For the given figure, find the measure of angle 𝐵𝐴𝐶, in degrees, to two decimal places.
The first thing we need to do is identify angle 𝐵𝐴𝐶, which is here. We can call this angle 𝜃. In relation to this angle, we know an opposite side length and the hypotenuse length. We know that this is the hypotenuse because it’s the side length that’s opposite the right angle. In right-angle trigonometry, the relationship of opposite over the hypotenuse is sin of 𝜃.
We sometimes use the memory device SOHCAHTOA to represent the relationships of sine, cosine, and tangent. Sine is the opposite over the hypotenuse. That means we can say that sin of 𝜃 equals eight over 13. In order to get 𝜃 by itself, we need to take the sin inverse of the sin of 𝜃. And if we take the sine inverse of the left side of the equation, we need to take the sine inverse of the right side of the equation. The sin inverse of sin of 𝜃 just equals 𝜃. This angle measure is equal to the sin inverse of eight over 13.
At this point, we’ll need a calculator. In a calculator, you’ll find the sin inverse of eight over 13. Notice that, in my calculator, it’s showing in degrees. This is because I know that I want to find the measure of this angle in degrees. Once I’ve checked that everything is in degrees, I hit enter and I get
37.97987224 [37.97987244]. That means 𝜃 equals 37.97987244 degrees.
We’re rounding to two decimal places. That’s the hundredths place. And so we consider the digit to the right of the hundredths place the deciding digit. This deciding digit is larger than five. And that means we’ll round up. The seven in the hundredths place rounds up to an eight. And so rounded to two decimal places, 𝜃 equals 37.98 degrees.
Now if your calculator gives you something like 0.66, that means it’s operating in radians. We can also visually tell that the angle 𝜃 is larger than half a degree. If that happens to you, you need to find in your calculator settings where you change it from degrees to radians. And make sure that you’ve selected degrees for your function.
The measure of angle 𝐵𝐴𝐶 equals 37.98 degrees.