Trigonometry Examples- Practice Problems with Solutions

What You're Getting Into

This is for people who need actual trigonometry examples, not another vague explanation of what sine means. I'll walk through real problems, show the math step-by-step, and give you practice questions to test yourself. That's it. No fluff.

The Basics You Need to Know First

Before touching any example, you need these three definitions locked in your head. If you don't know these cold, nothing else makes sense.

SOH CAH TOA

This is the only memory trick you need:

• SOH: Sine = Opposite ÷ Hypotenuse
• CAH: Cosine = Adjacent ÷ Hypotenuse
• TOA: Tangent = Opposite ÷ Adjacent

The hypotenuse is always across from the right angle. The opposite side sits across from your angle of interest. The adjacent side sits next to your angle (but isn't the hypotenuse).

The Unit Circle Quick Reference

These values come up constantly:

• sin(0°) = 0, cos(0°) = 1
• sin(30°) = 0.5, cos(30°) = √3/2
• sin(45°) = √2/2, cos(45°) = √2/2
• sin(60°) = √3/2, cos(60°) = 0.5
• sin(90°) = 1, cos(90°) = 0

If you forget these, you can always derive them from a 30-60-90 or 45-45-90 triangle.

Trigonometry Examples with Full Solutions

Example 1: Finding a Missing Side

Problem: A right triangle has an angle of 35° and a hypotenuse of 12 units. Find the length of the side opposite to the 35° angle.

Solution:

You're looking for the opposite side. You know the hypotenuse. That means Sine is your weapon.

sin(35°) = Opposite ÷ 12

Solving for Opposite:

Opposite = 12 × sin(35°)

Opposite = 12 × 0.574

Opposite ≈ 6.89 units

Example 2: Finding an Angle

Problem: In a right triangle, the side adjacent to your angle is 8 units and the hypotenuse is 10 units. What is the angle?

Solution:

You have adjacent and hypotenuse. That's Cosine.

cos(θ) = 8 ÷ 10 = 0.8

θ = cos⁻¹(0.8)

θ ≈ 36.87°

Most calculators have a cos⁻¹ button (or arccos). If your answer doesn't match, make sure your calculator is in degree mode.

Example 3: Two-Angle Problem

Problem: A ladder leans against a wall, forming a 70° angle with the ground. The base of the ladder sits 4 feet from the wall. How long is the ladder?

Solution:

Draw this out. The ground is adjacent to the angle. The wall is opposite. The ladder is the hypotenuse.

You know the adjacent side (4 ft) and need the hypotenuse. Use Cosine.

cos(70°) = 4 ÷ Hypotenuse

Hypotenuse = 4 ÷ cos(70°)

Hypotenuse = 4 ÷ 0.342

Hypotenuse ≈ 11.7 feet

Example 4: Word Problem with Tangent

Problem: You're standing 50 meters from the base of a tree. The angle of elevation to the top is 42°. How tall is the tree?

Solution:

Your distance from the tree (50m) is the adjacent side. The tree's height is the opposite side. This screams Tangent.

tan(42°) = Opposite ÷ 50

Opposite = 50 × tan(42°)

Opposite = 50 × 0.900

Opposite ≈ 45 meters

Comparing Trig Functions

Here's a quick reference table for the three main functions. Keep this handy.

The inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) do the opposite: they take a ratio and give you an angle.

Practice Problems

Try these before checking the answers. No peeking.

Problem 1: A right triangle has a 55° angle and a side opposite that angle measuring 7 units. Find the hypotenuse length.

Problem 2: A ramp makes a 12° angle with the ground. If the ramp's length is 15 feet, how high does it reach?

Problem 3: From a point on the ground 100 feet from a building, the angle of elevation to the roof is 28°. How tall is the building?

Solutions to Practice Problems

Problem 1 Answer:

sin(55°) = 7 ÷ Hypotenuse

Hypotenuse = 7 ÷ sin(55°)

Hypotenuse = 7 ÷ 0.819

Hypotenuse ≈ 8.55 units

Problem 2 Answer:

The ramp is the hypotenuse. The height is opposite the 12° angle. Use Sine.

sin(12°) = Height ÷ 15

Height = 15 × sin(12°)

Height = 15 × 0.208

Height ≈ 3.12 feet

Problem 3 Answer:

100 feet is adjacent. Building height is opposite. Use Tangent.

tan(28°) = Height ÷ 100

Height = 100 × tan(28°)

Height = 100 × 0.532

Height ≈ 53.2 feet

Common Mistakes That Kill Your Answers

• Using the wrong side: Always label your triangle first. Opposite? Adjacent? Hypotenuse? Wrong identification = wrong answer every time.
• Calculator in wrong mode: Degrees vs. radians. Check your settings. 90° in radians is π/2, not 90. If your answer looks insane, this is probably why.
• Inverse confusion: sin(30°) = 0.5, but sin⁻¹(0.5) = 30°. These are opposite operations. Mixing them up gives you garbage.
• Rounding too early: Keep full precision through your calculation. Only round at the final step. Intermediate rounding compounds errors.

Getting Started: Your Action Plan

Here's how to actually learn this stuff:

1. Draw the triangle for every problem. Don't skip this. Visual learners who draw their triangles get these problems right. People who try to solve in their heads get them wrong.
2. Label the sides relative to your target angle. Mark the hypotenuse, opposite, and adjacent.
3. Pick your function based on what you know. Two sides known = Sine, Cosine, or Tangent. One side and one angle = the inverse function.
4. Set up the equation using SOH CAH TOA.
5. Solve algebraically before plugging in numbers. It's cleaner and fewer mistakes.
6. Check your answer by plugging it back in. Does sin(θ) = opposite/hypotenuse work? If not, something's wrong.

Practice with 10-15 problems. By problem 10, this will feel automatic. By problem 20, you'll wonder why you ever struggled.