Line segment 𝐴𝐵 is a diameter of a circle with a radius of 62 and a half centimeters. Point 𝐶 is on the circumference of the circle where line segment 𝐴𝐶 is perpendicular to line segment 𝐶𝐵 and 𝐴𝐶 equals 75 centimeters. Find the exact values of cos 𝐴 and sin 𝐵.
The first thing we can do is label this image with the information we know. The image already has points 𝐴, 𝐵, and 𝐶. It also already has the perpendicular intersection of 𝐴𝐶 and 𝐶𝐵. That would be this right angle. We’re given that the radius is 62 and a half. We could use that to find the diameter. The diameter equals two times the radius. Two times 62 and a half equals 125. And that means, line segment 𝐴𝐵 measures 125 centimeters. 𝐴𝐶 measures 75 centimeters. We want to know the cos of 𝐴 and the sin of 𝐵.
We sometimes use the memory device SOH CAH TOA to help us remember the sine, cosine, and tangent relationships. The sin of 𝜃 is its opposite side length over the hypotenuse. And the cos of 𝜃 equals the adjacent side length over the hypotenuse.
Let’s first consider the cos of angle 𝐴. The adjacent side length to angle 𝐴 measures 75 centimeters. 75 should be the numerator. The hypotenuse of a right triangle is its longest side. And it’s always opposite the right angle. In this case, the hypotenuse is 125. So, 125 is the denominator. We can reduce this fraction. Both 75 and 125 are divisible by 25. So, we divide the numerator and the denominator by 25. 75 divided by 25 equals three. And 125 divided by 25 equals five. The cos of 𝐴 is three-fifths.
Now we’ll consider sin of angle 𝐵. The opposite side length of angle 𝐵 is 75 centimeters. 75 is the numerator. And because angle 𝐵 is part of the same triangle as angle 𝐴, they have the same hypotenuse, which measures 125. 75 over 125 can, again, be reduced to three-fifths. The cos of 𝐴 and the sin of 𝐵 are equal to each other. They’re both three over five.