Double Angle and Half Angle Formulas

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A series of free, online Trigonometry Video Lessons.
Videos, worksheets, and activities to help Trigonometry students.

In this lesson, we will learn

• the double angle formulas
• the half angle identities or formulas
• the angle inclination of a line

What are the Double-Angle Formulas?

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos²(x) - sin²(x) = 1 - 2sin²(x) = 2cos²(x) - 1/2
\(\tan (2x) = \frac{{2\tan (x)}}{{1 - {{\tan }^2}(x)}}\)

How to derive the Double-Angle Formulas?

The double angles sin(2x) and cos(2x) can be rewritten as sin(x + x) and cos(x + x).
Applying the cosine and sine addition formulas, we find that
sin(2x) = 2sinxcosx.
cos(2x) = cos2x − sin2x.
The cosine double angle formula is cos(2x) = cos2x − sin2x.
Combining this formula with the Pythagorean Identity, cos²(x) + sin²(x) = 1, two other forms appear: cos(2x) = 2cos²(x) − 1 and cos(2x ) = 1 − 2sin²(x). The derivation of the double angle identities for sine and cosine, followed by some examples. What are the Half-Angle Formulas?

\(\begin{array}{l}\sin \left( {\frac{u}{2}} \right) = \pm \sqrt {\frac{{1 - \cos u}}{2}} \\\cos \left( {\frac{u}{2}} \right) = \pm \sqrt {\frac{{1 + \cos u}}{2}} \\\tan \left( {\frac{u}{2}} \right) = \frac{{1 - \cos u}}{{\sin u}} = \frac{{\sin u}}{{1 + \cos u}}\end{array}\)

How to use the power reduction formulas to derive the half-angle formulas?

The half angle identities come from the power reduction formulas using the key substitution u = x/2 twice, once on the left and right sides of the equation.
With half angle identities, on the left side, this yields (after a square root) cos(x/2) or sin(x/2); on the right side cos(2x) becomes cos(x) because 2(1/2) = 1.
For a problem like sin(π/12), remember that x/2 = π/12, or x = π/6, when substituting into the identity. The derivations of the half-angle identities for both sine and cosine, plus listing the tangent ones. Then a couple of examples using the identities.

Angle Inclination of a Line

The angle inclination of a line is the angle formed by the intersection of the line and the x-axis. Using a horizontal "run" of 1 and m for slope, the angle of inclination, θ = tan⁻¹(m), or m = tan(θ). Therefore, if the angle or the slope is known, the other can be found using one of the equations. If the angle of inclination is negative, then the slope of the line is also negative.

Find the equation of a line given the angle of inclination.