Question

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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. $\sin\theta=\frac{\sqrt{3}}{2}$

Explanation:

Step1: Recall cosine special angle

$\cos\theta = \frac{1}{2}$ at $\theta = 60^\circ$ (reference angle).

Step2: Identify quadrants for positive cosine

Cosine is positive in Q1/Q4. Q1: $\theta = 60^\circ$; Q4: $\theta = 360^\circ - 60^\circ = 300^\circ$.

Step3: Recall tangent special angle

$\tan\theta = -1$: reference angle is $45^\circ$ (since $\tan45^\circ=1$).

Step4: Identify quadrants for negative tangent

Tangent is negative in Q2/Q4. Q2: $\theta = 180^\circ - 45^\circ = 135^\circ$; Q4: $\theta = 360^\circ - 45^\circ = 315^\circ$.

Step5: Recall sine special angle

$\sin\theta = \frac{\sqrt{3}}{2}$ at $\theta = 60^\circ$ (reference angle).

Step6: Identify quadrants for positive sine

Sine is positive in Q1/Q2. Q1: $\theta = 60^\circ$; Q2: $\theta = 180^\circ - 60^\circ = 120^\circ$.

Answer:

1. $60^\circ, 300^\circ$
2. $135^\circ, 315^\circ$
3. $60^\circ, 120^\circ$