Question

Angle A is the complement of angle B. Which equation about the 2 angles must be true? A. Sin A = Sin B B. Sin A = Cos A C. Cos B = Sin B D. Cos A = Sin B Explain.

Answer 1

Equation D is true for all `A`, `B in RR`, `A+B = pi/2`.

Explanation:

Two angles `alpha_1` and `alpha_2` are complementary if and only if `alpha_1+alpha_2 = pi/2`.

Taking each equation separately:

A. `sin A = sin B`

Let's prove that this is wrong by finding a counterexample.

Let `A= pi/3` and `B=pi/6`. Clearly, `A+B = pi/2`. By assuming equation A to be true, we must have

`sin(pi/3)=sin(pi/6)=> sqrt3/2 = 1/2 -= "False"`. So relation A is false.

B. `sin A = cosA`

Let's use the same counterexample. Let `A=pi/3`.

`sin(pi/3)=cos(pi/3) => sqrt3/2 = 1/2 -= "False"`. Relation B is false as well.

C. `cos B = sin B`

This is exactly the same as relation B, so it must be false.

D. `cos A = Sin B`

We know that `A+B = pi/2 => A=pi/2 - B`.

Relation D is equivalent to saying

`cos(pi/2 - B) = sin B`

We usually meet angles `B` and `pi/2-B` in right triangles, so, let's imagine a right triangle with `C` being the right angle and one of the angles is `B`.

From this image and by using the definitions of sine and cosine, we can derive that:

`sin B ="opposite"/"hypotenuse"= b/c`

`cosA = cos(pi/2 - B) ="adjacent"/"hypotenuse"=b/c`

Thus, we can see that `cos(pi/2 - B) = sin B`, or, equivanlently,

`color(purple)(cos A = sin B`, if `A` and `B` are complementary angles.

This proves that relation D is true.