Question

b. for each question below draw the terminal arms, state the quadrant 1 reference angle and then state the values for the missing angle. do not use a calculator.

1. $\cos\theta=\frac{1}{2}$
2. $\tan\theta=-1$
3. $\sin\theta=\frac{\sqrt{3}}{2}$
4. $\sin\theta=\frac{-\sqrt{2}}{2}$
5. $\cos\theta=\frac{-\sqrt{3}}{2}$
6. $\tan\theta=0$
7. $\sin\theta=\frac{1}{2}$
8. $\tan\theta=\sqrt{3}$
9. $\sin\theta=\frac{-\sqrt{3}}{2}$
10. $\tan\theta=\frac{1}{\sqrt{3}}$
11. $\cos\theta=0$
12. $\sin\theta=-1$

Explanation:

Step1: Identify quadrants for cosθ=1/2

cosθ positive in Q1/Q4; ref angle 60°
Q1: θ=60°, Q4: θ=300°

Step2: Identify quadrants for tanθ=-1

tanθ negative in Q2/Q4; ref angle 45°
Q2: θ=135°, Q4: θ=315°

Step3: Identify quadrants for sinθ=√3/2

sinθ positive in Q1/Q2; ref angle 60°
Q1: θ=60°, Q2: θ=120°

Step4: Identify quadrants for sinθ=-√2/2

sinθ negative in Q3/Q4; ref angle 45°
Q3: θ=225°, Q4: θ=315°

Step5: Identify quadrants for cosθ=-√3/2

cosθ negative in Q2/Q3; ref angle 30°
Q2: θ=150°, Q3: θ=210°

Step6: Identify angles for tanθ=0

tanθ=0 when sinθ=0: θ=0°, 180°, 360°

Step7: Identify quadrants for sinθ=1/2

sinθ positive in Q1/Q2; ref angle 30°
Q1: θ=30°, Q2: θ=150°

Step8: Identify quadrants for tanθ=√3

tanθ positive in Q1/Q3; ref angle 60°
Q1: θ=60°, Q3: θ=240°

Step9: Identify quadrants for sinθ=-√3/2

sinθ negative in Q3/Q4; ref angle 60°
Q3: θ=240°, Q4: θ=300°

Step10: Identify quadrants for tanθ=1/√3

tanθ positive in Q1/Q3; ref angle 30°
Q1: θ=30°, Q3: θ=210°

Step11: Identify angles for cosθ=0

cosθ=0 when θ=90°, 270°

Step12: Identify angle for sinθ=-1

sinθ=-1 at θ=270°

Answer:

1. Q1/Q4, ref 60°, θ=60°/300°
2. Q2/Q4, ref 45°, θ=135°/315°
3. Q1/Q2, ref 60°, θ=60°/120°
4. Q3/Q4, ref 45°, θ=225°/315°
5. Q2/Q3, ref 30°, θ=150°/210°
6. Axis, ref 0°, θ=0°/180°/360°
7. Q1/Q2, ref 30°, θ=30°/150°
8. Q1/Q3, ref 60°, θ=60°/240°
9. Q3/Q4, ref 60°, θ=240°/300°
10. Q1/Q3, ref 30°, θ=30°/210°
11. Axis, ref 90°, θ=90°/270°
12. Axis, ref 90°, θ=270°