6. The point $Q (- 3, 6)$ is on the terminal arm of an angle, $θ$ .
a) Draw this angle in standard position.
b) Determine the exact distance from the origin to point $Q$ .
c) Determine the exact values for $\sin θ$ , $\cos θ$ , and $\tan θ$ .
d) Determine the value of $θ$ .

Answer

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Answer

$θ = 180^{\circ} - \arctan (2)$

Steps

Step 1 ： $Q (- 3, 6)$ is on the terminal arm of angle $θ$ in standard position.

Step 2 ：Calculate the distance from the origin to point $Q$ using the distance formula: $d = \sqrt{(- 3)^{2} + 6^{2}}$

Step 3 ： $d = \sqrt{9 + 36} = \sqrt{45} = 3 \sqrt{5}$

Step 4 ： $\sin θ = \frac{6}{3 \sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}$

Step 5 ： $\cos θ = \frac{- 3}{3 \sqrt{5}} = \frac{- 1}{\sqrt{5}} = \frac{- \sqrt{5}}{5}$

Step 6 ： $\tan θ = \frac{\sin θ}{\cos θ} = \frac{\frac{2 \sqrt{5}}{5}}{\frac{- \sqrt{5}}{5}} = - 2$

Step 7 ： $θ = \arctan (- 2)$ in Quadrant II

Step 8 ： $θ = 180^{\circ} - \arctan (2)$