Applications of Trigonometry

Apply trigonometry to real-world angle and distance problems. Math 10-C Alberta mathematics.

Lesson 3.4 of Trigonometry in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Angles of Elevation and Depression. Learn to identify angles of elevation and depression from a horizontal line and understand why the two angles are always equal.
• The Pyramid Problem (3D Geometry). Apply trigonometry to a three-dimensional pyramid to find the angle a triangular face makes with the base.
• Two-Triangle Problems. Solve real-world problems that involve two right triangles sharing a common side, using the result from the first triangle as the input for the second.

Inside the lesson — a free preview

Definitions

The Key Rule — Always from the Horizontal. Both angles are always measured from a horizontal line — not from the ground and not from a vertical.

Angle of Elevation: measured upward from the horizontal to your line of sight. You are looking up.

Angle of Depression: measured downward from the horizontal to your line of sight. You are looking down.

A Key Relationship

Elevation Equals Depression — Always. The angle of elevation from point A looking up to point B is always equal to the angle of depression from point B looking down to point A.

This is because they are alternate interior angles formed by two parallel horizontal lines cut by the same line of sight (the transversal).

Practical Consequence. You can always convert between the two angles.

If the angle of depression from a tower to a boat is 15°, then the angle of elevation from the boat to the tower is also 15°.

You only ever need to know one of the two — they are interchangeable.

Worked example: Finding an Angle in a Pyramid

Key Strategy — Reduce to 2D. Three-dimensional geometry problems can always be reduced to a two-dimensional right triangle by taking the right cross-section.

The Problem — Muttart Conservatory, Edmonton. The Muttart Conservatory has square-based pyramids that are 24 m high with a base side length of 26 m.

What angle does the triangular face make with the base at the midpoint of one side?

The Key Insight — Finding the Right Triangle. The height of the pyramid (24 m) meets the base at its center.

The distance from the center to the midpoint of one side is half the side length: (26)/(2) = 13 m.

Slicing through the apex and the midpoint of one base side creates a right triangle with:

• Vertical side (opposite to x) = 24 m

• Horizontal side (adjacent to x) = 13 m

• Right angle where the height meets the base

Solution. Step 1 — Label the sides:

From angle x at the base: opposite = 24 m, adjacent = 13 m.

Step 2 — Choose the ratio:

Opposite and adjacent are known, so use .

Step 3 — Set up the equation:

x = (24)/(13)

Step 4 — Solve for x:

x = ^(-1)((24)/(13))

x = ^(-1)(1.846)

Answer: x ≈ 61.6°

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