CHAPTER 10
Right Triangle Trigonometry
Trigonometry means “triangle measurement.” It uses ideas from similarity to provide a means of finding measurements of sides or angles in right triangles.
Similar Right Triangles
To be certain that two triangles are similar, all that is necessary is to show that two angles in one triangle are congruent to two angles of the other. Because all right triangles have a right angle, they’re halfway to being similar. All that’s needed is to find a pair of acute angles, one from each triangle, that are congruent. That means that if you have a right triangle with a 23° angle, it’s similar to every other right triangle with a 23° angle. It’s as if the universe of right triangles is divided into families of similar right triangles, based on the size of the acute angles. In the figure below, ΔABK ~ ΔACJ ~ ΔADI ~ ΔAEH ~ ΔAFG. All of these triangles are similar because each has a right angle and each contains ∠A.
Let’s focus on ΔABK and ΔAFG, the smallest and largest of the group. Obviously ∠A ≅ ∠A, and the right angles ∠ABK and ∠AFG are congruent, so by the Third Angle theorem, ∠AKB ≅ ∠AGF. Corresponding sides are in proportion, and the proportion for these two triangles would say 
. The last ratio in that proportion is the ratio of the two hypotenuses, and the middle ratio compares the sides opposite ∠A. Sides 
and 
, whose lengths are compared in the first ratio, are the sides adjacent to ∠A. Of course, ∠A is between two sides, but one is always called the hypotenuse, so the non-hypotenuse side is referred to as the adjacent side. The proportion above says 
. Whatever scale factor enlarges or reduces the adjacent sides affects the other sides the same way.
If we focus on just two of the ratios at a time, and apply properties of proportions, 
could become 
, and 
could become 
. For example, 
says the ratio of the adjacent side to the opposite side in little ΔABK is the same as the ratio of the adjacent side to the opposite in big ΔAFG.
There are many different ways of organizing the information about corresponding sides into proportions. The key is to have a plan for what you want to compare, and hold to that for the entire proportion.
The area of mathematics labeled trigonometry is based in similar triangles. Use properties of similarity to complete these exercises.
1. Write a similarity statement that makes clear the correspondence between the similar right triangles below.
2. If ΔDOG ~ ΔCAT, write the extended proportion that describes the relationships among the sides.
3. ΔARM ~ ΔLEG. If AR = 22 cm, RM = 17 cm, and LE = 33 cm, find the length of 
.
4. ΔPQR ~ ΔMLK. If PQ = 21 inches, QR = 25 inches, and LK = 5 inches, find the length of ML. What is the scale factor which should be applied to ΔPQR to produce ΔMLK?
5. If the triangles in exercise 4 are right triangles, find the length of each hypotenuse.
6. In each of the similar right triangles below, what is the ratio of the leg opposite ∠A to the leg adjacent to ∠A?
7. In each of the similar right triangles below, what is the ratio of the leg opposite ∠X to the hypotenuse?
Basic Ratios
As the last few exercises suggest, a particular ratio of opposite side to hypotenuse, or adjacent side to opposite side, or whatever, turns out to be the same for every triangle in a particular family. If your right triangle has a 60° angle, the ratio of the side to adjacent to the 60° angle to the hypotenuse is 
. It may be that the adjacent side is 3 inches and the hypotenuse is 6 inches, or it may be that the adjacent side is 20 miles and the hypotenuse is 40 miles. It will always reduce to . Not every ratio will be such a tidy fraction. If your angle is 53° instead of 60°, the ratio will be an irrational number approximately equal to 0.545, but just as the ratio will be for the entire 60° family, it will be approximately 0.545 for the entire 53° family.
If every family of right triangles has a set value for each of the six possible ratios, then it makes sense to record those values to make them easier to use. First, we name the ratios to make them easier to talk about.
EXAMPLE
Instead of “for every right triangle in the 60° family, the ratio of the adjacent side to the hypotenuse is ” we say cos(60°) = . Cosine is the name of that ratio, it’s abbreviated as cos, and 60° is the “family.”
The values for all these ratios were recorded in tables, and you could look up the values you wanted to use (and you still can.) Now, scientific calculators or graphing calculators can provide the values for you. You’ll see keys for sin, cos, and tan on your calculator. If you want the value for the sine, cosine, or tangent for any family of right triangles, those keys will provide that information.
Use the right triangle below and the definitions of the trig ratios to find the ratios in questions 1 through 6.
1. sin(∠R)
2. cos(∠T)
3. sec(∠T)
4. cot(∠R)
5. cot(∠T)
6. tan(∠R)
7. The trig ratios are defined for the acute angles of a right triangle. What would happen if you tried to apply the definitions to the right angle?
8. Use your calculator to find each of the following, rounded to the nearest thousandth.
a) sin(30°)
b) cos(45°)
c) tan(82°)
d) cos(11°)
e) sin(75°)
EXAMPLE
The similar triangles method says if ΔABC ~ ΔXYZ and you have enough information to set up a proportion, you can solve for an unknown side. If 
, then 15x = 45 and x = 3.
The trigonometry method says if you have a right triangle with an acute angle of 59°, you know that the side opposite the 59° angle is 5 and you are looking for the adjacent side, you can write 
and you can find out that tan(59°) is approximately 1.66. That takes the place of the ratio from ΔABC, and you can check that 15 ÷ 9 is very close to the value the calculator gives for tan(59°).
Finding a Side
Because you’re able to call up the value of a particular ratio for any family, you can find the length of an unknown side without needing to have two similar triangles. You just need to have one right triangle, know the measurement of one acute angle, and know the length of one side.
To use trig ratios to find a missing side in a right triangle:
Use the right triangle below and the definitions of the trig ratios to find the ratios in questions 1 through 6.
1. Use a calculator to find each of the following to the nearest thousandth.
a) sin(78°)
b) cos(15°)
c) cos(43°)
d) tan(62°)
e) sin(21°)
2. Right triangle ΔXYZ has hypotenuse 
, which measures 38 cm. If m∠X = 30°, find the length of leg 
and the length of 
.
3. In right triangle ΔABC, ∠C measures 79° and side measures 35 cm. Find the length of hypotenuse 
to the nearest tenth.
4. Right triangle ΔRST has a 45° angle and a leg 12 inches long. Find the length of the hypotenuse, to the nearest tenth of an inch.
5. ΔXYZ is a right triangle with m∠X = 40° and one leg XY= 16 in. Find the length of the other leg, 
.
6. Rectangle ABCD has a diagonal BD = 12 inches, and ∠BDA measures 15°. Find the length of 
.
7. How long is diagonal 
in rectangle WXYZ if m∠WXZ = 47° and XW = 5 meters?
Finding an Angle
Trig ratios give you a quick way to find an unknown side of a right triangle, as long as you know the measure of one acute angle of the right triangle. But what if you don’t know the measure of the angle? What if the measure of the angle is exactly what you’re trying to find?
You can do that with a little more information. You’ll need to know two sides of the right triangle, and you’ll need the inverse operation. You’ve met other inverses; square roots undo the work of squaring, for example. When you write cos(60) = 
, you’re saying the right triangles in the 60° family have a ratio of adjacent side to hypotenuse that reduces to 
. What you need now is to say “what angle has a cosine value of 
?” You can write that as 
. The −1 is not an exponent but a symbol for the inverse or opposite process. To find the angle that has a sine equal to 
, you want 
.
There is another notation for the inverse of the trig relationships that is sometimes used. Instead of 
, the same idea could be expressed as 
, 
could be written as 
, and 
could be arctan(1).
To find the measure of an acute angle of a right triangle:
EXAMPLE
ΔABC has right angle ∠B. Hypotenuse 
measures 14 cm and leg 
measures 8 cm. To find the measure of ∠C, notice that you have the hypotenuse and the side opposite ∠C. The opposite and hypotenuse can create the sine ratio. 
. Simplifying the fraction is not absolutely necessary, so use 
if you prefer. Write 
, and then find 
above the sin key on your calculator. Type 2nd and then sin to get . Type 4/7, close the parentheses, and press ENTER. 
when rounded to the nearest thousandth.
Use inverse trig functions to complete these questions. Round answers as indicated.
1. Find each of the following to the nearest degree.
a) 
b) 
c) 
d) 
e) 
2. In right triangle ΔABC, leg measures 39 cm and leg 
measures 62 cm. Find the measure of ∠A to the nearest tenth of a degree.
3. The hypotenuse of right triangle ∠JKL is JL = 23 inches. If leg 
measures 14 inches, what is the measure of ∠L to the nearest tenth of a degree?
4. Find the measure of the smaller acute angle of right triangle ΔXYZ if the legs have measures of XY = 35 cm and YZ = 40 cm. Round to the nearest tenth of a degree.
5. Find the measure of the larger acute angle of right triangle ΔABC if hypotenuse AC = 20 inches and shorter leg AB = 10 inches.
6. Find the measures of the acute angles of a right triangle if the hypotenuse is 
cm and a leg measures 17 cm.
7. A right triangle has sides that are a 3-4-5 Pythagorean triple. Find the measure of the acute angles to the nearest tenth of a degree.
Reciprocal and Complementary Relationships
When the trig ratios were named and defined, you may have noticed that there were two ratios that involved each pair of sides. The six different ratios are three pairs of reciprocals, and like all reciprocal pairs, their product is 1.
The reciprocals give you a choice when you’re trying to find an unknown side. If you know the opposite and are looking for the adjacent, you can use
or 
, whichever you prefer. Each has an advantage. If you are looking for AB, you may prefer to have the variable in the numerator, because it’s faster to solve. On the other hand, using sin, cos, or tan means you have a key on your calculator that will give you that value. You’ll have to work a bit to get the value of sec, csc, or cot.
EXAMPLE
To find the value of csc(37°) on your calculator, use 1 ÷ sin(37°). To find sec(49°), enter 1 ÷ cos(49°). If you need cot(65°), you want 1 ÷ tan(65°).
The “co” in the names of three of the six trig ratios comes from “complementary.” Two angles are complementary if their measures add to 90°, and the two acute angles in a right triangle are complementary. You may have noticed that the sine of one angle in a right triangle is the same value as the cosine of the other angle. The sine of one angle is the cosine of its complement. The complementary relationships are sine and cosine, secant and cosecant, and tangent and cotangent.
If you’re trying to find the measure of an acute angle in ΔABC, and you know the lengths of leg and hypotenuse , you can use 
or 
, depending on which angle you want to find. The cosine of ∠A is the sine of ∠C.
In this set of questions, use your calculator when necessary for arithmetic, but do not use it to find trig ratios. Instead, deduce the values of trig ratios from the information given and the rules in the lesson.
1. If sin(∠A) = 
, what is csc(∠A)?
2. If sec(∠R) = 
, what is cos(∠R)?
3. If tan(∠X) = 3.1, what is cot(∠X)?
4. If you know the values of sin(∠L) and cos(∠L), explain how you could find tan(∠L).
5. If cos(60°) = 
and sin(60°) = 
, what is tan(60°)?
6. If tan(45°) = 1 and sin(45°) = 
, what is cos(45°)?
7. If cot(40°) ≈ 1.192 and sin(40°) = 0.643, find cos(40°) to the nearest thousandth.
8. If tan(75°) = 3.732, what is cot(15°)?
9. If cos(60°) = 0.5, what is sin(30°)?
10. If tan(75°) = 3.732, what is cot(75°)?
11. If ΔABC is a right triangle with right angle ∠B and sin(∠A) = 0.3, find the value, to the nearest thousandth, of:
a) cos(∠A)
b) tan(∠A)
c) cot(∠A)
d) sec(∠A)
e) csc(∠A)
f) sin(∠C)
g) cos(∠C)
h) tan(∠C)
i) cot(∠C)
j) sec(∠C)
k) csc(∠C)
Pythagorean Identity
The famous rule known as the Pythagorean theorem says that if a and b are the lengths of the legs of a right triangle, and c is the length of the hypotenuse, then 
. All of the trig ratios involve the lengths of sides of right triangles, and the Pythagorean theorem provides some information about the relationships among the ratios. Consider right triangle ΔRST with right angle ∠S, and label legs 
and 
as a and b, and hypotenuse 
as c. Think about the trig ratios for ∠T.
Square each ratio by squaring the numerator and squaring the denominator.
They have a common denominator, so look at what happens when they’re added.
The Pythagorean theorem says .
For any acute angle, the square of the sine of the angle plus the square of the cosine of the angle always equals one.
This observation, known as the Pythagorean identity, allows you to find the cosine of an angle if you know its sine, or find the sine if you know the cosine.
EXAMPLE
If you know that sin(30°) = 
, then:
For now, only the positive square root will make sense in your work, but later the negative option will play a role.
While 
is what is usually thought of as the Pythagorean identity, it is actually one of three. Divide each term by 
to get an identity involving cotangent and cosecant.
Or divide the original by 
to get an identity involving tangent and secant.
Modeling with Trigonometry
Right triangle trigonometry can often provide tools for modeling real-life situations and solving problems. When those situations are described, the description often includes “angle of elevation” or “angle of depression.” These angles provide important information but may or may not actually be in the right triangle.
EXAMPLE
Imagine you are standing on the field at a stadium. You look straight ahead to the goal post at the end of the field. Then you notice a flag flying over the stadium. You raise your eyes to look up at the flag. The angle of elevation is the angle between your line of sight straight ahead and your line of sight up to the flag. Depending on what you’re looking for, the angle of elevation may be an acute angle of your right triangle.
On the other hand, suppose you are standing on the roof of an office building, looking straight ahead at the city skyline. You realize you can see the park and a fountain in the park that you particularly like. You shift your gaze down to the fountain. The angle of depression is the angle between your straight-ahead sight line and your sight line to the fountain. The angle of depression may not seem to be an angle of the right triangle formed by the ground, the building and your line of sight, but your straight-ahead line is parallel to the ground, so there is an angle by the fountain that is equal to the angle of depression.
For each of the questions in this section, sketch a figure and label it with the given information. Then write and solve an equation to find the requested information.
1. From a point 250 yards from the foot of a building, the angle of elevation to the top of the building is 35°. How tall is the building to the nearest foot? (3 feet = 1 yard)
2. The angle of depression from the top of a security tower to the nearby building entrance is 42°. If the tower is 25 feet high, how far is it from the entrance to the base of the tower to the nearest tenth of a foot?
3. To the nearest degree, what angle does a stairway make with the floor if the steps have a tread of 9 inches and a rise of 7 inches?
4. From onboard a ship at sea, the angle of elevation to the top of a lighthouse on the shore is 7°. If the lighthouse is known to be 165 feet high, how far from shore is the ship?
5. A ladder 25 feet long makes an angle of 12° with the wall of a building. How far from the wall is the foot of the ladder?
6. From the top of a ski slope, Marta sees the lodge at the base of the slope, at an angle of depression of 20°. If signs by the ski trail tell Marta she is now at an elevation of 1,500 feet, how far will Marta have to ski down the slope to reach the lodge?
7. From a point 100 feet from the base of the schoolhouse, the angle of elevation to the bottom of a flagpole on the roof of the school house is 40°. Find the height of the schoolhouse.
8. If the angle of elevation to the top of the flagpole in question 7 is 42.6°, how tall is the flagpole?