Multiple Choice Questions
Trigonometry and Trigonometric Identities
Topic: Trigonometric Identities
Grade: 10
Question 1:
Which of the following is an identity for sin(2θ)?
a) sin^2(θ) + cos^2(θ)
b) 2sin(θ)cos(θ)
c) sin^2(θ) – cos^2(θ)
d) tan(θ) + cot(θ)
Answer: b) 2sin(θ)cos(θ)
Explanation: The identity for sin(2θ) is 2sin(θ)cos(θ). This can be proven using the double-angle identity for sine: sin(2θ) = 2sin(θ)cos(θ). For example, if θ = 30°, sin(2θ) = sin(60°) = 2sin(30°)cos(30°) = 2(0.5)(√3/2) = √3.
Question 2:
What is the value of cos^2(θ) – sin^2(θ)?
a) 1
b) cos(2θ)
c) 2cos(θ)sin(θ)
d) tan(θ) – cot(θ)
Answer: a) 1
Explanation: The given expression can be simplified using the Pythagorean identity for sine and cosine: cos^2(θ) – sin^2(θ) = cos^2(θ) – (1 – cos^2(θ)) = cos^2(θ) – 1 + cos^2(θ) = 2cos^2(θ) – 1. Since the Pythagorean identity cos^2(θ) + sin^2(θ) = 1, we can substitute this value to get 2cos^2(θ) – 1 = 2(1 – sin^2(θ)) – 1 = 2 – 2sin^2(θ) – 1 = 1 – 2sin^2(θ). Therefore, the expression is equal to 1.
Question 3:
What is the period of the function f(x) = sin(3x)?
a) π/3
b) π
c) 2Ï€/3
d) 2Ï€/9
Answer: b) π
Explanation: The period of a function is the distance between two consecutive points where the function repeats itself. For a sine or cosine function, the period is given by 2π divided by the coefficient of x inside the trigonometric function. In this case, the coefficient is 3, so the period is 2π/3. Therefore, the correct answer is b) π.
Question 4:
Which of the following is equivalent to tan(θ)cos(θ)?
a) sin(θ) – cos(θ)
b) sin(θ) + cos(θ)
c) sin(θ)/cos(θ)
d) sin(θ)cos(θ)
Answer: c) sin(θ)/cos(θ)
Explanation: The given expression can be rewritten using the identity tan(θ) = sin(θ)/cos(θ). Therefore, tan(θ)cos(θ) = (sin(θ)/cos(θ))cos(θ) = sin(θ).
Question 5:
What is the value of sin(Ï€/6)?
a) 1/2
b) √3/2
c) 1
d) 0
Answer: a) 1/2
Explanation: The value of sin(π/6) can be determined using the unit circle or the special right triangle with angles 30°, 60°, and 90°. In this case, sin(π/6) = 1/2, which corresponds to the y-coordinate of the point on the unit circle or the side opposite the angle in the triangle. Therefore, the correct answer is a) 1/2.