Sharpen your understanding of trigonometric relationships and complex number concepts with this engaging quiz designed to assess key skills at an intermediate level. Explore applications, properties, and calculations related to angles, identities, and the complex plane.
What is one solution for x in the interval [0, 2π) to the equation 2sin(x) = 1?
Explanation: Dividing both sides by 2 gives sin(x) = 1/2, and the angle x = π/6 satisfies this equation within the given interval. The option π/2 gives sin(π/2) = 1, which is too large, and π/4 yields sin(π/4) = √2/2, not 1/2. The distractor 2π/3 gives sin(2π/3) = √3/2, not 1/2. Only π/6 is correct for this equation.
Which of the following is the correct expansion for cos(a + b)?
Explanation: The cosine addition formula is cos(a + b) = cos(a)cos(b) - sin(a)sin(b), making the first option correct. The second and fourth choices are versions of the sine addition formula, not cosine. The third option has the sign incorrect; it should be a minus, not a plus.
If z₁ = 2 + 3i and z₂ = 1 - 2i, what is the product z₁ × z₂?
Explanation: Multiplying (2 + 3i)(1 - 2i), we use distribution: 2×1 + 2×(-2i) + 3i×1 + 3i×(-2i) = 2 - 4i + 3i - 6i^2. Since i^2 = -1, -6i^2 = +6. Adding real and imaginary parts: (2+6) + (-4i+3i) = 8 - i. The other options result from common calculation errors or sign mix-ups.
Which value of θ in degrees results in sin(θ) = cos(θ)?
Explanation: The sine and cosine of 45 degrees are both equal to √2/2, so θ = 45° satisfies the equation. For 30°, sine and cosine are 1/2 and √3/2 respectively. At 60°, their values are reversed but not equal. At 90°, sin(90) = 1 and cos(90) = 0. Only 45° is correct.
What is the modulus of the complex number z = -3 + 4i?
Explanation: The modulus is calculated as √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5. The distractor 1 comes from subtracting instead of adding the squares. Seven is given by incorrectly adding the numbers rather than squaring. Four is the imaginary part only, not the modulus.