Question

Find the exact value of each of the remaining trigonometric functions of \(\theta\). $$ \cot \theta=-2, \quad \sec \theta>0 $$

Short answer

The exact values are: \(\sin \theta = \frac{1}{\sqrt{5}}, \cos \theta = -\frac{2}{\sqrt{5}}, \tan \theta = -\frac{1}{2}, \sec \theta = -\frac{\sqrt{5}}{2}, \csc \theta = \sqrt{5}\)

Step-by-step solution

- Define the given information

Given that \( \cot \theta = -2 \) and \( \sec \theta > 0 \). \(\cot \theta\) is the ratio of cosine to sine, which means \(\frac{\cos \theta}{\sin \theta} = -2\). Since \(\sec \theta\) is positive, it indicates that \(\cos \theta\) is also positive.

- Use Pythagorean identity

Recall \( \cot \theta = \frac{\cos \theta}{\sin \theta} = -2 \). This means \(\cos \theta = -2 \sin \theta.\) Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to express this in terms of sine and cosine.

- Substitute \( \cos \theta = -2 \sin \theta\)

Substituting into the identity: \[ \sin^2 \theta + (-2 \sin \theta)^2 = 1 \] This simplifies to: \[ \sin^2 \theta + 4 \sin^2 \theta = 1 \]\ \[5 \sin^2 \theta = 1 \] Solving for \(\sin \theta\) gives: \[ \sin \theta = \pm \frac{1}{\sqrt{5}} \]

- Determine the correct sign of \( \sin \theta \) and \( \cos \theta \)

Since \( \sec \theta > 0 \) and \( \cot \theta < 0 \), \( \sin \theta \) must be positive and \( \cos \theta \) is negative. Thus, \(\sin \theta = \frac{1}{\sqrt{5}}\) and \(\cos \theta = -2 \sin \theta = -2 \cdot \frac{1}{\sqrt{5}} = -\frac{2}{\sqrt{5}}\)

- Calculate remaining trigonometric functions

Now that \(\sin \theta\) and \(\cos \theta\) are known, find the remaining trig functions: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{\sqrt{5}}}{-\frac{2}{\sqrt{5}}} = -\frac{1}{2} \] \[ \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{5}}{-2} = -\frac{\sqrt{5}}{2} \] \[ \csc \theta = \frac{1}{\sin \theta} = \sqrt{5} \] \[ \cot \theta \text{ is already given as } -2.\]

Key concepts

Cotangent

The cotangent function, denoted as \(\text{cot} \, \theta\), is defined as the ratio of the cosine and sine of an angle. Mathematically, this is written as: \[ \cot \, \theta = \frac{\cos \, \theta}{\sin \, \theta} \] In our problem, we have \(\text{cot} \, \theta = -2\). This means the ratio of cosine to sine is -2. If we let \(\text{sin} \, \theta\) be \(\frac{1}{\root{5}}\), then \(\text{cos} \, \theta\) must be \(-2 \times \frac{1}{\root{5}} = \- \frac{2}{\root{5}}\). It's crucial to consider the positive or negative signs based on the quadrant in which \(\theta\) lies.

Secant

The secant function, denoted as \(\text{sec} \theta\), is the reciprocal of the cosine function. This can be expressed as: \[ \sec \theta = \frac{1}{\cos \theta} \] In our problem, we're given that \(\text{sec} \theta > 0\), which indicates \(\text{cos} \theta\) must be positive. However, it can also be negative if it does not change the sign of \(\text{sec} \theta\). Once we determined \(\text{cos} \theta = -\frac{2}{\root{5}}\), we computed \(\text{sec} \theta\) as follows: \[ \sec \theta = \frac{1}{-\frac{2}{\root{5}}} = -\frac{\root{5}}{2} \].

Pythagorean identity

The Pythagorean Identity is one of the fundamental identities in trigonometry. It states that for any angle \(\theta\), the squared sine and cosine functions add up to 1: \[ \sin^2 \theta + \cos^2 \theta = 1\text{.} \] This identity was used in the problem when we substituted \(\text{cos} \theta = -2 \text{sin} \theta\) into the equation, giving: \[ \sin^2 \theta + 4 \text{sin}^2 \theta = 1 \] Simplifying, we find: \[ 5 \text{sin}^2 \theta = 1 \] Thus, \(\text{sin} \theta = \pm \frac{1}{\root{5}}\) and because \(\text{sec} \theta > 0 \text{ and} \text{cot} \theta < 0\), \(\text{sin} \theta\) must be positive.

Sine

The sine function, represented as \(\text{sin} \theta\), gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. In this problem, we found that: \[ \sin \theta = \frac{1}{\root{5}} \text{ or } \- \frac{1}{\root{5}} \] Based on the signs provided by \(\text{cot} \theta\) and \(\text{sec} \theta\), the correct value is \(\text{sin} \theta = \frac{1}{\root{5}}\). This tells us the ratio of the opposite side to the hypotenuse for the angle \(\theta\).

Cosine

The cosine function, denoted as \(\text{cos} \theta\), is the ratio of the adjacent side of a right triangle to its hypotenuse. Since \(\text{cot} \theta= -2 \) and \(\text{cos} \theta\) is negative in our problem, we found that: \[ \cos \theta = -2 \times \frac{1}{\root{5}} = -\frac{2}{\root{5}} \] This means the adjacent side of the triangle is twice as large as the opposite side but in the negative direction. Understanding the \(\text{cosine} \theta\)'s negative result helps identify the proper quadrant for \(\theta\).