tan 2A (Double-Angle Formula)

tan 2A (Double-Angle Formula)

How to solve tan 2A (Double-angle formula) problems: formula, proof, examples, and their solutions.

Formula

tan 2A = (2 tan A) / (1 - tan^2 A)

tan 2A = 2 tan A / (1 - tan2 A)

Proof

tan 2A (Double-Angle Formula): Proof of the Formula

tan 2A = tan (A + A)
= (tan A + tan A) / (1 - tan A tan A)

tan (A + B)

Example 1

If cos theta = -2/3 and pi/2 <= theta <= pi, find the value of tan 2*theta.

π/2 ≦ θπ
So θ is in quadrant II.

cos θ = -2/3
And cosine: CAH.

So draw a right triangle in quadrant II
whose adjacent side is -2
and whose hypotenuse is 3.

The triangle above is a right triangle.

So (blue side)2 + (-2)2 = 32.

Pythagorean theorem

The brown angle is the reference angle of ∠θ.

And tangent: TOA.

So tan θ = tan (brown)
= √5/(-2).

tan θ = -√5/2

So tan 2θ = [2⋅(-√5/2)] / [1 - (-√5/2)2].

Complex fraction

Example 2

Find the value of m. Angle between y = (1/2)x and the x-axis: theta. Angle between y = mx and y = (1/2)x: theta.

Recall that the slope of a line
is the tangent of the central angle.

Trigonometric ratio - tangent

So tan θ = 1/2.
And tan 2θ = m.

So tan 2θ = m
= [2⋅(1/2)] / [1 - (1/2)2].

Complex fraction