Trigonometry Advanced Trigonometric Concepts 1 — Questions and Answers
Question 1: Which of the following is the double-angle identity for sine?
- sin(2θ) = 2sin(θ) cos(θ) (Correct answer)
- sin(2θ)=sin 2 (θ−cos 2 (θ)
- sin(2θ)=1−2cos 2 (θ)
- sin(2θ)=1−sin 2(θ)
Correct answer: sin(2θ) = 2sin(θ) cos(θ)
The double-angle identity for sine is a fundamental trigonometric formula used to express the sine of twice an angle (2θ) in terms of the sine and cosine of the original angle (θ). This identity, sin(2θ) = 2sin(θ) cos(θ), is derived from the angle sum formula for sine by setting both angles to θ. It is widely used in solving trigonometric equations and simplifying expressions.
Question 2: What is the exact value of cos −1(0)?
- 0
- π/2 (Correct answer)
- π
- 2π
Correct answer: π/2
The expression cos⁻¹(0) asks for the angle whose cosine is 0. The range of the arccosine function is typically defined as [0, π] (or 0 to 180 degrees). Within this range, the only angle whose cosine is 0 is π/2 radians (or 90 degrees), which corresponds to the point (0,1) on the unit circle.
Question 3: Which of the following is equivalent to 1+tan2(𝜃)?
- sec2(𝜃) (Correct answer)
- csc 2(θ)
- cot 2(θ)
- sin 2(θ)+cos 2(θ)
Correct answer: sec2(𝜃)
This is one of the fundamental Pythagorean trigonometric identities. It can be derived by taking the primary identity, sin²(𝜃) + cos²(𝜃) = 1, and dividing every term by cos²(𝜃). This transformation yields (sin²(𝜃)/cos²(𝜃)) + (cos²(𝜃)/cos²(𝜃)) = 1/cos²(𝜃), which simplifies to tan²(𝜃) + 1 = sec²(𝜃).
Question 4: Simplify cos2(𝜃)−sin2(𝜃).
- sin(2𝜃)
- cos(2θ) (Correct answer)
- 1−2sin2(θ)
- 2cos2(θ)−1
Correct answer: cos(2θ)
The expression cos²(𝜃)−sin²(𝜃) is a direct form of the double-angle identity for cosine. This identity states that cos(2𝜃) can be expressed in three ways: cos²(𝜃) - sin²(𝜃), 2cos²(𝜃) - 1, or 1 - 2sin²(𝜃). Therefore, the given expression simplifies directly to cos(2𝜃).
Question 5: Solve the equation 2sin(𝑥)cos(𝑥)= √3/2 for 0≤𝑥≤2𝜋.
- x=π/6,5π/6
- x=π/3,2π/3
- x=π/6,7π/6
- x=π/3,4π/3 (Correct answer)
Correct answer: x=π/3,4π/3
The equation 2sin(x)cos(x) = √3/2 can be simplified using the double-angle identity for sine, which states that 2sin(x)cos(x) = sin(2x). So, the equation becomes sin(2x) = √3/2. Within the interval [0, 4π] for 2x (corresponding to [0, 2π] for x), the angles whose sine is √3/2 are π/3, 2π/3, 7π/3, and 8π/3. Dividing these by 2 gives the solutions for x as π/6, π/3, 7π/6, and 4π/3. Option D, x=π/3, 4π/3, correctly lists two of these valid solutions.
Trigonometry Advanced Trigonometric Concepts 1
Which of the following is the double-angle identity for sine?