Video Transcript
๐ด๐ต๐ถ is a right-angled triangle
at ๐ต where the measure of angle ๐ถ is 62 degrees and ๐ด๐ถ is 17 centimeters. Find the lengths of ๐ด๐ต and ๐ต๐ถ
giving the answer to two decimal places and the measure of angle ๐ด giving the
answer to the nearest degree.
Letโs begin by drawing a sketch of
this triangle. Weโre told that it is right-angled
at ๐ต. So ๐ต is the vertex by the right
angle and the other two vertices are ๐ด and ๐ถ. The other information weโre given
is that the measure of angle ๐ถ is 62 degrees and ๐ด๐ถ is 17 centimeters. Weโre asked to find the length of
๐ด๐ต and ๐ต๐ถ. Those are the other two sides of
the triangle. So weโll call them ๐ฅ centimeters
and ๐ฆ centimeters. And weโll also ask to find the
measure of angle ๐ด.
Now, actually, we can work out the
measure of angle ๐ด straightaway because we have a triangle in which we know the
other two angles. The angle sum in any triangle is
180 degrees, so we can work out the measure of the third angle by subtracting the
other two from 180 degrees. That gives 28 degrees. Now, letโs think about how weโre
going to find the lengths of the other two sides of this triangle. Weโll begin by labeling all three
sides in relation to the angle of 62 degrees. ๐ด๐ถ is the hypotenuse, ๐ด๐ต which
weโre calling ๐ฅ centimeters is the opposite, and ๐ต๐ถ is the adjacent.
Weโll then recall the acronym
SOHCAHTOA to help us decide which trigonometric ratio we need to calculate the
length of each side. Starting with ๐ด๐ต, first of all,
the side we want to calculate is the opposite, and the side we know is the
hypotenuse. So weโre going to use the sine
ratio. This tells us that sin of an angle
๐ is equal to the opposite divided by the hypotenuse. Substituting the values for this
triangle, we have sin of 62 degrees is equal to ๐ฅ over 17. We solve for ๐ฅ by multiplying both
sides of the equation by 17 giving ๐ฅ equals 17 sin 62 degrees. Evaluating gives 15.0101 which we
round to 15.01.
To calculate the second side, ๐ต๐ถ,
we have a choice. As we now know the length of two
sides in this right triangle, we could calculate the length of the third side by
applying the Pythagorean theorem. But as weโre focusing on
trigonometry here, letโs instead calculate ๐ต๐ถ using the trigonometric ratios. This time, the side we want to
calculate is the adjacent and the side we were originally given is the
hypotenuse. So weโre going to use the cosine
ratio. Alternatively, we could use the
side weโve just calculated, which would give the pair O and A. So weโd be using the tan ratio. But it makes sense to use the value
we were originally given in case you made any mistakes when calculating the length
of the opposite.
Substituting 62 degrees for ๐, ๐ฆ
for the adjacent, and 17 for the hypotenuse gives cos of 62 degrees equals ๐ฆ over
17. We can then multiply both sides of
the equation by 17 to give ๐ฆ equals 17 cos 62 degrees and evaluate on our
calculators, making sure theyโre in degree mode. We then round to two decimal
places, giving 7.98. So weโve completed the problem. The length of ๐ด๐ต is 15.01
centimeters. The length of ๐ต๐ถ is 7.98
centimeters, each to two decimal places. And the measure of angle ๐ด is 28
degrees.
