Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the given value and the trigonometric identities to find the remaining trigonometric functions of the angle. $$\sec \theta=-\frac{4}{3}, \cot \theta > 0$$

Step-by-Step Solution

Answer

The trigonometric functions based on the given exercise are: \( \sin \theta = -\frac{\sqrt{7}}{4}\), \( \cos \theta = -\frac{3}{4}\), \( \tan \theta = \frac{\sqrt{7}}{3}\), \( \csc \theta = -\frac{4}{\sqrt{7}}\), \( \sec \theta = -\frac{4}{3}\), and \( \cot \theta = \frac{3}{\sqrt{7}}\).

Step 1: Determine the Quadrant

Since \(\sec \theta=-\frac{4}{3}\), this means that \(\cos \theta\) is negative (because \(\sec \theta = \frac{1}{\cos \theta}\)). \(\cot \theta\) is positive and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), to make \(\cot \theta > 0\), \(\sin \theta\) must also be negative. Therefore, the angle \(\theta\) is in quadrant III, where both cosine and sine values are negative.

Step 2: Calculate Cosine

We have obtained that \(\sec(θ) = -\frac{4}{3}\). Thus, we can find \(\cos \theta\) as the reciprocal of \(\sec \theta\), so \(\cos \theta = -\frac{3}{4}\) .

Step 3: Calculate Sine

We can find \(\sin \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), solved for \(\sin \theta\). Substitute \(\cos \theta = -\frac{3}{4}\) to find \(\sin \theta = -\frac{\sqrt{1-\cos^2 \theta}}{1}\), which simplifies to \(\sin \theta = -\frac{\sqrt{1-\left(-\frac{3}{4}\right)^2}}{1} = -\frac{\sqrt{1-\left(\frac{9}{16}\right)}}{1} = -\frac{\sqrt{7}}{4}\).

Step 4: Calculate remaining trigonometric functions

Given \(\sin \theta = -\frac{\sqrt{7}}{4}\) and \(\cos \theta = -\frac{3}{4}\), we can calculate \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \left(-\frac{\sqrt{7}}{4}\right)\left(-\frac{4}{3}\right) = \frac{\sqrt{7}}{3}\) ; and \( \csc \theta = \frac{1}{\sin \theta} = -\frac{4}{\sqrt{7}}\) . We already have \( \sec \theta = -\frac{4}{3}\) and \( \cot \theta > 0 = \frac{\cos \theta}{\sin \theta} = \frac{-3/4}{-\sqrt{7}/4} = \frac{3}{\sqrt{7}}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical expressions equating one trigonometric function to another. These identities are crucial for simplifying and transforming trigonometric expressions.
In this exercise, we're using the identity that relates cosine and secant: \(\sec \theta = \frac{1}{\cos \theta}\). This shows that secant is the reciprocal of cosine.
Another identity used here is the Pythagorean identity, which will be explained later. Understanding these identities helps us derive unknown trigonometric values from known quantities.

Quadrants in Trigonometry

The unit circle is divided into four quadrants. Each quadrant corresponds to a specific set of signs for trigonometric functions:

• Quadrant I: All functions are positive.
• Quadrant II: Sine is positive; cosine and tangent are negative.
• Quadrant III: Tangent is positive; sine and cosine are negative.
• Quadrant IV: Cosine is positive; sine and tangent are negative.

For this problem, since \(\sec \theta = -\frac{4}{3}\), cosine is negative, indicating that the angle \(\theta\) is not in Quadrants I or IV. Given that \(\cot \theta > 0\), and knowing \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), both sine and cosine must be negative for cotangent to remain positive. This information places \(\theta\) in Quadrant III.

Pythagorean Identity

One of the fundamental trigonometric identities is the Pythagorean identity. It states:
\[\sin^2 \theta + \cos^2 \theta = 1\]
This identity is derived from the Pythagorean Theorem applied to a right triangle inscribed in the unit circle.
In the solution, we use this identity to find \(\sin \theta\). With \(\cos \theta = -\frac{3}{4}\), we substitute and solve \(\sin^2 \theta = 1 - \cos^2 \theta\). This simplifies to \(\sin^2 \theta = 1 - \left(-\frac{3}{4}\right)^2\), resulting in \(\sin \theta = -\frac{\sqrt{7}}{4}\). The sign is determined based on the quadrant.

Trigonometric Values Calculation

Once we have \(\cos \theta\) and \(\sin \theta\), calculating the remaining trigonometric functions becomes straightforward.

• \(\tan \theta\) is found using \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
• \(\csc \theta\) is the reciprocal of sine: \(\csc \theta = \frac{1}{\sin \theta}\).
• \(\cot \theta\) is the reciprocal of tangent or can be calculated using \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Given \(\sin \theta = -\frac{\sqrt{7}}{4}\) and \(\cos \theta = -\frac{3}{4}\), we compute:
- \(\tan \theta = \frac{\sqrt{7}}{3}\).
- \(\csc \theta = -\frac{4}{\sqrt{7}}\).
- \(\cot \theta = \frac{3}{\sqrt{7}}\).
These calculations demonstrate the interconnected nature of trigonometric functions.