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 37.97987244. That means 𝜃 equals 37.97987244
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