Questions: If θ = -3π/4, then sin(θ) = □ cos(θ) = □ Give exact values. No decimals allowed!

Solution Steps

The given angle is \(\theta = \frac{-3\pi}{4}\). To find the reference angle, we first convert the negative angle to a positive angle by adding \(2\pi\):

\[ \theta + 2\pi = \frac{-3\pi}{4} + \frac{8\pi}{4} = \frac{5\pi}{4} \]

The reference angle for \(\frac{5\pi}{4}\) is \(\pi - \frac{5\pi}{4} = \frac{\pi}{4}\).

The angle \(\frac{5\pi}{4}\) is in the third quadrant. In the third quadrant, the sine and cosine functions are both negative.

The sine of the reference angle \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\). Since \(\theta\) is in the third quadrant, \(\sin(\theta)\) is negative:

\[ \sin\left(\frac{-3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

The cosine of the reference angle \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\). Since \(\theta\) is in the third quadrant, \(\cos(\theta)\) is negative:

\[ \cos\left(\frac{-3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \]

Final Answer