If sin θ = 1/2 and cos θ = sqrt(3)/2 for some acute angle θ, then tan θ equals which value?

If sin θ = 1/2 and cos θ = sqrt(3)/2 for some acute angle θ, then tan θ equals which value?

• A. sqrt(3)/3
• B. 1
• C. sqrt(3)
• D. 2

Answer: (A)

Tan is the ratio of sine to cosine, so tan θ = sin θ / cos θ. With sin θ = 1/2 and cos θ = √3/2, the ratio is (1/2) ÷ (√3/2) = (1/2) × (2/√3) = 1/√3, which can be written as √3/3. Since θ is acute, both sine and cosine are positive, so the positive value is correct. This corresponds to a 30° angle in a 30-60-90 triangle, where tan 30° = 1/√3 = √3/3.

Tan is the ratio of sine to cosine, so tan θ = sin θ / cos θ. With sin θ = 1/2 and cos θ = √3/2, the ratio is (1/2) ÷ (√3/2) = (1/2) × (2/√3) = 1/√3, which can be written as √3/3. Since θ is acute, both sine and cosine are positive, so the positive value is correct. This corresponds to a 30° angle in a 30-60-90 triangle, where tan 30° = 1/√3 = √3/3.