Trigonometry Core Concepts 1 — Questions and Answers
Question 1: How many radians are in a full circle?
- π/2
- 2π (Correct answer)
- π
- 4π
Correct answer: 2π
In trigonometry, a full circle represents one complete rotation. In radian measure, this full rotation is defined as 2π radians. This is because the circumference of a unit circle (radius 1) is 2π, and radians are defined as the ratio of arc length to radius.
Question 2: In the unit circle, what are the coordinates of the point at 90∘?
- (1,0)
- (0,1) (Correct answer)
- (−1,0)
- (0,−1)
Correct answer: (0,1)
On the unit circle, the coordinates (x, y) correspond to (cos θ, sin θ) for a given angle θ. At 90 degrees, the angle points directly along the positive y-axis. Therefore, the x-coordinate (cosine) is 0, and the y-coordinate (sine) is 1, resulting in the point (0,1).
Question 3: What is the value of sin(30∘)?
- 0.707
- 1.0
- 0.5 (Correct answer)
- 0.866
Correct answer: 0.5
The sine of an angle in a right-angled triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For a 30-degree angle, this specific ratio is 1/2. Therefore, sin(30°) equals 0.5.
Question 4: What is the tangent of 45∘?
- 1.2
- 1 (Correct answer)
- 3
- 7
Correct answer: 1
The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle (tan θ = sin θ / cos θ). For 45 degrees, both sin(45°) and cos(45°) have a value of √2/2. Therefore, tan(45°) = (√2/2) / (√2/2) = 1.
Question 5: Convert 120∘ to radians.
- 2π/3 (Correct answer)
- π/2
- 3π/4
- 5π/6
Correct answer: 2π/3
To convert degrees to radians, you multiply the degree measure by the conversion factor π/180. So, 120° * (π/180°) = 120π/180. Simplifying this fraction by dividing both numerator and denominator by 60 gives 2π/3 radians.
Trigonometry Core Concepts 1
How many radians are in a full circle?