Half Angle
Compute sin(θ/2), cos(θ/2), tan(θ/2) using half angle formulas. Used in calculus and Weierstrass substitution — with step-by-step breakdown, identity verification, and interactive charts.
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Why: Understanding half angle helps you make better, data-driven decisions.
How: Enter Angle (θ), Unit to calculate results.
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Half Angle Value Breakdown
sin(θ/2), cos(θ/2), tan(θ/2)
sin²(θ/2) vs cos²(θ/2)
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • sin(θ/2) = ±√((1−cosθ)/2) — the ± depends on which quadrant θ/2 lies in
- • cos(θ/2) = ±√((1+cosθ)/2) — derived from the double-angle formula cos(2α) = 2cos²α − 1 with α = θ/2
- • tan(θ/2) = sinθ/(1+cosθ) = (1−cosθ)/sinθ — sign-free forms; use when cosθ ≠ −1 or sinθ ≠ 0
- • Half-angle formulas are essential for the Weierstrass substitution t = tan(θ/2), which converts any rational function of sin/cos to a rational function of t
- • Used in calculus for integrating √(1±cosθ), √(1±sinθ), and rational trig integrals
Did You Know?
How Half Angle Formulas Work
Half-angle identities express trig functions of θ/2 in terms of cosθ (and sometimes sinθ). They are the inverse of double-angle: if double-angle gives 2θ from θ, half-angle gives θ/2 from θ.
Derivation from Double Angle
From cos(2α) = 1 − 2sin²α, set α = θ/2: cosθ = 1 − 2sin²(θ/2), so sin²(θ/2) = (1−cosθ)/2. Similarly, cos(2α) = 2cos²α−1 gives cos²(θ/2) = (1+cosθ)/2.
Weierstrass Substitution
Let t = tan(θ/2). Then sinθ = 2t/(1+t²), cosθ = (1−t²)/(1+t²), dθ = 2dt/(1+t²). Any rational function of sinθ and cosθ becomes a rational function of t, integrable by partial fractions.
Sign Convention
The ± in √((1±cosθ)/2) is resolved by the quadrant of θ/2. sin(θ/2) ≥ 0 when θ/2 is in Q1 or Q2; cos(θ/2) ≥ 0 when θ/2 is in Q1 or Q4. This calculator uses the principal (non-negative) square root for 0° ≤ θ/2 ≤ 180°.
Expert Tips
Use tan(θ/2) for Sign-Free
tan(θ/2) = sinθ/(1+cosθ) has no ± ambiguity. Use when cosθ ≠ −1. The Double Angle Calculator gives the inverse relationship.
Weierstrass for Tough Integrals
∫dx/(3+5cosx) and similar become ∫R(t)dt after t = tan(x/2). See Trig Identities for more.
Check the Quadrant
θ = 120° gives θ/2 = 60° (Q1, all positive). θ = 240° gives θ/2 = 120° (Q2, sin positive, cos negative).
Connect to Sum Formulas
Half-angle can be derived from sum formulas: cos(θ/2+θ/2) = cos²(θ/2)−sin²(θ/2). The Sum & Difference Calculator explores these.
Half Angle vs Other Methods
| Feature | This Calculator | Direct sin(θ/2) | Manual Formula |
|---|---|---|---|
| All 3 half-angle forms | ✅ | ❌ | ✅ Slow |
| Quadrant & sign handling | ✅ | ❌ | ⚠️ Manual |
| Identity verification | ✅ | ❌ | ✅ |
| Weierstrass context | ✅ | ❌ | ⚠️ |
| Charts & visualization | ✅ | ❌ | ❌ |
| Step-by-step breakdown | ✅ | ❌ | ✅ |
| Degrees and radians | ✅ | ✅ | ✅ |
| Preset examples | ✅ | ❌ | ❌ |
Frequently Asked Questions
What is sin(θ/2) in terms of cosθ?
sin(θ/2) = ±√((1−cosθ)/2). The ± is determined by the quadrant of θ/2. For 0° ≤ θ/2 ≤ 180°, sin(θ/2) ≥ 0.
Why does cos(θ/2) use 1+cosθ?
From cos(2α) = 2cos²α−1 with α = θ/2: cosθ = 2cos²(θ/2)−1, so cos²(θ/2) = (1+cosθ)/2. The plus appears because we solve for cos²(θ/2).
When is tan(θ/2) undefined?
tan(θ/2) = sinθ/(1+cosθ) is undefined when 1+cosθ = 0, i.e. cosθ = −1, so θ = 180°, 540°, etc. Alternatively, (1−cosθ)/sinθ is undefined when sinθ = 0.
What is the Weierstrass substitution?
t = tan(θ/2) converts any rational function of sinθ and cosθ into a rational function of t. Then sinθ = 2t/(1+t²), cosθ = (1−t²)/(1+t²), dθ = 2dt/(1+t²).
How do I integrate √(1−cosθ)?
Use 1−cosθ = 2sin²(θ/2). Then √(1−cosθ) = √2|sin(θ/2)|. For 0 ≤ θ ≤ 2π, sin(θ/2) ≥ 0 when 0 ≤ θ ≤ 2π, so the integral becomes √2∫sin(θ/2)dθ = −2√2 cos(θ/2) + C.
What is the relationship to double angle?
Half-angle is the inverse: double angle gives sin(2θ) from sinθ and cosθ; half-angle gives sin(θ/2) from cosθ. They are derived from the same identity cos(2α) = 1−2sin²α.
How do I choose the sign for sin(θ/2)?
sin(θ/2) is positive when θ/2 is in Q1 (0°–90°) or Q2 (90°–180°), negative in Q3 or Q4. Check which quadrant (θ/2) mod 360 falls into.
Where are half-angle formulas used?
Calculus (Weierstrass substitution, integrating √(1±cosθ)), solving trig equations, and evaluating trig at non-standard angles like 15°, 22.5°.
Half Angle by the Numbers
Official & Educational Sources
Disclaimer: This calculator provides results based on standard IEEE 754 floating-point arithmetic. Results are accurate to approximately 15 significant digits. For mission-critical applications (aerospace, medical devices), always verify with certified computational tools. Not a substitute for professional engineering analysis.
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