Video Transcript
For the given figure, find the
measure of angle 𝜃 in degrees to two decimal places.
Looking at the figure, we can see
that we have a right triangle, in which 𝜃 represents the measure of one of the
non-right angles. We’ve also been given the lengths
of two of the sides of the right triangle. They are three and eight units. We can therefore approach this
problem using trigonometry.
Our first step in any problem
involving trigonometry is to label the three sides of the triangle in relation to
the angle 𝜃. The side directly opposite the
right angle is the hypotenuse, which we’ll abbreviate to H. The side directly opposite the
angle 𝜃 is called the opposite, abbreviated to O. And the side between the angle 𝜃
and the right angle is the adjacent, which we’ll abbreviate to A.
We’ll now recall the acronym
SOHCAHTOA to help us decide which of the three trigonometric ratios we need to use
in this problem. The two side lengths we’ve been
given are the adjacent and the hypotenuse. So we’re going to be using the
cosine ratio. The cos of an angle 𝜃 is defined
to be the length of the adjacent side divided by the length of the hypotenuse. Substituting the values for this
triangle, we have that cos of 𝜃 is equal to three over eight, or three-eighths.
Now we need to find the value of
𝜃, which means we need to apply the inverse cosine function. This is the function that
essentially does the opposite of the cosine function. It says if cos of 𝜃 is equal to
three-eighths, then what is 𝜃? We have 𝜃 is equal to the inverse
cos of three-eighths.
We can then evaluate this on our
calculators, making sure they’re in degree mode. To access the inverse cosine
function, we usually need to press shift and then the cos button on our
calculators. Evaluating gives 67.975. And then we round to two decimal
places as specified in the question. So, by applying the inverse cosine
function in this right triangle, we found that the measure of angle 𝜃 to two
decimal places is 67.98 degrees.
