Ratios, identities, complementary angles MCQs

Q1. If sin θ = 3/5 and θ is acute, then (1 − tan θ)/(1 + tan θ) equals:

• A.1/7
• B.1/3
• C.7/25
• D.5/7

Answer: A. 1/7

From the stem we have: 3; 5; 1; 1. The unknown asked is: the unknown asked in the stem. Governing law - tan θ = 3/4. This is the equation that links the given quantities to the unknown (math, chapter 'Introduction to Trigonometry'). Substituting: tan θ = 3/4 → (1 − 3/4)/(1 + 3/4) = (1/4)/(7/4) = 1/7. Therefore the answer is A) 1/7.

Q2. If tan θ + 1/tan θ = 2 sec θ, then sin θ equals (θ in (0, π/2)):

• A.1/2
• B.1
• C.√3/2
• D.0

Answer: A. 1/2

Q3. If cos²θ + cos²(60° − θ) + cos²(60° + θ) = k, then k equals:

• A.3/2
• B.1
• C.2
• D.1/2

Answer: A. 3/2

Data from the problem: 60 °. Required: the unknown asked in the stem. Governing law - Using 2 cos²x = 1 + cos 2x: sum = (3 + cos 2θ + cos(120° − 2θ) + cos(120° + 2θ))/2. This is the equation that links the given quantities to the unknown (math, chapter 'Introduction to Trigonometry'). Substituting: Using 2 cos²x = 1 + cos 2x: sum = (3 + cos 2θ + cos(120° − 2θ) + cos(120° + 2θ))/2 → The sum of three cosines equally spaced at 120° apart is 0, so sum = 3/2. So the answer is A) 3/2.