7) Find the angle \( \theta \) closest to \( 1000^{\circ} \) such that \( \sin \theta=-\frac{1}{2} \). Round to the nearest degree If \( \sin \theta=0.7431 \) and \( \cos \theta=-0.6691 \), find the measure of \( \angle \theta \). Round to the nearest

To find the angle \( \theta \) closest to \( 1000^{\circ} \) such that \( \sin \theta = -\frac{1}{2} \), we first need to determine the reference angles where the sine function is negative. The sine function is negative in the third and fourth quadrants. The reference angle for \( \sin \theta = \frac{1}{2} \) is \( 30^{\circ} \). Therefore, the angles in the third and fourth quadrants where \( \sin \theta = -\frac{1}{2} \) are: - In the third quadrant: \[ \theta = 180^{\circ} + 30^{\circ} = 210^{\circ} \] - In the fourth quadrant: \[ \theta = 360^{\circ} - 30^{\circ} = 330^{\circ} \] Now, we can find the angles that are equivalent to \( 210^{\circ} \) and \( 330^{\circ} \) by adding multiples of \( 360^{\circ} \) to these angles to get close to \( 1000^{\circ} \). Calculating for \( 210^{\circ} \): \[ 210^{\circ} + 360^{\circ} \times n \quad \text{for } n = 0, 1, 2, \ldots \] - For \( n = 0 \): \( 210^{\circ} \) - For \( n = 1 \): \( 210^{\circ} + 360^{\circ} = 570^{\circ} \) - For \( n = 2 \): \( 210^{\circ} + 720^{\circ} = 930^{\circ} \) - For \( n = 3 \): \( 210^{\circ} + 1080^{\circ} = 1290^{\circ} \) Calculating for \( 330^{\circ} \): \[ 330^{\circ} + 360^{\circ} \times n \quad \text{for } n = 0, 1, 2, \ldots \] - For \( n = 0 \): \( 330^{\circ} \) - For \( n = 1 \): \( 330^{\circ} + 360^{\circ} = 690^{\circ} \) - For \( n = 2 \): \( 330^{\circ} + 720^{\circ} = 1050^{\circ} \) - For \( n = 3 \): \( 330^{\circ} + 1080^{\circ} = 1410^{\circ} \) Now we compare these angles to find the one closest to \( 1000^{\circ} \): - \( 930^{\circ} \) (from \( 210^{\circ} \)) is \( 70^{\circ} \) away from \( 1000^{\circ} \). - \( 1050^{\circ} \) (from \( 330^{\circ} \)) is \( 50^{\circ} \) away from \( 1000^{\circ} \). Thus, the angle closest to \( 1000^{\circ} \) such that \( \sin \theta = -\frac{1}{2} \) is: \[ \theta = 1050^{\circ} \] --- Next, we need to find the measure of \( \angle \theta \) given \( \sin \theta = 0.7431 \) and \( \cos \theta = -0.6691 \). To find \( \theta \), we can use the inverse tangent function to find the angle in the correct quadrant. Since \( \sin \theta > 0 \) and \( \cos \theta < 0 \), \( \theta \) is in the second quadrant. First, we can find the reference angle using the sine function: \[ \theta = \arcsin(0.7431) \] Calculating this gives: \[ \theta \approx 48.5^{\circ} \] Since we are in the second quadrant, we find the actual angle: \[ \theta = 180^{\circ} - 48.5^{\circ} \approx 131.5^{\circ} \] Rounding to the nearest degree, we have: \[ \theta \approx 132^{\circ} \] Thus, the measure of \( \angle \theta \) is: \[ \theta \approx 132^{\circ} \]