What is double or half angle formula?
What is double or half angle formula?
Key Equations
| Double-angle formulas | sin(2θ)=2sinθcosθcos(2θ)=cos2θ−sin2θ=1−2sin2θ=2cos2θ−1tan(2θ)=2tanθ1−tan2θ |
|---|---|
| Reduction formulas | sin2θ=1−cos(2θ)2cos2θ=1+cos(2θ)2tan2θ=1−cos(2θ)1+cos(2θ) |
| Half-angle formulas | sinα2=±√1−cosα2cosα2=±√1+cosα2tanα2=±√1−cosα1+cosα=sinα1+cosα=1−cosαsinα |
How do you calculate half angle?
How to Use Half-Angle Identities to Evaluate a Trig Function
- Rewrite the trig function and the angle as half of a unit circle value.
- Determine the sign of the trig function.
- Substitute the angle value into the right identity.
- Replace cos x with its actual value.
- Simplify the half-angle formula to solve.
How do you find half angle identity tan?
We can use the half-angle formula for tangent: tan θ2=√1−cos θ1+cos θ. Since tan θ is in the first quadrant, so is tan θ2. We can take the inverse tangent to find the angle: tan−1(0.57)≈29.7°.
Where do double angle identities come from?
Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent.
How do you find the half angle of a double angle formula?
Double‐Angle and Half‐Angle Identities
- Using the Pythagorean identity, sin 2 α+cos 2α=1, two additional cosine identities can be derived.
- and.
- The half‐angle identities for the sine and cosine are derived from two of the cosine identities described earlier.
How do you find the half angle?
The half-angle formula for sine is derived as follows:
- sin2θ=1−cos(2θ)2sin2(α2)=1−(cos2⋅α2)2=1−cosα2sin(α2)=±√1−cosα2.
- cos2θ=1+cos(2θ)2cos2(α2)=1+cos(2⋅α2)2=1+cosα2cos(π2)=±√1+cosα2.
- tan2θ=1−cos(2θ)1+cos(2θ)tan2(α2)=1−cos(2⋅α2)1+cos(2⋅α2)tan(α2)=±√1−cosα1+cosα