Half Angle Calculator – Guide & Formulas
Calculate half angle values for sin(θ/2), cos(θ/2), and tan(θ/2) instantly. Enter an angle to see half angle results with step-by-step derivations.
The Half Angle Calculator computes sin(θ/2), cos(θ/2), and tan(θ/2) for any given angle. Enter an angle in degrees or radians to see all three half angle results with step-by-step derivations and the underlying identities.
Key Takeaway
Use the free Half Angle Calculator to calculate half angle values for sin(θ/2), cos(θ/2), and tan(θ/2) instantly. enter an angle to see half angle results with step-by-step derivations. Get instant results with step-by-step explanations.
How to Use the Half Angle Calculator
- Enter an angle value θ in the input field.
- Select degrees or radians as the angle unit.
- View all three half angle results: sin(θ/2), cos(θ/2), and tan(θ/2).
- Review the step-by-step derivation using half angle identities.
The Formula
Variable Definitions
- θ/2: Half the input angle
- sin(θ/2): Sine of the half angle (sign depends on quadrant)
- cos(θ/2): Cosine of the half angle (sign depends on quadrant)
- tan(θ/2): Tangent of the half angle
- cos(θ): Cosine of the original angle (needed for the formula)
Example: Half Angle of 60°
Find sin(30°), cos(30°), and tan(30°) using half angle formulas with θ = 60°.
- Step 1: Identify θ = 60°, so θ/2 = 30°. Compute cos(60°) = 0.5.
- Step 2: sin(30°) = √((1 - cos(60°)) / 2) = √((1 - 0.5) / 2) = √(0.25) = 0.5.
- Step 3: cos(30°) = √((1 + cos(60°)) / 2) = √((1 + 0.5) / 2) = √(0.75) = 0.866.
- Step 4: tan(30°) = sin(60°) / (1 + cos(60°)) = 0.866 / 1.5 = 0.577.
Frequently Asked Questions
What are half angle formulas?
Half angle formulas express trigonometric functions of θ/2 in terms of functions of θ. The three main formulas are: sin(θ/2) = ±√((1 - cos(θ))/2), cos(θ/2) = ±√((1 + cos(θ))/2), and tan(θ/2) = sin(θ)/(1 + cos(θ)).
How do I derive the half angle formula for sine?
Start with the double angle formula cos(2α) = 1 - 2sin²(α). Let α = θ/2 to get cos(θ) = 1 - 2sin²(θ/2). Rearranging gives sin²(θ/2) = (1 - cos(θ))/2, so sin(θ/2) = ±√((1 - cos(θ))/2).
When do I use the positive or negative sign?
The sign depends on the quadrant where θ/2 lies. If θ/2 is in Q1 or Q2, use the positive sign for sine. If θ/2 is in Q3 or Q4, use the negative sign. For cosine, Q1 and Q4 use positive, Q2 and Q3 use negative.
How do I derive the half angle formula for tangent?
From sin(θ/2) = ±√((1 - cos(θ))/2) and cos(θ/2) = ±√((1 + cos(θ))/2), divide to get tan(θ/2) = sin(θ/2)/cos(θ/2). Simplifying gives tan(θ/2) = sin(θ)/(1 + cos(θ)) = (1 - cos(θ))/sin(θ).
What is the tan half angle formula in terms of sin and cos?
There are two common forms: tan(θ/2) = sin(θ)/(1 + cos(θ)) and tan(θ/2) = (1 - cos(θ))/sin(θ). Both are equivalent and useful in different situations.
How are half angle formulas used in calculus?
Half angle formulas are used to reduce powers of trigonometric functions (e.g., ∫sin²(x)dx uses the power-reducing identity derived from half angle), simplify integrals, and solve trigonometric equations.
What is the relationship between half angle and double angle?
Half angle formulas are the reverse of double angle formulas. If sin(2α) = 2sin(α)cos(α), then setting α = θ/2 gives sin(θ) = 2sin(θ/2)cos(θ/2). The half angle formulas isolate sin(θ/2) and cos(θ/2).
How do I find tan(θ/2) if I only know tan(θ)?
Use the identity tan(θ/2) = (1 - cos(θ))/sin(θ). First find sin(θ) = tan(θ)/√(1 + tan²(θ)) and cos(θ) = 1/√(1 + tan²(θ)), then substitute into the formula.
What is Weierstrass substitution?
Weierstrass substitution (also called the t-substitution) uses t = tan(θ/2) to convert trigonometric functions into rational functions of t: sin(θ) = 2t/(1 + t²), cos(θ) = (1 - t²)/(1 + t²). This is extremely useful for integrating rational functions of sin and cos.
What are common half angle values?
Common values: sin(15°) = (√6 - √2)/4 ≈ 0.259, cos(15°) = (√6 + √2)/4 ≈ 0.966, tan(15°) = 2 - √3 ≈ 0.268 (from θ = 30°). Also sin(22.5°) = √(2 - √2)/2 ≈ 0.383 (from θ = 45°).