How do you solve sin 2theta -1 = cos 2 theta?

3 Answers
Jul 14, 2016

pi/4 and pi/2

Explanation:

Re-write the equation:
sin 2t - cos 2t = 1
Use the trig identity:
sin a - cos a = sqrt2sin (a - pi/4).
Replace a by 2t -->
sin 2t - cos 2t = sqrt2 sin (2t - pi/4) = 1
sin (2t - pi/4) = 1/sqrt2 = sqrt2/2
Trig table and unit circle -->
sqrt2/2 is sin ((pi)/4) and in the same time sin ((3pi)/4). There for:
a. (2t - pi/4) = pi/4
2t = pi/4 + pi/4 = pi/2 --> t = pi/4
b. (2t - pi/4) = (3pi)/4
2t = (3pi)/4 + pi/4 = pi --> t = pi/2
Check
a. t = pi/4 --> 2t = pi/2 --> sin 2t = 1 and cos 2t = 0
We have 1 = 1 . OK
b. t = pi/2 --> 2t = pi --> sin 2t = 0, and -cos 2t = -(-1) = 1.
We have 1 = 1. OK

Jul 14, 2016

2sinthetacostheta - 1 = 1 - 2sin^2theta

2sinthetacostheta - 1 - 1 + 2sin^2theta = 0

2(sinthetacostheta - 1 + sin^2theta) = 0

Since sin^2theta - 1 = -cos^2theta:

sinthetacostheta - cos^2theta = 0

costheta(sin theta - costheta) = 0

Solving costheta = 0 first:

theta = 90˚, 270˚

Solving sintheta - costheta = 0:

sqrt((sin theta - costheta)^2) = 0

sqrt(sin^2theta - 2sinthetacostheta + cos^2theta) = 0

(sqrt(sin^2theta - 2sinthetacostheta + cos^2theta))^2 = 0^2

sin^2theta + cos^2theta - 2sinthetacostheta = 0

Since sin^2theta + cos^2theta = 1 and 2sinthetacostheta = sin2theta:

1 - sin2theta = 0

1 = sin2theta

theta = 45˚

Hence, the solutions are theta = 45˚, 90˚ and 270˚.

Hopefully this helps!

Jul 14, 2016

sin2theta-1=cos2theta

=>2sinthetacostheta-(1+cos2theta)=0

=>2sinthetacostheta-2cos^2theta=0

=>2costheta(sintheta-costheta)=0

:.costheta=0 and sintheta-costheta=0

When costheta=0

then theta=npi+pi/2

Again when sintheta-costheta=0

=>tantheta=1=tan(pi/4)

Then theta=npi+pi/4

where n=0,+-1,+-2,+-3...
.