\frac{AC}{AB}$

So, $\cos A=\sin B$.

Since, $\angle A$ and $\angle B$ form a complementary pair, we can say that the word “cosine” is related to the phrase “complement’s sine”.

Also,

  $\angle \text{A +}\angle \text{B = 90}°$ .

  $\begin{array}{l}\Rightarrow \angle A=90°-\angle B,\\ \text{taking cosine on both sides, we get,}\\ \text{ cos }A=\text{cos (90}°-B),\\ \Rightarrow \cos A=\sin B\text{ [since cos (90}°-B)=\sin B].\square \end{array}$