Lesson Objectives

- Learn how to solve a right triangle given an angle and a side
- Learn how to solve a right triangle given two sides
- Learn how to solve word problems involving angles of elevation
- Learn how to solve word problems involving angles of depression

## How to Solve Right Triangles

In order to solve a right triangle, we need to find the measures of all of the angles and all of the sides of the triangle.

Example #1: Solve the right triangle. Round answers to the nearest tenth. To solve our right triangle, we need to find the measures of the remaining sides and angles. Let's start by finding the value for a. To do this, we can think about the sine of 37°. $$\text{sin}\hspace{.1em}θ=\frac{\text{opp}}{\text{hyp}}$$ $$\text{sin}\hspace{.1em}37°=\frac{\text{a}}{14}$$ We can solve for a by multiplying both sides by our denominator of 14: $$\require{cancel}14 \cdot \text{sin}\hspace{.1em}37°=\cancel{14}\cdot \frac{\text{a}}{\cancel{14}}$$ $$a=14 \cdot \text{sin}\hspace{.1em}37°$$ $$a ≈ 8.4$$ Let's update our triangle: Next, let's find the value for b. Instead of using the calculated value for a, we will continue to use the values given to us in the problem. This will help us to obtain a more accurate answer since the value of a is an approximation. To find b, we will think about the cosine of 37°. $$\text{cos}\hspace{.1em}θ=\frac{\text{adj}}{\text{hyp}}$$ $$\text{cos}\hspace{.1em}37°=\frac{\text{b}}{14}$$ We can solve for b by multiplying both sides by our denominator of 14: $$\require{cancel}14 \cdot \text{cos}\hspace{.1em}37°=\cancel{14}\cdot \frac{\text{b}}{\cancel{14}}$$ $$b=14 \cdot \text{cos}\hspace{.1em}37°$$ $$b ≈ 11.2$$ Let's update our triangle: Our last task is to find the measure of angle B. Recall from the angle sum property of a triangle that the sum of the measures of the angles of any triangle is 180°. In our example, we know we have one angle that is 90° from our right angle symbol "∟". This means the sum of the two remaining angles must be 90°. Additionally, we are given the measure of angle A as 37°. This tells us that the measure of angle B can be found as: $$\text{m}\hspace{.1em}∠\hspace{.1em}B = 90° - 37°=53°$$ Let's update our triangle:

Example #2: Solve the right triangle. Round answers to the nearest tenth. Let's begin by finding the value for a.

Pythagorean Formula: $$a^2 + b^2=c^2$$ For the Pythagorean Formula, a and b represent the lengths of the two shorter sides, known as legs. Our c represents the longest side or the hypotenuse. If we relate our Pythagorean formula to the current problem, we obtain: $$4^2 + a^2=15.4^2$$ Simplify: $$16 + a^2=237.16$$ Subtract 16 away from each side: $$a^2=221.16$$ Isolate a by taking the square root of each side. Here we only need the principal square root: $$a=\sqrt{221.16}$$ Since this value is not exact, we will use the approximate symbol: $$a ≈ 14.9$$ Let's update our triangle: Now, we want to find the measure of angle A and the measure of angle B. Let's begin with the measure of our angle A. To perform this action, we will use our inverse sine function. Recall: $$\text{cos}\hspace{.1em}(θ)=\frac{\text{adj}}{\text{hyp}}$$ $$\text{cos}^{-1}\hspace{.1em}\left(\frac{\text{adj}}{\text{hyp}}\right)=θ$$ Let's apply this to our angle A. We will use the cosine function in order to get a more accurate answer. Again, to maintain accuracy, try to use as much given information as possible. $$\text{cos}\hspace{.1em}(A)=\frac{4}{15.4}$$ $$\text{cos}^{-1}\hspace{.1em}\left(\frac{4}{15.4}\right)=A$$ $$\text{m}\hspace{.1em}∠ \hspace{.1em}A ≈ 74.9°$$ Now that we have the measure of our angle A, we can use the angle sum property of triangles once again to find our remaining angle measure. The measure of angle B will be 90° - 74.9°: $$\text{m}\hspace{.1em}∠\hspace{.1em}B = 90° - 74.9° ≈ 15.1°$$

Example #3: Solve each word problem.

When Max is 123 feet away from the base of the school's flagpole, the angle of elevation to the top of the flagpole is 26° 40'. Find the height of the flagpole if his eyes are 5.30 feet above the ground.

To solve this problem, let's first draw an image to depict our problem. Here, we need to find the height of the flagpole. We can do this by finding the side opposite of our angle A, which is labeled as side a, and adding this to 5.30 feet. Since we know the length of the side that is adjacent to our angle A, we can use the tangent of our angle A. $$\text{tan}\hspace{.1em}A=\frac{\text{opp}}{\text{adj}}$$ $$\text{tan}\hspace{.1em}26° 40'=\frac{a}{123}$$ We can clear our denominator by multiplying both sides by 123: $$a=123 \cdot \text{tan}\hspace{.1em}26° 40'$$ (If using a TI-83 or TI-84 hit 2nd ANGLE to choose degrees or minutes) $$a ≈ 61.8 \text{ft}$$ We are not done since 61.8 feet is the length of our side a. To get to the ground, we need to add another 5.30 feet. $$61.8 \text{ft}+ 5.30 \text{ft}=67.1 \text{ft}$$ We can say that the height of the flagpole is approximately 67.1 feet.

### Solving a Right Triangle Given an Angle and a Side

For our first task, we will learn how to solve a right triangle given an angle and a side. Let's look at an example.Example #1: Solve the right triangle. Round answers to the nearest tenth. To solve our right triangle, we need to find the measures of the remaining sides and angles. Let's start by finding the value for a. To do this, we can think about the sine of 37°. $$\text{sin}\hspace{.1em}θ=\frac{\text{opp}}{\text{hyp}}$$ $$\text{sin}\hspace{.1em}37°=\frac{\text{a}}{14}$$ We can solve for a by multiplying both sides by our denominator of 14: $$\require{cancel}14 \cdot \text{sin}\hspace{.1em}37°=\cancel{14}\cdot \frac{\text{a}}{\cancel{14}}$$ $$a=14 \cdot \text{sin}\hspace{.1em}37°$$ $$a ≈ 8.4$$ Let's update our triangle: Next, let's find the value for b. Instead of using the calculated value for a, we will continue to use the values given to us in the problem. This will help us to obtain a more accurate answer since the value of a is an approximation. To find b, we will think about the cosine of 37°. $$\text{cos}\hspace{.1em}θ=\frac{\text{adj}}{\text{hyp}}$$ $$\text{cos}\hspace{.1em}37°=\frac{\text{b}}{14}$$ We can solve for b by multiplying both sides by our denominator of 14: $$\require{cancel}14 \cdot \text{cos}\hspace{.1em}37°=\cancel{14}\cdot \frac{\text{b}}{\cancel{14}}$$ $$b=14 \cdot \text{cos}\hspace{.1em}37°$$ $$b ≈ 11.2$$ Let's update our triangle: Our last task is to find the measure of angle B. Recall from the angle sum property of a triangle that the sum of the measures of the angles of any triangle is 180°. In our example, we know we have one angle that is 90° from our right angle symbol "∟". This means the sum of the two remaining angles must be 90°. Additionally, we are given the measure of angle A as 37°. This tells us that the measure of angle B can be found as: $$\text{m}\hspace{.1em}∠\hspace{.1em}B = 90° - 37°=53°$$ Let's update our triangle:

### Solving a Right Triangle Given Two Sides

Let's now look at an example of how to solve a right triangle given two sides.Example #2: Solve the right triangle. Round answers to the nearest tenth. Let's begin by finding the value for a.

Pythagorean Formula: $$a^2 + b^2=c^2$$ For the Pythagorean Formula, a and b represent the lengths of the two shorter sides, known as legs. Our c represents the longest side or the hypotenuse. If we relate our Pythagorean formula to the current problem, we obtain: $$4^2 + a^2=15.4^2$$ Simplify: $$16 + a^2=237.16$$ Subtract 16 away from each side: $$a^2=221.16$$ Isolate a by taking the square root of each side. Here we only need the principal square root: $$a=\sqrt{221.16}$$ Since this value is not exact, we will use the approximate symbol: $$a ≈ 14.9$$ Let's update our triangle: Now, we want to find the measure of angle A and the measure of angle B. Let's begin with the measure of our angle A. To perform this action, we will use our inverse sine function. Recall: $$\text{cos}\hspace{.1em}(θ)=\frac{\text{adj}}{\text{hyp}}$$ $$\text{cos}^{-1}\hspace{.1em}\left(\frac{\text{adj}}{\text{hyp}}\right)=θ$$ Let's apply this to our angle A. We will use the cosine function in order to get a more accurate answer. Again, to maintain accuracy, try to use as much given information as possible. $$\text{cos}\hspace{.1em}(A)=\frac{4}{15.4}$$ $$\text{cos}^{-1}\hspace{.1em}\left(\frac{4}{15.4}\right)=A$$ $$\text{m}\hspace{.1em}∠ \hspace{.1em}A ≈ 74.9°$$ Now that we have the measure of our angle A, we can use the angle sum property of triangles once again to find our remaining angle measure. The measure of angle B will be 90° - 74.9°: $$\text{m}\hspace{.1em}∠\hspace{.1em}B = 90° - 74.9° ≈ 15.1°$$

## How to Find Angles of Elevation or Depression

When solving application problems that involve right triangles, the angle of elevation from point X to point Y is the acute angle formed by ray XY and the horizontal ray with an endpoint at X. Additionally, the angle of depression from point X to point Y is the acute angle formed by ray XY and a horizontal ray with endpoint X. When we discuss the angle of elevation or the angle of depression, these are both measured between the line of sight and a horizontal line. Let's look at an example.Example #3: Solve each word problem.

When Max is 123 feet away from the base of the school's flagpole, the angle of elevation to the top of the flagpole is 26° 40'. Find the height of the flagpole if his eyes are 5.30 feet above the ground.

To solve this problem, let's first draw an image to depict our problem. Here, we need to find the height of the flagpole. We can do this by finding the side opposite of our angle A, which is labeled as side a, and adding this to 5.30 feet. Since we know the length of the side that is adjacent to our angle A, we can use the tangent of our angle A. $$\text{tan}\hspace{.1em}A=\frac{\text{opp}}{\text{adj}}$$ $$\text{tan}\hspace{.1em}26° 40'=\frac{a}{123}$$ We can clear our denominator by multiplying both sides by 123: $$a=123 \cdot \text{tan}\hspace{.1em}26° 40'$$ (If using a TI-83 or TI-84 hit 2nd ANGLE to choose degrees or minutes) $$a ≈ 61.8 \text{ft}$$ We are not done since 61.8 feet is the length of our side a. To get to the ground, we need to add another 5.30 feet. $$61.8 \text{ft}+ 5.30 \text{ft}=67.1 \text{ft}$$ We can say that the height of the flagpole is approximately 67.1 feet.

#### Skills Check:

Example #1

Find the measure of side b. Round to the nearest tenth.

Please choose the best answer.

A

b ≈ 3.2

B

b ≈ 5.7

C

b ≈ 8.1

D

b ≈ 6.1

E

b ≈ 4.9

Example #2

Find the measure of angle θ. Round to the nearest tenth.

Please choose the best answer.

A

θ ≈ 31.9°

B

θ ≈ 15.4°

C

θ ≈ 26.7°

D

θ ≈ 24.3°

E

θ ≈ 29.7°

Example #3

Solve each word problem.

A fire truck with a ladder that is 13.5 feet long is resting against the wall of a building. Note, the ladder sits on top of the fire truck. The ladder makes an angle of 43° 50' with the horizontal. Find the distance the ladder goes above the fire truck. Round to the nearest hundredth.

Please choose the best answer.

A

8.72 feet

B

20.02 feet

C

9.35 feet

D

19.15 feet

E

13.12 feet

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