Question

Determine the exact measure of all angles that satisfy the following. Draw a diagram for each. a) \(\sin \theta=-\frac{1}{2}\) in the domain \(\mathbf{0} \leq \boldsymbol{\theta}<2 \pi\) b) \(\cot \theta=1\) in the domain \(-\pi \leq \theta<2 \pi\) c) \(\sec \theta=2\) in the domain \(-180^{\circ} \leq \theta<90^{\circ}\) d) \((\cos \theta)^{2}=1\) in the domain \(-360^{\circ} \leq \theta<360^{\circ}\)

Short answer

a) \( \theta = \frac{7\pi}{6}, \frac{11\pi}{6} \) b) \( \theta = \frac{\pi}{4}, \frac{5\pi}{4}, -\frac{7\pi}{4}, -\frac{3\pi}{4} \) c) \( \theta = 60^{\circ}, -300^{\circ} \) d) \( 0^{\circ}, 180^{\circ}, -180^{\circ}, 360^{\circ}, -360^{\circ} \)

Step-by-step solution

Understand the sine function

We need to find \(\theta\) such that \(\theta \) satisfies \(\sin \theta = -\frac{1}{2}\) in the domain \(\mathbf{0 \leq \boldsymbol{\theta} \< 2 \pi}\). The sine function is negative in the third and fourth quadrants.

Find reference angle for \( \sin \theta = -\frac{1}{2}\)

The reference angle for \( \sin^{-1} \big(\frac{1}{2}\big) \) is \(\frac{\pi}{6}\). Using symmetry properties of the sine function, the solutions in the given domain are: \(\theta = \pi + \frac{\pi}{6}\) and \(\theta = 2\pi - \frac{\pi}{6}\). Thus, \(\theta = \frac{7\pi}{6}\) and \(\theta = \frac{11\pi}{6}\).

Interpret cotangent function

We need to find \( \theta \) such that \(\cot \theta = 1\). Recall \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This implies \( \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{\pi}{4} - 2\pi, \frac{5\pi}{4} - 2\pi \) are solutions in the given domain.

Verify solutions for cotangent function

Thus, \(\theta = \frac{\pi}{4}, \frac{5\pi}{4}, -\frac{7\pi}{4}, -\frac{3\pi}{4}\). These are the angles that satisfy the condition.

Understand the secant function

We need to find \( \theta \) such that \(\sec \theta = 2\). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), indicating that \( \cos \theta = \frac{1}{2}\). We look for solutions in the given domain.

Find reference angle for \( \sec \theta = 2\)

The reference angle for \( \cos^{-1} \big(\frac{1}{2}\big) \) is \( \frac{\pi}{3} \) or \( \theta = 60^{\circ} \). In the given domain, the solutions are \( \theta = 60^{\circ}, -300^{\circ} \) (Recall \( \theta = -300^{\circ} \) because it repeats every 360 degrees).

Understand cosine squared function

We need to find \( \theta \) such that \( \cos^2 \theta = 1\). This implies \( \cos \theta = 1 \) or \( \cos \theta = -1 \).

Find all solutions for \( \cos^2 \theta = 1\)

This means \( \cos \theta = 1\) at \(\theta = 0, 2\pi, -2\pi, -4\pi \) and \( \cos \theta = -1\) at \(\pi, -\pi, 3\pi, -3\pi\). So, in the given domain \( -360^{\circ} \le \theta \< 360^{\circ} \), the solutions are \( 0^{\circ} , 180^{\circ} , -180^{\circ} , 360^{\circ} , -360^{\circ} \).

Key concepts

sine function

The sine function, represented as \(\sin\ \theta\)\, is a fundamental trigonometric function. It measures the y-coordinate of a point on the unit circle. When we solve for \(\sin\ \theta\ = -\frac{1}{2}\)\ in the interval from 0 to 2\pi (0 to 360 degrees), we're looking for all angles where the y-coordinate is -1/2. \ \Sine is negative in the third and fourth quadrants of the unit circle. This means our possible solutions are found at angles where the reference angle in each quadrant results in the sine value of -1/2. The reference angle for \(\sin^{-1}(\frac{1}{2})\) is \(\frac{\pi}{6}\) (or 30 degrees). Using this reference, we identify the exact angles in our specified domain: \(\theta = +\pi + \frac{\pi}{6} \)and \(\theta = 2\pi - \frac{\pi}{6} \), which simplifies to \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\)\.

cotangent function

The cotangent function is the reciprocal of the tangent function. It can be expressed as \(\cot\ \theta = \frac{\cos\ \theta}{\sin\ \theta}\), making it a measure of the ratio of the adjacent side to the opposite side in a right triangle. To solve for \(\cot\ \theta = 1\), we look for where this ratio equals 1. \ \Cotangent equals 1 at angles where the sine and cosine functions are equal. This happens at \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\), or their periodic equivalents in the specified domain \[-\pi \leq \theta \< 2\pi\]. Adding multiples of \(\pm2\pi\), we can find more values within the extended range provided. These values are \(\theta = \frac{\pi}{4}, \frac{5\pi}{4}, -\frac{7\pi}{4}\), and \(\frac{3\pi}{4}\)\.

secant function

The secant function is the reciprocal of the cosine function. Represented as \(\sec\ \theta = \frac{1}{\cos\ \theta}\), it measures how far the point on the unit circle is from the origin horizontally. To solve for \(\sec\ \theta = 2\), we need \(\cos\ \theta = \frac{1}{2}\)\. \ \The reference angle for \(\cos^{-1} (\frac{1}{2})\) is \(\frac{\pi}{3}\) or 60 degrees. Within the specified domain of \[-180° \leq \theta \< 90° \], these conditions give us solutions at \(\theta = 60°\) and \(\theta = -300°\) because of periodic repetition in the circular context of trigonometric functions\.\

cosine squared function

The function of cosine squared, denoted as \((\cos\ \theta)^{2}\)\, involves squaring the cosine of an angle. Solving for \((\cos \theta)^{2} = 1\)\, we're looking for points where the cosine value is either 1 or -1. \ \Cosine equals 1 and -1 at specific key angles. within the entire domain from \(-360° \leq \theta \< 360°\), we find that \(\cos\ \theta = 1\) at \(\theta = 0°, 2\pi, -2\pi, -4\pi\) and \(\cos\ \theta = -1\) at \(\theta = \pi, -\theta, 3\pi, -3\pi\)\. Therefore, within our prescribed domain, the solutions we have are \((0°, 180°, -180°, 360°, -360°)\).\