Chapter 8.6, Problem 25WE

To determine

To prove: if R is any acute angle, ${(\sin R)}^2+{(\cos R)}^2=1$.

Answer to Problem 25WE

If R is any acute angle, ${(\sin R)}^2+{(\cos R)}^2=1$ .

Explanation of Solution

Given information: Hint: From any point on one side of $\angle R$ , draw a perpendicular to the other side.

Calculation:

Let’s draw a perpendicular to the other side from any point on one side of $\angle R$ .

  

Let’s say the side shown in figure has length $l$ ,

Therefore, the other side will have length, $l\cos R$ and the perpendicular line will have length, $l\sin R$

Now, these three lines will make a right − angled triangle. From the Pythagorean theorem,

  $\begin{array}{l}{\left(l\sin R\right)}^2+{\left(l\cos R\right)}^2=l^2,\\ \Rightarrow l^2{\sin }^{2}R+l^2{\cos }^{2}R=l^2,\\ $ \Rightarrow {(\sin R)}^2+{(\cos R)}^2=1.\square $\end{array}$