Lesson 4: Solving Right Triangles and Applications
Estimated time: 30-35 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Solve a right triangle (find all missing sides and angles)
- Choose the correct trig function to set up an equation
- Solve angle of elevation and angle of depression problems
- Apply right triangle trig to navigation and surveying problems
What Does "Solve a Triangle" Mean?
To solve a right triangle means to find all unknown sides and all unknown angles. Since one angle is always 90°, you need to find the other two angles and any missing sides.
Strategy: Given a right triangle with at least one side and one acute angle (or two sides), use SOH-CAH-TOA and the Pythagorean theorem to find everything else.
Solving When Given an Angle and a Side
Example 1: Finding a Missing Side
In right triangle ABC with C = 90°, angle A = 35° and hypotenuse c = 20. Find side a (opposite A).
Solution:
We know the hypotenuse and want the opposite side. Use sine:
sin 35° = a/20
a = 20 sin 35° = 20(0.5736) ≈ 11.47
Example 2: Finding the Other Side and Angle
Continuing from Example 1, find side b (adjacent to A) and angle B.
Side b: cos 35° = b/20, so b = 20 cos 35° ≈ 20(0.8192) ≈ 16.38
Angle B: B = 90° − 35° = 55°
Solving When Given Two Sides
Example 3: Finding an Angle from Two Sides
A right triangle has legs a = 6 and b = 10. Find angle A (opposite side a).
Solution:
tan A = opposite/adjacent = 6/10 = 0.6
A = tan¹(0.6) ≈ 30.96°
B = 90° − 30.96° ≈ 59.04°
Hypotenuse: c = √(36 + 100) = √136 ≈ 11.66
Angles of Elevation and Depression
Angle of Elevation: The angle measured upward from the horizontal to a line of sight to an object above.
Angle of Depression: The angle measured downward from the horizontal to a line of sight to an object below.
Key insight: the angle of elevation from point A to point B equals the angle of depression from B to A (alternate interior angles).
Example 4: Angle of Elevation
A person standing 50 meters from a building looks up at an angle of elevation of 60° to see the top. How tall is the building? (Assume eye level is at ground level for simplicity.)
Solution:
tan 60° = height / 50
height = 50 tan 60° = 50√3 ≈ 86.6 meters
Example 5: Angle of Depression
From the top of a 200-foot lighthouse, the angle of depression to a boat is 25°. How far is the boat from the base of the lighthouse?
Solution:
The angle of depression from the top equals the angle of elevation from the boat: 25°.
tan 25° = 200 / d, so d = 200 / tan 25° = 200 / 0.4663 ≈ 429 feet
Navigation and Bearing Problems
Example 6: Simple Bearing Problem
A ship sails on a bearing of N 30° E for 100 miles. How far north and how far east has it traveled?
Solution:
The bearing N 30° E means 30° east of due north. Draw a right triangle with the 100-mile path as hypotenuse.
Northward distance = 100 cos 30° = 100(√3/2) ≈ 86.6 miles
Eastward distance = 100 sin 30° = 100(1/2) = 50 miles
Check Your Understanding
1. In a right triangle, angle A = 42° and the side adjacent to A is 15. Find the opposite side.
2. A ladder 20 ft long leans against a wall making a 70° angle with the ground. How high up the wall does it reach?
3. From a point 100 m from a tree, the angle of elevation to the top is 28°. Find the height.
Key Takeaways
- Solving a right triangle means finding all missing sides and angles.
- Choose the trig ratio that involves the known and unknown quantities.
- Use inverse trig functions (sin¹, cos¹, tan¹) to find angles from side ratios.
- Angle of elevation is measured up from horizontal; angle of depression is measured down.
- Always draw a diagram for application problems.