Trig Ratios Multiple Choice Test- Practice Questions with Answer Key
Trig Ratios Multiple Choice Test: Practice Questions with Answer Key
You're here because you need to practice trig ratios. Not read another 2,000-word explanation of why sine matters. Fine. Here are 15 multiple choice questions that actually test what you know. Answers at the bottom.
Quick Trig Ratios Refresher
Before you dive in, make sure you're solid on SOHCAHTOA:
- SOH: Sine = Opposite ÷ Hypotenuse
- CAH: Cosine = Adjacent ÷ Hypotenuse
- TOA: Tangent = Opposite ÷ Adjacent
The hypotenuse is always across from the right angle. The opposite side is across from your angle. The adjacent side touches your angle but isn't the hypotenuse.
Practice Questions
Questions 1-5: Basic SOHCAHTOA
Question 1: In a right triangle, angle A has opposite side = 3 and hypotenuse = 5. What is sin(A)?
- A) 3/5
- B) 5/3
- C) 4/5
- D) 5/4
Question 2: If cos(θ) = 7/25, what is the length of the adjacent side if the hypotenuse is 25?
- A) 7
- B) 24
- C) 18
- D) 25
Question 3: A right triangle has legs of 6 and 8. What is tan(θ) where θ is the angle opposite the side of length 6?
- A) 6/8
- B) 8/6
- C) 10/6
- D) 6/10
Question 4: Which trig ratio should you use to find angle A if you know the opposite side is 5 and the adjacent side is 12?
- A) sin(A) = 5/12
- B) cos(A) = 5/12
- C) tan(A) = 5/12
- D) sin(A) = 12/5
Question 5: In a 45-45-90 triangle, if each leg is 1, what is tan(45°)?
- A) 0
- B) 1
- C) √2
- D) 1/2
Questions 6-10: Finding Angles
Question 6: If sin(θ) = 0.5, what is θ (in degrees) assuming θ is an acute angle?
- A) 30°
- B) 45°
- C) 60°
- D) 90°
Question 7: A ladder leans against a wall forming a 60° angle with the ground. If the ladder is 20 feet long, how far is the base from the wall?
- A) 10 feet
- B) 10√3 feet
- C) 20√3 feet
- D) 20/√3 feet
Question 8: What is cos(30°) equal to?
- A) 1/2
- B) √3/2
- C) √2/2
- D) 1
Question 9: In a right triangle, if the angle of elevation to the top of a tree is 45° and you're standing 30 meters away, how tall is the tree (approximately)?
- A) 15 meters
- B) 21 meters
- C) 30 meters
- D) 42 meters
Question 10: Which of these is NOT a valid value for sine or cosine of an acute angle?
- A) 0.3
- B) 0.9
- C) 1.2
- D) 0.5
Questions 11-15: Mixed Applications
Question 11: A ramp makes a 15° angle with the ground. If the ramp is 12 feet long, how high is the top of the ramp off the ground?
- A) 12 sin(15°) ≈ 3.1 feet
- B) 12 cos(15°) ≈ 11.6 feet
- C) 12 tan(15°) ≈ 3.2 feet
- D) 12/15 = 0.8 feet
Question 12: If tan(θ) = 1, what could θ be?
- A) 30°
- B) 45°
- C) 60°
- D) Both B and C
Question 13: A right triangle has hypotenuse = 13 and one leg = 5. What is tan of the angle opposite the side of length 5?
- A) 5/12
- B) 12/5
- C) 5/13
- D) 13/5
Question 14: Which statement is always true for any acute angle θ in a right triangle?
- A) sin(θ) > tan(θ)
- B) cos(θ) < 1
- C) tan(θ) > 1
- D) sin(θ) = cos(90° - θ)
Question 15: If sin(θ) = x, what is cos(90° - θ)?
- A) x
- B) 1/x
- C) √(1-x²)
- D) x - 1
Answer Key
| Question | Answer | Quick Explanation |
|---|---|---|
| 1 | A) 3/5 | Directly from SOH: opposite/hypotenuse |
| 2 | A) 7 | Adjacent is the numerator in cos ratio |
| 3 | A) 6/8 | TOA: opposite (6) ÷ adjacent (8) |
| 4 | C) tan(A) = 5/12 | Opposite/adjacent = tan |
| 5 | B) 1 | In 45-45-90, legs are equal, so tan = 1 |
| 6 | A) 30° | sin(30°) = 0.5 is a standard value |
| 7 | B) 10√3 feet | cos(60°) = adjacent/20, adjacent = 10 |
| 8 | B) √3/2 | Cos(30°) = √3/2 from 30-60-90 triangle |
| 9 | C) 30 meters | tan(45°) = 1, so height = distance = 30m |
| 10 | C) 1.2 | Sine and cosine must be between 0 and 1 |
| 11 | A) ≈3.1 feet | Opposite = 12 × sin(15°) |
| 12 | B) 45° | tan(45°) = 1, tan(60°) = √3, not 1 |
| 13 | A) 5/12 | Third side = √(169-25) = 12 |
| 14 | B) cos(θ) < 1 | Cosine of acute angles is always less than 1 |
| 15 | A) x | Co-function identity: sin(θ) = cos(90°-θ) |
How to Use These Questions
Don't just read through the answers. Here's what actually works:
- Time yourself. Give yourself 20-25 minutes for 15 questions. This mirrors real test conditions.
- Grade yourself harsh. Partial credit doesn't exist on the actual test.
- Redo every wrong question before checking the next answer. Figure out if you made a calculation error or a concept error.
- Track which question types you miss. If you're bombing anything involving 30-60-90 triangles, that's your gap.
Common Mistakes on Trig Ratio Tests
| Mistake | What You Actually Did | How to Fix It |
|---|---|---|
| Used wrong ratio | Used sin when you needed tan | Circle the given sides, then ask: which two do I have? |
| Flipped the ratio | Put adjacent/hypotenuse when you needed hypotenuse/adjacent | Say SOHCAHTOA out loud. Write it down every time. |
| Identified wrong side | Called the hypotenuse "opposite" | The hypotenuse is always the longest side, across from the right angle. |
| Forgot to use inverse trig | Gave up when asked for the angle | sin, cos, and tan give ratios. arcsin, arccos, arctan give angles. |
| Rounded too early | Got 0.866, called it 0.87, then picked wrong answer | Keep exact values until the end, or use more decimal places. |
What to Study Next
If you got fewer than 12 correct, go back and review:
- The definitions of sin, cos, tan with diagrams
- How to find the third side using Pythagorean theorem
- Standard angles: 30°, 45°, 60° and their trig values
If you got 12-14 correct, you're solid. Do a few more mixed practice sets to build speed.
If you got 15 correct, you're probably over-prepared. Go take the test.
That's it. No summary paragraph. No "you've got this." Just practice until it's automatic.