Math Trigonometry and the Unit Circle — Questions and Answers
Question 1: In a right triangle, the side opposite a 30° angle is 5. What is the length of the hypotenuse?
- 10 (Correct answer)
- 5√3
- 5√2
- 2.5
Correct answer: 10
sin(30°) = opposite/hypotenuse = 1/2. So hypotenuse = opposite ÷ sin(30°) = 5 ÷ (1/2) = 10.
Question 2: What is the value of cos(60°)?
- 0.5 (Correct answer)
- √3/2
- 1
- √2/2
Correct answer: 0.5
cos(60°) = 1/2 = 0.5. This is a standard value from the unit circle / 30-60-90 triangle.
Question 3: Which of the following is equivalent to tan(θ)?
- sin(θ)/cos(θ) (Correct answer)
- cos(θ)/sin(θ)
- 1/sin(θ)
- sin²(θ) + cos²(θ)
Correct answer: sin(θ)/cos(θ)
By definition, tan(θ) = opposite/adjacent = (opposite/hypotenuse)/(adjacent/hypotenuse) = sin(θ)/cos(θ).
Question 4: An angle of 3Ï€/2 radians is equivalent to how many degrees?
- 270° (Correct answer)
- 180°
- 90°
- 360°
Correct answer: 270°
Multiply by 180/π: (3π/2) × (180/π) = (3 × 180)/2 = 270°.
Question 5: A ladder leans against a wall making a 65° angle with the ground. The ladder is 10 feet long. How high up the wall does it reach? (sin 65° ≈ 0.906)
- 9.06 feet (Correct answer)
- 4.23 feet
- 10.9 feet
- 6.54 feet
Correct answer: 9.06 feet
The height (opposite side) = hypotenuse × sin(65°) = 10 × 0.906 = 9.06 feet.
Question 6: Which of the following is a Pythagorean identity?
- sin²(θ) + cos²(θ) = 1 (Correct answer)
- sin(θ) + cos(θ) = 1
- tan²(θ) + 1 = sin²(θ)
- sin(θ) = cos(90° + θ)
Correct answer: sin²(θ) + cos²(θ) = 1
sin²(θ) + cos²(θ) = 1 is the fundamental Pythagorean identity derived from the Pythagorean theorem applied to the unit circle.
Math Trigonometry and the Unit Circle
In a right triangle, the side opposite a 30° angle is 5.
What is the length of the hypotenuse?