Trig half angle identities or functions actually involved in those trigonometric functions which have half angles in them. The square root of the first 2 functions sine & cosine either negative or positive totally depends upon the existence of angle in a quadrant. Learn more about **Trig Identities** at trigidentities.info.

Here comes the comprehensive table which depicts clearly the half-angle identities of all the basic trigonometric identities. Explore more about **Inverse trig identities**.

Contents

## Derivation of Trig Half-Angle Identities

Today we are going to derive following trig half-angle formulas.

**Half Angle Formula – Sine**

**cos 2***θ*= 1− 2sin2*θ*

**Now, if we let**

**θ****= α/2**

**then ****2****θ**** = ****α**** and our formula becomes:**

**cos***α***=****1****−****2****sin****2****(****2**)*α*

**We now solve for**

**Sin (α/2)****2 sin****2****(****2***α***)****=1−cos***α**sin**ˆ**2 (2**α**) = 1 – cos**α*

Solving gives us the following **sine of a half-angle** identity:

** Now let talk about the positive or negative sign of Sin (α/2).**

**If the angle lies in the first quadrant then all positive means sine half angle identity will be positive.****And if it is in 3**^{rd }or 4^{th}quadrant we will introduce a negative sign with the sine half angle identity.

**Half Angle Formula – Cosine**

**Simply by using a similar process, With the same substitutions, we did above. Now we have to substitute these values into the following Trig identity. **

**cos 2****θ****= 2cos**^{2}**θ****– 1**

After substituting the values We

**Cos***α*= 2 cosˆ2(α/2)-1 —–(1)

Now you need to reverse the equation.

Reverse the equation:

**2 cosˆ2(α/2)-1 = Cos***α —–(2)*

*Add both eq (1) & (2)*

**2 cosˆ2(α/2) = 1+cos***α*

*Divide by 2 on both sides *

**cosˆ2(α/2) = 1+cos***α/2*

Taking square root on both sides we got **cosine half angle formula** now

** let’s talk about the positive or negative sign of Cosine (α/2).**

**If the angle lies in first or 4th quadrant then Cosine(α/2) will be positive****And if it is in 2**^{nd }or 3^{rd}quadrant we will introduce a negative sign with the cosine half angle identity.