Question

Find the exact value of each of the remaining trigonometric functions of \(\theta\). $$ \cot \theta=\frac{4}{3}, \quad \cos \theta<0 $$

Short answer

The remaining trigonometric functions of \(\theta\) are \sin \theta = \frac{3}{5} \quad \cos \theta = -\frac{4}{5}, \quad \tan \theta = \frac{3}{4}, \quad \sec \theta = -\frac{5}{4}, \quad \csc \theta = \frac{5}{3}. \

Step-by-step solution

Understand Given Information

Given \ \(\cot \theta = \frac{4}{3}\) and \(\cos \theta < 0\). This information tells us that \(\theta\) is in the second quadrant because \(\cot \theta = \frac{4}{3}\) is positive, and in the second quadrant, \(\cos \theta\) is negative.

Express \( \tan \theta \)

Since \(\cot \theta = \frac{1}{\tan \theta}\), we have: \tan \theta = \frac{1}{\cot \theta} = \frac{1}{\frac{4}{3}} = \frac{3}{4}

Find \( \text{sin} \theta \) and \( \text{cos} \theta \) using the Pythagorean Identity

We use the relationship \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). \ Let \( \sin \theta = 3k \) and \( \cos \theta = 4k \). Using Pythagorean Theorem: \( (\sin \theta)^2 + (\cos \theta)^2 = 1 \) \Substitute the expressions: \(3k)^2 + (4k)^2 = 1 \ 9k^2 + 16k^2 = 1 \25k^2 = 1 \k^2 = \frac{1}{25} \k = \pm \frac{1}{5} \Since \$\theta\$ is in the second quadrant, \ \sin \theta is positive and \ \cos \theta is negative: \sin \theta = \3\left(\frac{1}{5}\right) = \frac{3}{5}\ \cos \theta = \4\left(\frac{-1}{5}\) = -\frac{4}{5}\.

Calculate Remaining Trigonometric Functions

Now that we have \(\sin \theta\) and \(\cos \theta\), we can find the other trigonometric functions: \sin \theta = \frac{3}{5}, \quad \cos \theta = -\frac{4}{5}, \quad \tan \theta = \frac{3}{4} \cot \theta = \frac{4}{3}, \quad \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}, \quad \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}\

Key concepts

cotangent

Cotangent, denoted as \(\text{cot}\theta\), is one of the six primary trigonometric functions. It is the reciprocal of the tangent function, so \(\text{cot}\theta = \frac{1}{\text{tan}\theta}\). In classical trigonometry, it is defined in a right triangle where \(\theta\) is one of the angles. Specifically, cotangent is the ratio of the length of the adjacent side to the length of the opposite side.Given \(\text{cot}\theta = \frac{4}{3}\), we start by finding \(\text{tan}\theta\) using the reciprocal relationship, resulting in \(\text{tan}\theta = \frac{3}{4}\).

second quadrant

In the unit circle, the coordinate plane is divided into four quadrants. The second quadrant lies between 90° and 180°, or between \(\frac{\pi}{2}\) and \(\pi\) radians. In this quadrant, sine values are positive, while cosine and tangent values are negative. Since the cosine function is negative in the second quadrant and the given \(\text{cot}\theta = \frac{4}{3}\) is positive, it confirms \(\theta\) is in the second quadrant, making \(\text{sin}\theta\) positive and \(\text{cos}\theta\) negative.

Pythagorean identity

The Pythagorean identity is a fundamental relation in trigonometry derived from the Pythagorean theorem. It states \( \text{sin}^2\theta + \text{cos}^2\theta = 1 \). By using this identity, we can find the missing trigonometric functions once we have either sine or cosine. In this scenario, to find \(\text{sin}\theta\) and \(\text{cos}\theta\), we set \( \text{sin}\theta = 3k \) and \(\text{cos}\theta = 4k \), and solve for \((3k)^2 + (4k)^2 = 1\). That leads us to \( 25k^2 = 1 \), so \k = \frac{1}{5}\. Hence, \(\text{sin}\theta = \frac{3}{5}\) and \(\text{cos}\theta = -\frac{4}{5}\) based on the quadrant's sign conventions.

exact value calculation

For precise trigonometric values, it's essential to comprehend the relationships and identities amongst the functions. Given \(\text{cot}\theta = \frac{4}{3}\), the exact values are calculated using known identities and relationships: \(\text{tan}\theta = \frac{3}{4}\), \(\text{sin}\theta = \frac{3}{5}\), and \(\text{cos}\theta = -\frac{4}{5}\). Furthermore, we calculate \(\text{sec}\theta = \frac{1}{\text{cos}\theta} = -\frac{5}{4}\) and \(\text{csc}\theta = \frac{1}{\text{sin}\theta} = \frac{5}{3}\). These exact values are derived step-by-step to ensure precision and comprehension.