Half Angle Identities
Calculator
👉 Proof of Half Angle Identities
Half Angle Identities calculator computes the half angle for sin (u/2), cos (u/2) and tan (u/2) for given angle using basic formulas:
sin(u/2) = + or - √ (1 – cos u) / 2.
cos(u/2) = + or - √ (1 + cos u) / 2.
The half-angle formulas allow the expression of trigonometric functions to determine the trigonometric values for another angle u/2 in terms of u.
The half-angle formulas are useful in finding the values of unknown trigonometric functions. The half-angle formulas for any angle u can be stated as follows:
sin(u/2) = + or - √ (1 – cos u) / 2.
In the case of u is in first and second quadrant then function is positive and in the case of u is in third and forth quadrant then function is negative.
cos(u/2) = + or - √ (1 + cos u) / 2.
In the case of u is in first and forth quadrant then function is positive and in the case of u is in second and third quadrant then function is negative.
Half Angle Formulas
Sine Half Angle Formula
sin(u/2) = ± √[(1 − cos(u))/2]
Cosine Half Angle Formula
cos(u/2) = ± √[(1 + cos(u))/2]
Tangent Half Angle Formula
tan(u/2) = sin(u)/(1 + cos(u))
Worked Example
Important Notes
- The sign (+ or −) depends on the quadrant of the angle u/2.
- If u/2 lies in Quadrant I or II, sine is positive.
- If u/2 lies in Quadrant III or IV, sine is negative.
- If u/2 lies in Quadrant I or IV, cosine is positive.
- If u/2 lies in Quadrant II or III, cosine is negative.
- tan(u/2) is undefined when 1 + cos(u) = 0.
Frequently Asked Questions
What is the half-angle formula?
Half-angle formulas express trigonometric functions of u/2 in terms of trigonometric functions of u.
Why are half-angle identities useful?
They help evaluate trigonometric functions of angles that are difficult to compute directly.
How do you know whether to use + or −?
Determine the quadrant containing the angle u/2 and use the sign of the corresponding trigonometric function in that quadrant.