What is the value of tan(θ/2)?
2023
What is the value of tan(θ/2)?
Answer: D. (sin θ)/(1 + cos θ) — Concept. The half-angle identities rewrite a full-angle sine or cosine in terms of θ/2: sin θ = 2 sin(θ/2) cos(θ/2), 1 + cos θ = 2 cos2(θ/2), and 1 − cos θ =…
- A.
(cos θ)/(1 − sin θ)
- B.
(sin θ)/(1 − cos θ)
- C.
(cos θ)/(1 − cos θ)
- D.
(sin θ)/(1 + cos θ)
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Correct answer: D
Concept. The half-angle identities rewrite a full-angle sine or cosine in terms of θ/2: sin θ = 2 sin(θ/2) cos(θ/2), 1 + cos θ = 2 cos2(θ/2), and 1 − cos θ = 2 sin2(θ/2). Combined with the definition tan x = sin x / cos x, any ratio assembled from sin θ and 1 ± cos θ collapses to a single half-angle function.
Application. Build the half-angle tangent up from its definition:
By definition, tan(θ/2) = sin(θ/2) / cos(θ/2).
Multiply numerator and denominator by 2 cos(θ/2), which is non-zero wherever tan(θ/2) is defined: tan(θ/2) = 2 sin(θ/2) cos(θ/2) / (2 cos2(θ/2)).
The numerator 2 sin(θ/2) cos(θ/2) is precisely sin θ.
The denominator 2 cos2(θ/2) is precisely 1 + cos θ.
Therefore tan(θ/2) = (sin θ)/(1 + cos θ).
Cross-check. Take θ = 90°: tan 45° = 1, and (sin 90°)/(1 + cos 90°) = 1/(1 + 0) = 1. Take θ = 60°: tan 30° = 1/√3, and (sin 60°)/(1 + cos 60°) = (√3/2)/(3/2) = 1/√3. Multiplying the definition by 2 sin(θ/2) instead of 2 cos(θ/2) puts 1 − cos θ in the numerator and sin θ in the denominator, giving the companion form tan(θ/2) = (1 − cos θ)/(sin θ).
Contrast. The other ratios assembled from the same pieces land on different half-angle functions:
Expression | Half-angle form |
|---|---|
(cos θ)/(1 − sin θ) | tan(π/4 + θ/2) |
(sin θ)/(1 − cos θ) | cot(θ/2) |
(cos θ)/(1 − cos θ) | ½ cosec2(θ/2) − 1 |
(sin θ)/(1 + cos θ) | tan(θ/2) |