tan(theta) * sin(theta) = 0.3823?

Question

need help solving for theta, i tried using identities but made it more complicated.

Answer 1

You need to get it down to a single function:

tan O = sin O/cosO, so replace that and get

sin O/ cos O * sin O = 0.3823

This is sin^2 O / cos O

Which in turn is (1 - cos^2 O)/cos O = .3823

Cross multiply: .3823 cos O = 1 - cos^2 O

You can make this into a quadratic equation equal to zero, and solve.

Answer 2

tan(t) sin(t) = 0.3823

[sin(t)/cos(t)] sin(t) = 0.3823

sin^2(t) / cos(t) = 0.3823

Multiply both sides by cos(t),

sin^2(t) = 0.3823 cos(t)

Convert sin^2(t) to 1 - cos^2(t).

1 - cos^2(t) = 0.3823cos(t)

Now, we move everything to the right hand side.

0 = cos^2(t) + 0.3823 cos(t) - 1

And now, we solve this as a quadratic.

cos(t) = [-0.3823 +/- sqrt( (0.3823)^2 - 4(1)(-1) )] / 2

cos(t) = [-0.3823 +/- sqrt( 4.14615329 ) ] / 2

That means we have two equations:

cos(t) = [-0.3823 + sqrt( 4.14615329 ) ] / 2

cos(t) = [-0.3823 - sqrt( 4.14615329 ) ] / 2

We solve each of them individually, and

t = arccos([-0.3823 + sqrt( 4.14615329 ) ] / 2)
t = arccos([-0.3823 - sqrt( 4.14615329 ) ] / 2)