Grade: Grade 10 Subject: Mathematics Unit: Trigonometry Basics SAT: Geometry+Trigonometry ACT: Math

Solving Right Triangles

📖 Learn

What Does "Solving a Triangle" Mean?

Solving a triangle means finding all unknown sides and angles. For a right triangle, you start with some known information (sides or angles) and use the Pythagorean theorem and trigonometric ratios to find everything else.

Tools for Solving Right Triangles

Tool When to Use It Formula
Pythagorean Theorem When you know two sides and need the third a² + b² = c² (where c is the hypotenuse)
sin θ When you have/need the opposite and hypotenuse sin θ = opposite/hypotenuse
cos θ When you have/need the adjacent and hypotenuse cos θ = adjacent/hypotenuse
tan θ When you have/need the opposite and adjacent tan θ = opposite/adjacent
Inverse trig functions When you know sides and need to find an angle θ = sin⁻¹(opp/hyp), θ = cos⁻¹(adj/hyp), θ = tan⁻¹(opp/adj)
Inverse Trigonometric Functions

To find an angle when you know the ratio:

  • sin⁻¹ (or arcsin): Given opp/hyp, find the angle
  • cos⁻¹ (or arccos): Given adj/hyp, find the angle
  • tan⁻¹ (or arctan): Given opp/adj, find the angle

On calculators, these are often labeled as "sin⁻¹", "cos⁻¹", "tan⁻¹" or "asin", "acos", "atan".

Strategies for Solving Right Triangles

Step-by-Step Approach:
  1. Identify what you know: List the given sides and angles
  2. Identify what you need: Which sides or angles are unknown?
  3. Find the third angle: Since one angle is 90°, the other two must sum to 90°
  4. Choose your method: Select the appropriate ratio or theorem
  5. Solve and check: Calculate and verify using another method if possible

Angles of Elevation and Depression

Key Terms

  • Angle of Elevation: The angle formed by a horizontal line and the line of sight looking UP to an object
  • Angle of Depression: The angle formed by a horizontal line and the line of sight looking DOWN to an object

These angles are always measured from the horizontal, and they are equal when looking between two points (alternate interior angles).

SAT/ACT Connection: Solving right triangles is a core skill tested on both exams. You may encounter word problems involving heights, distances, angles of elevation/depression, and real-world applications. Be comfortable moving between finding sides and finding angles.

💡 Examples

Example 1: Solving a Triangle Given Two Sides

Problem: In right triangle ABC with the right angle at C, AC = 5 and BC = 12. Find AB, angle A, and angle B.

Solution:

Step 1: Find AB using Pythagorean theorem

AB² = AC² + BC² = 5² + 12² = 25 + 144 = 169

AB = √169 = 13

Step 2: Find angle A (opposite to BC = 12)

tan A = opposite/adjacent = BC/AC = 12/5

A = tan⁻¹(12/5) = tan⁻¹(2.4) ≈ 67.4°

Step 3: Find angle B

B = 90° - A = 90° - 67.4° ≈ 22.6°

Answer: AB = 13, angle A ≈ 67.4°, angle B ≈ 22.6°

Example 2: Solving Given One Side and One Angle

Problem: In right triangle PQR with the right angle at R, angle P = 35° and PR = 8. Find QR and PQ.

Solution:

Step 1: Identify the sides relative to angle P

PR = 8 is adjacent to angle P

QR is opposite to angle P

PQ is the hypotenuse

Step 2: Find QR using tangent

tan 35° = QR/PR = QR/8

QR = 8 × tan 35° = 8 × 0.7002 ≈ 5.60

Step 3: Find PQ using cosine

cos 35° = PR/PQ = 8/PQ

PQ = 8/cos 35° = 8/0.8192 ≈ 9.77

Answer: QR ≈ 5.60, PQ ≈ 9.77

Example 3: Angle of Elevation

Problem: A person standing 100 feet from the base of a tree looks up at the top of the tree at an angle of elevation of 28°. How tall is the tree?

Solution:

The distance from the person to the tree (100 ft) is the adjacent side.

The height of the tree (h) is the opposite side.

tan 28° = opposite/adjacent = h/100

h = 100 × tan 28°

h = 100 × 0.5317 ≈ 53.17

Answer: The tree is approximately 53.2 feet tall

Example 4: Finding an Angle

Problem: A ramp rises 3 feet over a horizontal distance of 12 feet. What is the angle of inclination?

Solution:

The rise (3 ft) is the opposite side.

The run (12 ft) is the adjacent side.

tan θ = opposite/adjacent = 3/12 = 1/4 = 0.25

θ = tan⁻¹(0.25) ≈ 14.04°

Answer: The angle of inclination is approximately 14°

Example 5: Angle of Depression

Problem: From the top of a 200-foot cliff, the angle of depression to a boat is 25°. How far is the boat from the base of the cliff?

Solution:

The angle of depression from the cliff equals the angle of elevation from the boat (alternate interior angles).

The cliff height (200 ft) is the opposite side.

The distance from the boat to the cliff (d) is the adjacent side.

tan 25° = 200/d

d = 200/tan 25°

d = 200/0.4663 ≈ 428.9

Answer: The boat is approximately 429 feet from the base of the cliff

✏️ Practice

Try these problems on your own. Choose the best answer for each question.

1. In a right triangle with legs 6 and 8, what is the length of the hypotenuse?

A) 10

B) 14

C) 48

D) √14

2. A right triangle has a hypotenuse of 15 and one leg of 9. What is the other leg?

A) 6

B) 12

C) √306

D) 24

3. If tan θ = 4/3 in a right triangle, what is θ (approximately)?

A) 36.9°

B) 53.1°

C) 45°

D) 30°

4. From a point 80 meters from a building, the angle of elevation to the top is 40°. How tall is the building?

A) 80 × sin 40°

B) 80 × tan 40°

C) 80/tan 40°

D) 80 × cos 40°

5. In a right triangle, if one acute angle is 52°, what is the other acute angle?

A) 38°

B) 52°

C) 128°

D) 28°

6. A ladder 20 feet long leans against a wall. If the base is 8 feet from the wall, approximately what angle does the ladder make with the ground?

A) cos⁻¹(8/20)

B) sin⁻¹(8/20)

C) tan⁻¹(8/20)

D) tan⁻¹(20/8)

7. If sin A = 0.6 in a right triangle with hypotenuse 10, what is the length of the side opposite angle A?

A) 6

B) 8

C) 4

D) 0.6

8. From a lighthouse 150 feet above sea level, the angle of depression to a ship is 12°. How far is the ship from the base of the lighthouse?

A) 150 × tan 12°

B) 150/tan 12°

C) 150 × sin 12°

D) 150/sin 12°

9. In right triangle XYZ with right angle at Z, XY = 13 and XZ = 5. Find cos X.

A) 5/13

B) 12/13

C) 5/12

D) 13/5

10. A road rises 1 foot for every 20 feet of horizontal distance. What is the angle of elevation of the road?

A) About 2.9°

B) About 5°

C) About 0.05°

D) About 87°

Click to reveal answers
  1. A) 10 — √(6² + 8²) = √(36 + 64) = √100 = 10 (3-4-5 triple scaled by 2)
  2. B) 12 — √(15² - 9²) = √(225 - 81) = √144 = 12
  3. B) 53.1° — tan⁻¹(4/3) = tan⁻¹(1.333) ≈ 53.1°
  4. B) 80 × tan 40° — Height = adjacent × tan(angle) = 80 × tan 40°
  5. A) 38° — The two acute angles sum to 90°: 90° - 52° = 38°
  6. A) cos⁻¹(8/20) — cos(angle with ground) = adjacent/hypotenuse = 8/20
  7. A) 6 — sin A = opp/hyp, so opp = hyp × sin A = 10 × 0.6 = 6
  8. B) 150/tan 12° — tan 12° = 150/distance, so distance = 150/tan 12°
  9. A) 5/13 — cos X = adjacent/hypotenuse = XZ/XY = 5/13
  10. A) About 2.9° — tan θ = 1/20 = 0.05, so θ = tan⁻¹(0.05) ≈ 2.86°

✅ Check Your Understanding

Question 1: When should you use the Pythagorean theorem versus trigonometry to find a missing side?

Reveal Answer

Use the Pythagorean theorem when you know two sides and need to find the third side, regardless of whether you know any angles. Use trigonometry when you know one side and one acute angle and need to find another side. Trigonometry is essential when angle information is part of the problem, while the Pythagorean theorem works purely with side lengths.

Question 2: Why are angles of elevation and depression between the same two points always equal?

Reveal Answer

They are equal because they form alternate interior angles with a horizontal line. Imagine the horizontal lines from the top of a building and from a person on the ground are parallel. The line of sight between them acts as a transversal. The angle of depression (measured down from horizontal at the top) and the angle of elevation (measured up from horizontal at the bottom) are alternate interior angles, which are always equal when formed by a transversal crossing parallel lines.

Question 3: How can you check your answer when solving a right triangle?

Reveal Answer

There are several ways to verify your work: (1) Check that the three angles sum to 180°; (2) Verify that the sides satisfy the Pythagorean theorem (a² + b² = c²); (3) Calculate the same side using a different trig ratio and see if you get the same answer; (4) Make sure the largest side is opposite the largest angle, and the smallest side is opposite the smallest angle; (5) Confirm that all angles and sides are positive and reasonable for the context.

Question 4: In an angle of elevation/depression problem, how do you identify which sides are opposite and adjacent?

Reveal Answer

Draw a diagram first. The horizontal distance (along the ground) is typically the adjacent side to the angle of elevation or depression. The vertical distance (height) is typically the opposite side. The line of sight (the slanted line from observer to object) is the hypotenuse. Remember: the angle is always measured from the horizontal, so the side along the horizontal is adjacent, and the side perpendicular to it is opposite.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review