Trigonometry Study Guide: Tips for Acing Your Next Exam

Trigonometry can seem complicated at first, but the right formulas, a few reliable strategies, and regular practice can make it much easier. This study guide covers the essential ideas students should know before a trigonometry exam.




What Is Trigonometry?

Trigonometry is the study of relationships between angles and sides, especially in triangles. In right-triangle problems, the three most common trigonometric ratios are sine, cosine, and tangent.

1. Master SOH, CAH, TOA

For a right triangle, use these three relationships:

sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent


Memory Tip: SOH CAH TOA

SOH = Sine → Opposite ÷ Hypotenuse
CAH = Cosine → Adjacent ÷ Hypotenuse
TOA = Tangent → Opposite ÷ Adjacent

2. The Important Pythagorean Identity

One of the most useful trigonometric identities is:

sin2θ + cos2θ = 1

This identity is especially useful when you know one trigonometric ratio and need to find another.

3. Worked Example: Finding Cosine

Example 1

Given: sin θ = 3 ÷ 5

Use the identity:

sin2θ + cos2θ = 1

Substitute sin θ = 3 ÷ 5:

(3 ÷ 5)2 + cos2θ = 1

Calculate:

9 ÷ 25 + cos2θ = 1

Therefore:

cos2θ = 16 ÷ 25

For an acute angle, the positive square root gives:

cos θ = 4 ÷ 5

Check: (3 ÷ 5)2 + (4 ÷ 5)2 = 9 ÷ 25 + 16 ÷ 25 = 25 ÷ 25 = 1.

4. Know the Common Special Angles

Memorizing the basic values for 30°, 45°, and 60° can save time on many exam questions.

Angle sin cos tan
30° 1 ÷ 2 √3 ÷ 2 1 ÷ √3
45° √2 ÷ 2 √2 ÷ 2 1
60° √3 ÷ 2 1 ÷ 2 √3


5. Worked Example: A 3-4-5 Triangle

Example 2

Suppose a right triangle has an opposite side of 3 units, an adjacent side of 4 units, and a hypotenuse of 5 units.

sin θ = 3 ÷ 5 = 0.6
cos θ = 4 ÷ 5 = 0.8
tan θ = 3 ÷ 4 = 0.75

These values can also be checked with the Pythagorean theorem: 32 + 42 = 9 + 16 = 25 = 52.

6. Common Trigonometry Mistakes

  • Mixing up the opposite and adjacent sides.
  • Using tangent when sine or cosine is required.
  • Forgetting to check whether the calculator is in degree mode or radian mode.
  • Rounding too early during a multi-step problem.
  • Forgetting that the sign of a trigonometric value can depend on the quadrant.
  • Skipping the final answer check.

7. A Simple Exam Strategy

  1. Identify what you know. Write down the given sides, angle, or trigonometric ratio.
  2. Draw or label the triangle. Clearly mark the opposite, adjacent, and hypotenuse sides.
  3. Choose the formula. Think SOH, CAH, or TOA.
  4. Substitute carefully. Keep fractions organized.
  5. Check your calculator mode. Use degree mode when the problem uses degrees.
  6. Check the result. Ask whether the answer makes sense for the triangle.

8. Practice Questions

1. In a right triangle, the opposite side is 6 and the hypotenuse is 10. Find sin θ.

2. In a right triangle, the adjacent side is 12 and the hypotenuse is 13. Find cos θ.

3. In a right triangle, the opposite side is 5 and the adjacent side is 12. Find tan θ.

4. If sin θ = 5 ÷ 13 and θ is acute, find cos θ.

5. Find tan 45°.

6. Find sin 30°.

9. Answer Section

1. sin θ = 6 ÷ 10 = 3 ÷ 5 = 0.6

2. cos θ = 12 ÷ 13 ≈ 0.9231

3. tan θ = 5 ÷ 12 ≈ 0.4167

4. sin2θ + cos2θ = 1, so cos θ = 12 ÷ 13 for an acute angle.

5. tan 45° = 1

6. sin 30° = 1 ÷ 2 = 0.5

Final Takeaway

For your next trigonometry exam, remember three things:

1. Use SOH CAH TOA for right-triangle ratios.

2. Remember sin2θ + cos2θ = 1.

3. Always label your triangle, choose the correct formula, and check your answer.

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