Half Angle Formula With Fractions

We will brainstorm by looking at the Double Bending Formula for cosine.

Solve this for cos x, like and so.

The final pace to get the cos x lonely on the left side, we demand to apply the square root to both sides of the equation. In doing and so, we must place a plus-minus sign on the right side of the equation.

Let 2x = A, which means ten = ^A/₂. We can supercede the x-variables with ^A/_ii, like so.

We tin clean upwards the angle inside the foursquare root to get the last one-half-angle formula for cosine.

Here is some other Double Angle Formula for cosine.

Nosotros will solve it for sin 10 by using elementary algebra.

To simplify the left side of the equation, we can dissever both the numerator and the denominator past -one. In doing and so, both the numerator and the denominator will change to contrary expressions.

Now, nosotros can take the square root of both sides, which requires a plus-minus sign.

Like what was done in the section above for the cosine formula, let 2x = A, which means 10 =^A/₂. Nosotros can replace the x-variables with ^A/₂, like so.

Cleaning up the angle expression under the square root, we get...

The formula above is the one-half-angle formula for sine.

Say we had an bending similar 15°. If nosotros wanted the verbal value of this angle using any trigonometric function, we could non calculate it. xv° is non a special angle. However, we can calculate double its value, which allows us to use a one-half-angle formula.

Nosotros should expect at a specific example, like this.

Example: Summate the exact value of cos(xv°).

We can calculate the cos(30°). And so, we can use the half-angle formula for cosine. Offset like this.

Writing our problem similar this allows usa to use the half-angle formula for cosine, like so.

Using our knowledge of special angles, we know the exact value of cos(30°). We also know that the cosine of angles in the offset quadrant are positive ratios. So, in that location is no need for the plus-minus sign.