Question

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

Short answer

\sin \theta = -\frac{1}{2}, \ \cos \theta = -\frac{\sqrt{3}}{2}, \ \tan \theta = \frac{\sqrt{3}}{3}, \ \sec \theta = -\frac{2\sqrt{3}}{3}, \ \cot \theta = \sqrt{3}.

Step-by-step solution

- Identify Known Values

Given that \(\csc \theta = -2\), we can derive \(\sin \theta\) since \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, \(\sin \theta = -\frac{1}{2}\).

- Determine the Quadrant

Given that \(\sin \theta<0\) and \(\tan \theta>0\), \(\theta\) must be in the third quadrant because both conditions are satisfied only in this quadrant.

- Identify \(\cos \theta\)

To find \(\cos \theta\), use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\sin \theta = -\frac{1}{2}\): \((-\frac{1}{2})^2 + \cos^2 \theta = 1\), which simplifies to \(\frac{1}{4} + \cos^2 \theta = 1\) and \(\cos^2 \theta = \frac{3}{4}\). Since \(\theta\) is in the third quadrant, \(\cos \theta = -\frac{\sqrt{3}}{2}\).

- Calculate Remaining Trigonometric Functions

Calculate \(\tan \theta\) as the ratio \(\frac{\sin \theta}{\cos \theta} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\). Next, \(\sec \theta\) as the reciprocal of \(\cos \theta\) is \(\sec \theta = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\), and \(\cot \theta\) as the reciprocal of \(\tan \theta\) is \(\cot \theta = \sqrt{3}\).

Key concepts

Cosecant

The cosecant function, written as \(\text{csc} \theta\), is the reciprocal of the sine function. This means that \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). In our given problem, \(\text{csc} \theta = -2\). To find \(\text{sin} \theta\), we take the reciprocal: \(\text{sin} \theta = \frac{1}{\text{csc} \theta} = \frac{1}{-2} = -\frac{1}{2}\). Understanding this reciprocal relationship is fundamental in solving trigonometric problems.

Sine

The sine function, often written as \(\text{sin} \theta\), represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. From the given problem, we have \(\text{sin} \theta = -\frac{1}{2}\). This information tells us more about the angle \(\theta\) since sine is negative in certain quadrants. We'll use this relationship to delve further into identifying the quadrant and other trigonometric functions related to \(\theta\).

Tangent

The tangent function, denoted as \(\text{tan} \theta\), is the ratio of the sine of an angle to the cosine of the same angle. Mathematically, \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\). Given that \(\text{sin} \theta = -\frac{1}{2}\) and using the Pythagorean identity to find \(\text{cos} \theta\), we ultimately find that \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} = \frac{-\frac{1}{2}}{-\frac{\frac{\text{sqrt}(3)}}{2}} = \frac{1}{\text{sqrt}(3)} = \frac{\text{sqrt}(3)}{3}\). Knowing the value of tangent allows us to explore more about the properties of this angle.

Cosine

Cosine, represented as \(\text{cos} \theta\), is the ratio of the adjacent side to the hypotenuse in a right-angled triangle. We use the Pythagorean identity: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). Plugging in \(\text{sin} \theta = -\frac{1}{2}\), we get: \(\text{cos}^2 \theta = 1 - \text{sin}^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}\). Since \(\theta\) is in the third quadrant, where cosine values are negative, \(\text{cos} \theta = -\frac{\text{sqrt}(3)}{2}\). This calculation is key in understanding the cosine function.

Quadrant Identification

Identifying the quadrant in which an angle lies is crucial in trigonometry since the signs of trigonometric functions depend on the quadrant. From the given conditions \(\text{sin} \theta < 0\) and \(\text{tan} \theta > 0\), we can deduce that the angle \(\theta\) must be in the third quadrant. In this quadrant, both sine and cosine are negative, making the tangent positive because \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\). Understanding quadrant identification helps us determine the correct signs for trigonometric functions.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry given by \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). It relates the squares of the sine and cosine of an angle to 1. In our problem, knowing \(\text{sin} \theta = -\frac{1}{2}\) allowed us to find \(\text{cos} \theta\) through this identity: \(\text{cos}^2 \theta = 1 - \text{sin}^2 \theta = 1 - \frac{1}{4} = \frac{3}{4}\), thus \(\text{cos} \theta = -\frac{\text{sqrt}(3)}{2}\). Mastering this identity is essential for solving many trigonometric problems.