Question

1. for the point p(3, -3), state the exact primary trigonometric ratios associated with ∠θ. find the measure of ∠θ.

Explanation:

Step1: Identify x, y, calculate r

Given point \( P(3, -3) \), so \( x = 3 \), \( y = -3 \).
\( r = \sqrt{x^2 + y^2} = \sqrt{3^2 + (-3)^2} = \sqrt{18} = 3\sqrt{2} \).

Step2: Compute trigonometric ratios

\( \sin\theta = \frac{y}{r} = \frac{-3}{3\sqrt{2}} = -\frac{\sqrt{2}}{2} \)
\( \cos\theta = \frac{x}{r} = \frac{3}{3\sqrt{2}} = \frac{\sqrt{2}}{2} \)
\( \tan\theta = \frac{y}{x} = \frac{-3}{3} = -1 \)

Step3: Find measure of \( \angle\theta \)

Since \( \cos\theta > 0 \) and \( \sin\theta < 0 \), \( \theta \) is in Quadrant IV.
\( \theta = 360^\circ - 45^\circ = 315^\circ \) (or \( \frac{7\pi}{4} \) radians).

Answer:

Trigonometric ratios: \( \sin\theta = -\frac{\sqrt{2}}{2} \), \( \cos\theta = \frac{\sqrt{2}}{2} \), \( \tan\theta = -1 \);
Measure of \( \angle\theta \): \( 315^\circ \) (or \( \frac{7\pi}{4} \) radians)