Video Transcript
For the given figure, find the
measures of angle 𝐴𝐶𝐵 and angle 𝐵𝐴𝐶 in degrees to two decimal places.
We’ve been given a right triangle
in which we know the lengths of two of its sides. We can therefore approach this
problem using trigonometry. Our first step in a problem like
this is to label the sides of the triangle. But in order to do this, we need to
know which angle we’re labeling the sides in relation to. Let’s calculate angle 𝐴𝐶𝐵 first,
and we’ll label this on our diagram as angle 𝑥. The hypotenuse of a right triangle
is always the same. It’s the side directly opposite the
right angle. The opposite is the side directly
opposite the side we’re interested in. So the side opposite angle 𝑥 is
the side 𝐴𝐵. And finally, the adjacent is the
side between our angle and the right angle. It’s the side 𝐵𝐶.
We can now recall the acronym
SOHCAHTOA to help us decide which of the three trigonometric ratios — sine, cosine,
or tangent — we need to use to answer this question. The sides whose lengths we’re given
are the opposite and adjacent. So we’re going to be using the tan
ratio. For an angle 𝜃 in a right
triangle, this is defined as the length of the opposite divided by the length of the
adjacent. Substituting 𝑥 for the angle 𝜃,
four for the opposite, and five for the adjacent, we have the equation tan of 𝑥 is
equal to four-fifths.
To determine the value of 𝑥, we
need to apply the inverse tangent function, which says if tan of 𝑥 is equal to
four-fifths, then what is 𝑥? Evaluating this on our calculators,
making sure they’re in degree mode, gives 38.659. We can round this to two decimal
places, giving 38.66. So we found the measure of our
first angle.
To calculate the final angle in the
triangle, we have a choice of methods. We could use trigonometry again, or
we could use the fact that the angles in a triangle sum to 180 degrees. It’s more efficient to use the
second method. So we have that the measure of
angle 𝐵𝐴𝐶 is equal to 180 degrees minus 90 degrees for the right angle minus
38.66 degrees for angle 𝐴𝐶𝐵, which is equal to 51.34 degrees. So we found the measures of the two
angles.
If we did want to use trigonometry,
we’d have to relabel the sides of the triangle in relation to this angle, which
means the opposite and adjacent sides would swap round. We’d still be using the tangent
ratio, but this time we’d have tan of 𝑦 is equal to five over four. We’d then have 𝑦 is equal to the
inverse tan of five over four, which is indeed equal to 51.34 when rounded to two
decimal places.
Our answer to the problem then is
that the measure of angle 𝐴𝐶𝐵 is 38.66 degrees and the measure of angle 𝐵𝐴𝐶 is
51.34 degrees, each rounded to two decimal places.
