Trig Word Problems- Examples and Step-by-Step Solutions

What Trig Word Problems Actually Are

Trig word problems are just geometry problems dressed up in a story. Instead of saying "find the height of the triangle," they say "a ladder leans against a wall at 70° and reaches a window 12 feet up. How long is the ladder?"

The math hasn't changed. The language has. That's it.

Most students fail these problems not because they can't do trigonometry, but because they can't translate the words back into triangles. This guide fixes that.

The Core Skill: Drawing the Picture

Before you touch a calculator, draw a diagram. Always. Every time. No exceptions.

Most word problems describe a right triangle. Identify:

• Which angle is your reference angle
• Which side is opposite that angle
• Which side is adjacent to that angle
• Which side is the hypotenuse

Label your diagram with the given values. Cross out information that doesn't matter. This takes 30 seconds and prevents 90% of mistakes.

The Three Ratios You Actually Need

You need these three formulas. Memorize them. Know them so well you could write them in your sleep:

• SOH: sin(θ) = opposite ÷ hypotenuse
• CAH: cos(θ) = adjacent ÷ hypotenuse
• TOA: tan(θ) = opposite ÷ adjacent

Pick the formula that uses the two sides you know and the side you need to find. That's the whole game.

Example 1: The Ladder Problem

Problem: A 15-foot ladder leans against a building, making a 65° angle with the ground. How far is the base of the ladder from the building?

Step 1: Draw it. You have a right triangle. The ladder is the hypotenuse (15 ft). The angle at the ground is 65°. You need the distance from building to ladder base.

That's the adjacent side to the 65° angle.

Step 2: Choose your ratio. You know hypotenuse, you need adjacent. That's cosine:

cos(65°) = adjacent ÷ 15

Step 3: Solve. cos(65°) ≈ 0.4226

0.4226 = x ÷ 15

x = 0.4226 × 15

x ≈ 6.34 feet

Answer: The base is about 6.3 feet from the building.

Example 2: The Flagpole Problem

Problem: From a point 40 feet from the base of a flagpole, the angle of elevation to the top is 28°. How tall is the flagpole?

Step 1: Draw it. You have a right triangle. The distance from the point to the base is the adjacent side (40 ft). The flagpole is the opposite side. The angle at the observation point is 28°.

Step 2: Choose your ratio. You know adjacent, you need opposite. That's tangent:

tan(28°) = opposite ÷ 40

Step 3: Solve. tan(28°) ≈ 0.5317

0.5317 = x ÷ 40

x = 0.5317 × 40

x ≈ 21.3 feet

Answer: The flagpole is about 21.3 feet tall.

Example 3: The Angle of Depression Problem

Problem: A man stands on a cliff 200 feet above the ocean. He looks down at a boat at a 35° angle of depression. How far is the boat from the base of the cliff?

Critical rule: Angle of depression from horizontal equals angle of elevation from the boat to the man's eye. Draw a horizontal line from the man's eye parallel to the water. The angle of depression and angle of elevation are equal.

Step 1: Draw it. Your diagram shows a right triangle. The cliff height (200 ft) is the opposite side. The distance from cliff base to boat is the adjacent side. The angle at the man's eye is 35°.

Step 2: Choose your ratio. You know opposite, you need adjacent. That's tangent:

tan(35°) = 200 ÷ adjacent

Step 3: Solve. tan(35°) ≈ 0.7002

0.7002 = 200 ÷ x

x = 200 ÷ 0.7002

x ≈ 285.6 feet

Answer: The boat is about 286 feet from the base of the cliff.

Which Function Do I Use?

Here's a quick reference table:

Common Mistakes That Cost You Points

• Mixing up opposite and adjacent. Always check which angle you're referencing.
• Using the wrong ratio. SOH-CAH-TOA. If you can't remember, think: "Silly Old Harry, Can't Actually Help, Only Through" or make up your own.
• Forgetting to check if the answer makes sense. A ladder shorter than the wall it leans against? Impossible. A person 200 feet tall? Check your work.
• Rounding too early. Keep full calculator values until the final step.

Getting Started: Your Process

Follow this exact sequence every time:

1. Read once — get the general story
2. Read twice — identify what you're solving for
3. Draw the triangle — label known sides and angles
4. Mark the reference angle — circle it if you need to
5. Identify which sides are opposite, adjacent, hypotenuse
6. Pick SOH, CAH, or TOA based on what you know and need
7. Set up the equation
8. Solve — algebra, calculator, done
9. Check — does the number make sense?

That's it. Practice this process on 10 problems and it'll become automatic. The trigonometry is simple. The translation is the hard part, and now you know how to do it.