If 0 is greater than or equal to theta and theta is greater than or equal to 2 pi, find the values of theta that make each statement true. Keep your answers in terms of pi.

Question

a) cos theta= 1/2...... Theta= and

b) sin theta= 1/2...... theta= and

(there are two answers)

Answer 1

HI Audrey,

If 0 ≥ Θ ≥ 2π

Find the values of Θ that make each statement true.

Keep your answers in terms of pi.

a) cos theta= 1/2...... Theta= and

b) sin theta= 1/2...... theta= and

a) To determine all the values of theta for which the cosine is equal to 1/2, you have to understand the unit circle. The cosine is the x value of the coordinates of a unit circle. A reference angle of π/3 gives you a cosine of 1/2. You have a positive value for cosine when your x-value of your unit circle is positive. This happens in the I and IV quadrants. So in quadrant 1 you have cos(π/3) = 1/2, and in Quadrant 4, the reference angle of π/3 is from the angle of (2π-π/3) = 5π/3.

So your answer is π/3 and 5π/3

b) Sin theta = 1/2 when your reference angle is π/6. The sin is the y value of your unit circle. So the sin has a positive value when y is positive. This is in the I and II quadrant. So in quadrant I, you have sin(π/6)=1/2. In quadrant II, your reference angle is π/6. So your angle is (π-π/6)= 5π/6.

sin(5π/60= 1/2

So this answer is π/6 and 5π/6

Things to look up:

1. special triangles (30,60,90) (45,45,90)
2. The Unit circle and how it relates to sin, cos, tan
3. Reference angles in your unit circle

Hope this helps. Let me know if you have more questions :)

Answer 2

cosθ = 1/2

cosθ is positive in quadrants 1 and 4

θ = π/3 and 2π - π/3 = 5π/3

sinθ = 1/2

sinθ is positive in quadrants 1 and 2

θ = π/6 and π - π/6 = 5π/6