3.7 Calculating the Measure of an Angle in a Right Triangle

Leanne Thompson

3.7 Calculating the Measure of an Angle in a Right Triangle

In this lesson, we will go over how to find a missing angle in a right triangle using trigonometric ratios. To find a missing angle in a right triangle, you will need the length of two sides. To find the missing angle, you will use inverse trigonometric functions. The inverse trigonometric functions on your calculator look like sin^-1, cos^-1, and tan^-1.

Using the Trigonometric Ratios to Calculate the Measure of an Angle

Table 3.7.1
To find an angle on your calculator:
Step 1: Press the 2nd function Step 2: Press SIN, COS, or TAN Step 3: Type in the trigonometric ratio Step 4: Press ENTER

Example 1

Use the inverse trigonometric functions on your calculator to find the following angles. Round your final answer to one decimal place.

a) sin A = 0.5

b) cos B = 0.9735

c) tan C = 2.89358

Table 3.7.2
Solution a)	Solution b)	Solution c)
A = sin^-1(0.5) $\angle A$ = 30°	B = cos^-1(0.9735) $\angle B$ = 13.2°	C = tan^-1(2.89358) $\angle C$ = 70.9°

Practice 1

Use the inverse trigonometric functions on your calculator to find the following angles. Round your final answer to one decimal place.

a) sin X = 1

b) cos Y = 0.375

c) tan Z = 1.676767

d) sin C = 0

e) cos D = 0.95

f) tan F = 6.3333

Example 2

Find the missing $\angle T$ . Round your final answer to one decimal place.

Right triangle RTS. R is the right angle. T is angle theta. Side r is 11 and side t is 7.

Table 3.7.3
Steps	Solution
Identify which sides you have been given.	We have been given side r = 11, which is the hypotenuse, and side t = 7, which is the opposite side.
Identify which trigonometric ratio you will use (sin, cos, or tan).	We will use the sin ratio because we are working with the opposite and hypotenuse sides of the triangle.
Set up the trigonometric ratio.	sin θ° = $\frac{opp}{hyp}$ sin θ = $\frac{7}{11}$
Solve the ratio.	sin T = 0.6364 T = sin^-1(0.6364) $\angle T$ = 39.5°

Practice 2

a) Find angle X using trigonometric ratios. Round your final answer to one decimal place.

Right triangle YXZ. Y is the right angle. Side x is 12 and side y is 35.

b) Find angle Z using trigonometric ratios. Round your final answer to one decimal place.

Right triangle YXZ. Y is the right angle. Side x is 12 and side y is 35.

The following property can be used to find the measure of an angle in a triangle given that you know the other two angles.

Table 3.7.4
Sum of the Measures of the Angles of a Triangle
For any $\Delta ABC$ , the sum of the measures of the angles is $\text{180}$ °: $m\angle A+m\angle B+m\angle C=\text{180}$ °

Practice 3

Find all the missing unknown angles in the triangles below. Round your final answers to one decimal place.

a)

Right triangle BAC. B is the right angle. Side c is 5 and side a is 9.

b)

Right triangle DCE. D is the right angle. Side e is 9 and side c is 12.

Angle of Elevation and Angle of Depression

In the previous examples we found missing sides and missing angles in right triangles. We will now look at how we can use these skills to solve real-world examples. Many applications of trigonometric ratios involve an angle of elevation or an angle of depression.

Angle of Elevation

Angle of Depression

An angle of elevation is the angle between a horizontal line (often the ground) and the observer’s line of sight.

The angle of depression is the angle between a horizontal line (that is parallel to the ground) and the observer’s line of sight.

The diagrams above show that angle of elevation = angle of depression.

Practice 4

Answer the following questions. Draw a diagram for each question, and round your answer to the nearest tenth, where necessary.

a) The Harbour Centre in Vancouver is 146 m tall, and José is standing 31 metres away from the base of it. Calculate the angle of elevation from José to the top of the Harbour Centre. Draw a diagram.

b) The tallest tree in the world is a coast redwood. It is in Redwood National Park in California, and it measures 116 m tall. If the distance from the top of the tree to Roxanne’s feet is 153 m, what is the angle of elevation from where Roxanne is standing to the top of the tree?

c) Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend who is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?

d) Vallen is standing at the top of a building and waves to his friend, Gayoung, who is standing on the ground, 22 metres from the base of the building. If the straight line distance from Vallen to Gayoung is 68 metres, what is the angle of depression from Vallen to Gayoung?

Homework

Find angle θ using trigonometric ratios. Label the side lengths with the given values, and round your final answer to the nearest whole number.

a) side g = 10 m and side f = 12 m

b) side e = 9 km

c) side s = 24.4 in and side r = 31.2 in

d) side a = 2.1 ft and side b = 6 ft
Find all missing angles in each triangle. Round your answers to the nearest whole number.

a)

b)
Find the missing angle in each triangle. Round your answers to the nearest whole number.

a) Find $\angle$ Z.

b) Find $\angle$ E.

c) Find $\angle$ X.

d) Find $\angle$ O.
A submarine starts on the surface, and then dives at an unknown angle of depression. It travels diagonally a distance of 423 m, then stops to observe a fish. If the vertical distance from the surface is 315 m when the submarine is observing the fish, what is the angle of depression? Round to the nearest hundredth.
A staircase is 32 cm deep and 17 cm tall. What is the angle of elevation of the staircase rounded to the nearest tenth?
Find the angle of elevation in the right triangle, and round to the nearest whole number. The right angle is the angle between the two given side lengths.
Mira is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation rounded to the nearest tenth?
From the top of a wall, the angle of depression to a boy standing on the ground is 43°. If the wall is 24 m high, how far from the base of the wall is the boy? Round to one decimal place.
A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52°. How tall is the tree rounded to the nearest tenth?
A cat is standing at point C, which is 10 ft away from a building, making an angle of elevation of 36° with point B at the top of the building. What is the angle of depression from the top of the building to the cat?