Applications of Trigonometry

Now that you know the trigonometric ratios, let us put them to work solving real-world problems. How tall is that temple? How far away is that mountain peak? Trigonometry answers these questions without needing to climb or measure directly.

Angle of Elevation

When you look up at an object, the angle between the horizontal line of sight and your line of vision to the object is the angle of elevation.

Angle of Depression

When you look down at an object from a height, the angle between the horizontal and your downward line of sight is the angle of depression.

Key Fact: The angle of elevation from point A to point B equals the angle of depression from point B to point A (alternate interior angles).

Problem-Solving Strategy

1. Draw a clear diagram with the right triangle
2. Label the known values (angles, distances)
3. Identify what you need to find
4. Choose the right trig ratio (SOH-CAH-TOA)
5. Solve the equation

Worked Example 1: Finding Height

A student stands 30 m away from the base of a flagpole. The angle of elevation to the top is 60 degrees. Find the height.

tan(60) = height/30, so root3 = height/30

height = 30 x root3 = approximately 51.96 m

Worked Example 2: Finding Distance

From the top of a 45 m building, the angle of depression to a car is 30 degrees. How far is the car from the base?

tan(30) = 45/distance, so 1/root3 = 45/distance

distance = 45 x root3 = approximately 77.94 m

Worked Example 3: Two Angles

A person observes a tower at 30 degrees elevation from point A. After walking 50 m closer, the angle becomes 60 degrees. Find the height.

Let h = height and d = distance from second point.

From closer point: h = d x root3. From farther point: h = (d+50)/root3

Setting equal: d x root3 = (d+50)/root3, so 3d = d + 50, d = 25 m

h = 25 x root3 = approximately 43.3 m

Nepal Connection: Nepal's Department of Survey uses trigonometric methods to measure Himalayan peaks. The 2020 re-measurement confirmed Sagarmatha's height as 8,848.86 m -- a calculation that relies on the same trigonometry you are learning!

Key Takeaways

• Angle of elevation: looking up; angle of depression: looking down
• Always draw a diagram before solving
• Use tan when you have opposite and adjacent
• Practice problems with two observation points to build confidence