Practice Problems: Trigonometry Questions with Detailed Solutions
Ready to improve your trigonometry skills? This practice guide gives you step-by-step questions and detailed solutions covering sine, cosine, tangent, special angles, right triangles, and an important trigonometric identity.
The Main Idea
The best way to learn trigonometry is to practice the same problem-solving process repeatedly: identify what you know, choose the correct formula, substitute the values, calculate carefully, and check your answer.
1. The Three Essential Trigonometric Ratios
For a right triangle, the three basic ratios are sine, cosine, and tangent.
On a small phone screen, swipe horizontally inside an equation box when necessary.
Memory Tip: SOH CAH TOA
SOH = Sine → Opposite ÷ Hypotenuse
CAH = Cosine → Adjacent ÷ Hypotenuse
TOA = Tangent → Opposite ÷ Adjacent
2. Practice Question: Find Sine
Question 1
A right triangle has an opposite side of 6 units and a hypotenuse of 10 units. Find sin θ.
Step 1: Choose the formula.
Step 2: Substitute the values.
Step 3: Simplify.
3. Practice Question: Find Cosine
Question 2
A right triangle has an adjacent side of 12 units and a hypotenuse of 13 units. Find cos θ.
Step 1: Use the cosine formula.
Step 2: Substitute.
Answer:
4. Practice Question: Find Tangent
Question 3
A right triangle has an opposite side of 5 units and an adjacent side of 12 units. Find tan θ.
Step 1: Choose tangent because we know the opposite and adjacent sides.
Step 2: Substitute.
Answer:
5. The Pythagorean Trigonometric Identity
This identity is one of the most useful tools for solving trigonometry problems:
Swipe horizontally inside the equation box on a small phone if needed.
6. Practice Question: Use the Identity
Question 4
If sin θ = 5 ÷ 13 and θ is an acute angle, find cos θ.
Step 1: Start with the identity.
Step 2: Substitute sin θ = 5 ÷ 13.
Step 3: Square the fraction.
Step 4: Subtract 25 ÷ 169 from both sides.
Because θ is acute, cosine is positive:
Check: (5 ÷ 13)2 + (12 ÷ 13)2 = 25 ÷ 169 + 144 ÷ 169 = 169 ÷ 169 = 1.
7. Special-Angle Practice
Knowing the common special angles can make many exam questions much faster.
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1 ÷ 2 | √3 ÷ 2 | √3 ÷ 3 |
| 45° | √2 ÷ 2 | √2 ÷ 2 | 1 |
| 60° | √3 ÷ 2 | 1 ÷ 2 | √3 |
On a phone, swipe horizontally inside the table to see all columns clearly.
Memory Tip for Special Angles
The values at 30°, 45°, and 60° appear frequently in algebra and trigonometry. Practice writing them from memory until they become automatic.
8. More Practice Problems
Question 5: Find tan 45°.
Question 6: Find sin 30°.
Question 7: A right triangle has legs of 8 units and 15 units. Find the hypotenuse.
Question 8: A right triangle has an opposite side of 8 units and a hypotenuse of 17 units. Find sin θ.
Question 9: If cos θ = 4 ÷ 5 and θ is acute, find sin θ.
Question 10: A right triangle has an adjacent side of 7 units and an opposite side of 24 units. Find tan θ.
9. Answer Section
Question 5: tan 45° = 1.
Question 6: sin 30° = 1 ÷ 2 = 0.5.
Question 7: Use the Pythagorean theorem: 82 + 152 = 64 + 225 = 289. Since 172 = 289, the hypotenuse is 17 units.
Question 8: sin θ = opposite ÷ hypotenuse = 8 ÷ 17 ≈ 0.4706.
Question 9: sin2θ + cos2θ = 1. Therefore, sin2θ = 1 − 16 ÷ 25 = 9 ÷ 25. Since θ is acute, sin θ = 3 ÷ 5.
Question 10: tan θ = opposite ÷ adjacent = 24 ÷ 7 ≈ 3.4286.
10. Common Trigonometry Mistakes
- Confusing the opposite and adjacent sides.
- Using the wrong ratio from SOH CAH TOA.
- Forgetting that the hypotenuse is always opposite the right angle.
- Entering degrees into a calculator set to radians, or vice versa.
- Rounding too early in a multi-step calculation.
- Forgetting that square roots can have positive or negative possibilities when solving an identity; the angle information determines the correct sign.
- Not checking whether the final answer is reasonable.
11. The Fast Exam Method
- Read the question carefully.
- Draw the triangle if one is not provided.
- Label the sides as opposite, adjacent, and hypotenuse.
- Choose SOH, CAH, or TOA.
- Write the formula before inserting numbers.
- Calculate without unnecessary early rounding.
- Check the answer and calculator mode.
Final Takeaway
Trigonometry becomes much easier when you follow a consistent process.
Remember SOH CAH TOA, practice the important special angles, and keep sin2θ + cos2θ = 1 ready for identity problems.
The more problems you solve, the faster you will recognize which formula to use. Practice a few questions regularly rather than trying to memorize everything at once.
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