Trigonometry

Grade 10 · trigonometry · 20 practice problems · read aloud

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🔺 Trigonometry: The Study of Triangles

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It's incredibly useful for measuring heights and distances that are difficult to measure directly, like the height of a tree or the width of a river.

Key Concepts: SOH CAH TOA

In a right-angled triangle, we focus on one acute angle (θ). The three main ratios are:

  • Sine (sin): Opposite / Hypotenuse
  • Cosine (cos): Adjacent / Hypotenuse
  • Tangent (tan): Opposite / Adjacent

The memory aid SOH CAH TOA helps you remember these.

Step-by-Step Problem Solving

  1. Identify: Label the triangle's sides relative to the given angle (Opposite, Adjacent, Hypotenuse).
  2. Choose: Select the correct ratio (SOH, CAH, or TOA) that uses the two sides you know or want to find.
  3. Set Up: Write the equation.
  4. Solve: Calculate the unknown side or angle using a calculator.

Example 1: Finding a Missing Side

Find side 'x' (opposite to the 30° angle). The hypotenuse is 10 cm.

Step 1: Identify: Opposite = x, Hypotenuse = 10.

Step 2: Choose: We use SOH (Sine).

Step 3: Set Up: sin(30°) = x / 10

Step 4: Solve: x = 10 * sin(30°) = 10 * 0.5 = 5 cm.

Example 2: Finding a Missing Angle

Find angle θ. The side adjacent to θ is 4 units, and the hypotenuse is 8 units.

Step 1: Identify: Adjacent = 4, Hypotenuse = 8.

Step 2: Choose: We use CAH (Cosine).

Step 3: Set Up: cos(θ) = 4 / 8 = 0.5

Step 4: Solve: θ = cos⁻¹(0.5) = 60°.

⚠️ Common Mistakes

  • Wrong Ratio: Mixing up SOH CAH TOA. Always double-check which sides are involved.
  • Calculator Mode: Forgetting to set your calculator to "DEGREE" mode instead of "RADIAN."
  • Hypotenuse: Misidentifying the hypotenuse. It is always the longest side, opposite the right angle.

💡 Tips & Tricks

  • SOH CAH TOA: Practice saying it until it's automatic.
  • Angle Check: The sum of the two acute angles in a right triangle is always 90°.
  • Square Root: Remember the Pythagorean Theorem (a² + b² = c²) for finding a third side when you know two.

Practice Suggestions

1. Flashcards: Create cards for SOH CAH TOA and common angle values (sin 30°, cos 45°, etc.).
2. Draw It: For every problem, sketch the triangle and label the sides.
3. Word Problems: Practice with real-world scenarios like ladders leaning on walls or ramps.

Practice problems

6 of the 20, worked through step by step — try them before opening the answer.

1 In a right triangle, if tan(θ) = 7/9, find sin(θ).

Hint: Use the tangent ratio to identify the sides, then apply the Pythagorean theorem to find the hypotenuse before calculating the sine ratio.

Show the answer

Answer: 7/√130

  1. Given tan(θ) = 7/9, this means opposite side = 7 and adjacent side = 9.
  2. Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(7² + 9²) = √(49 + 81) = √130.
  3. sin(θ) = opposite/hypotenuse = 7/√130.

The answer is 7/√130.

2 In a right triangle, if tan(θ) = 15/20, find sin(θ).

Hint: Use the tangent ratio to identify the sides, then apply the Pythagorean theorem to find the hypotenuse before calculating the sine ratio.

Show the answer

Answer: 0.6

  1. Given tan(θ) = 15/20, this means opposite side = 15 and adjacent side = 20.
  2. Use the Pythagorean theorem to find the hypotenuse: hypotenuse^2 = 15^2 + 20^2 = 225 + 400 = 625.
  3. Hypotenuse = sqrt(625) = 25.
  4. sin(θ) = opposite/hypotenuse = 15/25 = 3/5 = 0.6.

The answer is 0.6.

3 In a right triangle, if tan(θ) = 7/9, find sin(θ) and cos(θ).

Hint: Use the tangent ratio to identify the opposite and adjacent sides, then apply the Pythagorean theorem to find the hypotenuse before calculating sine and cosine.

Show the answer

Answer: sin(θ) = 7/√130, cos(θ) = 9/√130

  1. Given tan(θ) = 7/9, this means opposite side = 7 and adjacent side = 9.
  2. Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(7² + 9²) = √(49 + 81) = √130.
  3. sin(θ) = opposite/hypotenuse = 7/√130.
  4. cos(θ) = adjacent/hypotenuse = 9/√130.

The answer is sin(θ) = 7/√130 and cos(θ) = 9/√130.

4 In a right triangle, if tan(θ) = 40/75, find θ to the nearest degree.

Hint: Use the inverse tangent function to find the angle when given the ratio of opposite to adjacent sides.

Show the answer

Answer: 28

  1. We are given tan(θ) = 40/75
  2. To find θ, use the inverse tangent function: θ = tan⁻¹(40/75)
  3. Calculate 40 ÷ 75 = 0.5333
  4. Use a calculator: tan⁻¹(0.5333) ≈ 28.07°
  5. Round to the nearest degree: 28° The angle θ is 28°.

5 In a right triangle, if the hypotenuse is 25 and one angle is 40°, find the adjacent side.

Hint: Identify which trigonometric ratio relates the adjacent side and hypotenuse in a right triangle.

Show the answer

Answer: 19.15

  1. In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: cos(angle) = adjacent/hypotenuse.
  2. We are given the hypotenuse = 25 and the angle = 40°.
  3. Set up the equation: cos(40°) = adjacent / 25.
  4. Solve for the adjacent side: adjacent = 25 * cos(40°).
  5. Using a calculator, cos(40°) ≈ 0.7660.
  6. Calculate: adjacent = 25 * 0.7660 = 19.15. The adjacent side is approximately 19.15.

6 In a right triangle, if the opposite side is 12 and the adjacent side is 35, find the hypotenuse.

Hint: Use the Pythagorean theorem to find the hypotenuse when both legs are known.

Show the answer

Answer: 37

  1. The Pythagorean theorem states: hypotenuse² = opposite² + adjacent²
  2. Substitute the given values: hypotenuse² = 12² + 35²
  3. Calculate the squares: 12² = 144, 35² = 1225
  4. Add the squares: 144 + 1225 = 1369
  5. Take the square root: sqrt(1369) = 37

The answer is 37.

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