sin2A = 2sinA cosA

cos2A = cos^2 A - sin^2 A

= 2cos^2 A - 1

= 1 - 2sin^2 A

tan2A = (2tanA)/(1 - tan^2 A)

These formulas can be proven using the Sum and Difference Formulae. Observe:

sin2A = sin(A + A) Use the Sum and Difference Formulae here.

sin2A = sinAcosA + cosAsinA

Answer: sin2A = 2sinAcosA

cos2A = cos(A + A) Use the Sum and Difference Formulae here.

cos2A = cosAcosA - sinAsinA

Answer: cos2A = cos^2 A - sin^2 A

tan2A = tan(A + A) Use the Sum and Difference Formulae here.

tan2A = (tanA + tanA)/(1 - tanAtanA)

Answer: tan2A = (2tanA)/(1 - tan^2 A)

Example

Find all solutions of 2cos x + sin 2x = 0.

Solution

Trig Frog suggests that thou begin by rewriting the equation in terms of functions of x - get rid of the 2x. So:

2cos x + sin 2x = 0

2cos x + 2sin x cos x = 0

2 cos x(1 + sin x) = 0 Factor out 2cos x.

cos x = 0, 1 + sin x = 0 Set the two factors equal to 0 and solve.

x = pi/2, 3pi/2 and x = 3pi/2 Obtain these function values from the Unit Circle.

Now, these solutions need to be over the interval [0, 2pi). So, write the general solution as well.

Answer: x = pi/2 + 2pin and x = 3pi/2 + 2pin