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

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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

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

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.

To find the exact values for all the solutions to the equation cos(theta) = sqrt(3)/2, we need to determine the angles in the interval -2pi <= theta <= 2pi that satisfy this equation.

First, let's recall the unit circle:

θ = 0° θ = 90° θ = 180° θ = 270°
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QIV | | | |
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QI ---- (1,0) ------ (0,1) ------ (-1,0) ------ (0,-1)
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QIII | | | |
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θ = -270° θ = -180° θ = -90° θ = 360°

Cosine is the x-coordinate of a point on the unit circle, so we are looking for angles θ where the x-coordinate is equal to sqrt(3)/2.

From the unit circle, we know that the x-coordinate is sqrt(3)/2 at angles 30° and 330° (or -30°). These correspond to the solutions θ = pi/6 and θ = -11pi/6.

However, the interval given is -2pi <= theta <= 2pi, so we need to find additional solutions within this range.

Since cosine is a periodic function, we can add or subtract multiples of 2pi to get equivalent solutions. Adding or subtracting 2pi will bring us back to the same x-coordinate.

To find other solutions, we can use the symmetry of the unit circle.

We can add 2pi to the initial solutions:

θ = pi/6 + 2pi = 13pi/6 and
θ = -11pi/6 + 2pi = pi/6

We can subtract 2pi from the initial solutions as well:

θ = pi/6 - 2pi = -11pi/6 and
θ = -11pi/6 - 2pi = -23pi/6

Therefore, the exact values for all the solutions to cos(theta) = sqrt(3)/2 in the interval -2pi <= theta <= 2pi are:

θ = pi/6, 13pi/6, -11pi/6, -23pi/6.