Posts

Finding an Angle’s Sine Given the Sine of Its Complement

This post about finding an angle’s sine given the sine of its complement is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.

Question

Triangle ABC is a right triangle. Sin B = 3/8. What is the value of sin A?


Solution

“Sin” is an abbreviation for sine. The sine of angle A in a right triangle means the ratio of the length of the side opposite angle A to the length of the hypotenuse (longest side).

You are asked to find an angle’s sine given the sine of its complement. A diagram should help. We can make some guesses about how triangle ABC looks. It’s conventional to name the right angle of a right triangle C, so right triangle ABC may look like this:

Right triangle ABC. Side c is the hypotenuse, the longest side.

Right triangle ABC

Right triangle ABC. Side AB, the hypotenuse, is labelled 8 units long and side AC is labelled 3 units long.

Right triangle ABC.

That the sine of angle B is 3/8 means the length of the side opposite angle divided by the length of the hypotenuse is 3/8. 

Those lengths may be 3 and 8 (or 30 and 80 or 0.3 and 0.8 or any combination in the proportion of 3/8).

The sine of angle is equal to the length of the side opposite angle A divided by the length of the hypotenuse. What is the length of the side opposite angle A?

Sidebar: Pythagorean Theorem.

The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the legs.

Right triangle ABC. The side opposite angle A is labelled a; the side opposite angle B is labelled b; the side opposite angle C is labelled c.

Right triangle ABC

 In other words, using the labels in the triangle at left, where each side is named for the angle opposite it, .

Back to the Question

We need to know a. Solve the Pythagorean equation above for a:

And we know that and . Then

Now the triangle looks like this:

Right triangle ABC. The side opposite angle B is labelled 3; the side opposite angle C is labelled 8; and the side opposite angle A is labelled "square root of 55."

Right triangle ABC.

The sine of A is the length of side opposite A divided by the length of the hypotenuse:

Read more about sine.

This question is similar to question number 20 in the sample questions for the Accuplacer Advanced Algebra and Functions test.