Question

Which relationship in the triangle must be true?

sin(B) = sin(a)
sin(B) = cos(90 -B)
cos(B) = sin(180-B)
cos(B) cos(A)

Answer 1

sin(B) = cos(90 -B)

Explanation:

In triangle ABC by using angle sum property, ∠A + ∠B + ∠C= 180°

∠A + ∠B + 90°= 180°

∠A + ∠B= 180°-90°

∠A + ∠B = 90°

∠A = 90°- ∠B

sin B = b/a.

cos A = b/a.

Hence, sin B = cos A

put the value of ∠A = 90°- ∠B in cos A

sin (B) = cos (90°-B)

Thus, the correct answer is option (2).

Answer 2

sin(B)=cos(90°-B)

Explanation:

we know that

In the right triangle of the figure

sin(B)=b/c -----> The sine of angle B is equal to divide the opposite side to angle B by the hypotenuse

cos(A)=b/c -----> The cosine of angle A is equal to divide the adjacent side to angle A by the hypotenuse

we have that

sin(B)=cos(A)

Remember that

A+B=90° -----> by complementary angles

so

A=90°-B

therefore

sin(B)=cos(A)

sin(B)=cos(90°-B)