Equations involving trigonometric ratios

Question

How would I simplify

$\cos (61) + \sin (29)$

Is it easy? I have never done something like this before, and cannot find many literature online about this topic.

Answer 1

Note that $61+29=90$, so $\sin(29)=\sin(90-61)$, and since $\sin(90-x)=\cos(x)$, you have

$$\cos(61)+\sin(29)=2\cos(61)$$

Also, $\cos(90-x)=\sin(x)$, so you also get

$$\cos(61)+\sin(29)=\cos(90-29)+\sin(29)=2\sin(29)$$

Answer 2

Are you familiar with the trigonometric identity:

$$\sin(90-a)=\cos(a)$$

So we have $\cos(61)=\sin(90-61)=\sin(29)$

So using $\cos(61)=\sin(29)$ we have

$\cos(61)+\sin(29)=\sin(29)+\sin(29)=2\sin(29)$