Find exact values for all the solutions to the equation cos theta = sqrt 3/2 for -2pi <= theta <= 2pi

Question

Theta =
Theta =
Theta =
Theta =

Answer 1

Make a real effort to memorize the trig ratios for these basic right-angled triangles
45-45-90 degree triangle ---> 1 -- 1 -- √2
30-60-90 degree triangle ---> 1 -- √3 -- 2

make a quick sketch of these triangles, It is easy to see where the values go. Remember, the smallest side is always the smallest side, the largest angle is opposite the largest side etc

so you are looking at cosØ = √3/2
then Ø must be 30° or π/6
but the cosine is positive in quads I and IV
so another solution would be 360-30 = 330° or 11π/6 radians

other angles occur at ±360° or ± 2π radians
so other answers are 30 - 360 or -330° or -11π/6
and 330-360 or -30° or -π/6

so in radians we have
-11π/6 , -π/6 , π/6 , and 11π/6

Answer 2

what values for theta(0<theta <2pi) satisfy the equation 2 sin theta cos theta+costheta=0

Answer 3

We can factor out cos(theta) from the first two terms, which gives:

2sin(theta)cos(theta) + cos(theta) = cos(theta)(2sin(theta) + 1) = 0

So either cos(theta) = 0 or 2sin(theta) + 1 = 0.

If cos(theta) = 0, then theta = π/2 or 3π/2.

If 2sin(theta) + 1 = 0, then sin(theta) = -1/2, which means theta = 7π/6 or 11π/6.

Therefore, the values of theta that satisfy the equation are:

π/2, 3π/2, 7π/6, and 11π/6.