Question

In ΔABC , angle C is a right angle.

Which statement must be true?






A.sin A = cos B
B.sin A = tan B
C.sin B = tan A
D.sin B = cos B

Answer 1

In a right-angled triangle such as ∆ABC, the sine of one acute angle is the cosine of the other acute angle. Therefore, the statement sin A = cos B must be true.

Step-by-step explanation:

In the context of a right-angled triangle, such as ∆ABC given that angle C is a right angle, certain trigonometric identities must apply. These relate the sine, cosine, and tangent functions of the angles within the triangle. In right-angled triangles, the sine of one acute angle is the cosine of the other because they are complementary angles (adding up to 90 degrees).

Therefore, the correct statement that must be true is sin A = cos B. This is because in a right-angled triangle the sides opposite to angles A and B are the adjacent and opposite sides for the other angle, respectively, when considering the right angle as a reference. Consequently, the ratio of the lengths of these sides (opposite to A and adjacent to B in respect to the hypotenuse) defines the sine of angle A and the cosine of angle B.

Answer 2

Answer: Choice A. sin(A) = cos(B)

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Step-by-step explanation:

The rule is that sin(A) = cos(B) if and only if A+B = 90.

Note how

• sin(A) = opposite/hypotenuse = BC/AB
• cos(B) = adjacent/hypotenuse = BC/AB

Since both result in the same fraction BC/AB, this helps us see why sin(A) = cos(B). Similarly, we can find that cos(A) = sin(B).

In the diagram below, the angles A and B are complementary, meaning they add to 90 degrees. So this trick only applies to right triangles.

The side lengths can be anything you want, as long as you're dealing with a right triangle.