Problem 91 - Algebra and Trigonometry with Analytic Geometry

Question

Use fundamental identities to find the values of the trigonometric functions for the given conditions. \(\sec \theta=-4\) and \(\csc \theta>0\)

Step-by-Step Solution

Answer

\(\sin \theta = \frac{\sqrt{15}}{4}, \cos \theta = -\frac{1}{4}, \tan \theta = -\sqrt{15}, \csc \theta = \frac{4\sqrt{15}}{15}, \sec \theta = -4, \cot \theta = -\frac{\sqrt{15}}{15}\).

Step 1: Convert Secant to Cosine

Since \(\sec \theta = -4\), we know that \(\cos \theta = \frac{1}{\sec \theta} = \frac{1}{-4} = -\frac{1}{4}\). This tells us \(\theta\) is in the second quadrant given \(\csc \theta > 0\).

Step 2: Identify Sine Using Pythagorean Identity

Use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = -\frac{1}{4}\):\[\sin^2 \theta + \left(-\frac{1}{4}\right)^2 = 1\]\[\sin^2 \theta + \frac{1}{16} = 1\]\[\sin^2 \theta = 1 - \frac{1}{16}\]\[\sin^2 \theta = \frac{15}{16}\]As \(\csc \theta > 0\), \(\sin \theta\) must be positive, thus \(\sin \theta = \frac{\sqrt{15}}{4}\).

Step 3: Calculate Cosecant

Cosecant is the reciprocal of sine:\[\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{15}}{4}} = \frac{4}{\sqrt{15}}\]Rationalize the denominator:\[\csc \theta = \frac{4\sqrt{15}}{15}\]

Step 4: Calculate Tangent Using Sine and Cosine

Tangent is sine divided by cosine:\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{15}}{4}}{-\frac{1}{4}} = -\sqrt{15}\]

Step 5: Calculate Cotangent as Reciprocal of Tangent

Since \(\tan \theta = -\sqrt{15}\), we find \(\cot \theta\) as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\sqrt{15}} = -\frac{\sqrt{15}}{15}\]

Step 6: Calculate Sine From Cosecant

Since \(\csc \theta = \frac{4\sqrt{15}}{15}\), verify \(\sin \theta\):\[\sin \theta = \frac{1}{\csc \theta} = \frac{15}{4\sqrt{15}} = \frac{\sqrt{15}}{4}\]This confirms our sine calculation is correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in trigonometry and are used to relate angles to side lengths in right-angled triangles. In this exercise, the following trigonometric functions are involved:

• Sine ( \( \sin \theta \)
• Cosine ( \( \cos \theta \)
• Secant ( \( \sec \theta \)
• Cosecant ( \( \csc \theta \)
• Tangent ( \( \tan \theta \)
• Cotangent ( \( \cot \theta \)

Conversion between these functions often involves reciprocal or quotient relationships, making identities crucial for calculations. For example, the secant of an angle is the reciprocal of its cosine, and the cosecant is the reciprocal of the sine. Understanding these relationships allows for solving complex trigonometric expressions and tackling various problems across the fields of mathematics and physics.

By converting secant to cosine and identifying related sine values, this exercise demonstrates the application of these essential functions.

Pythagorean Identity

The Pythagorean identity is a vital piece of trigonometry, providing the relationship between the square of sine and cosine:\[\sin^2 \theta + \cos^2 \theta = 1\]This identity is derived from the Pythagorean theorem and applies to any angle \(\theta\). It forms the foundation for solving trigonometric expressions involving sine and cosine. In the exercise, given that \( \cos \theta = - \frac{1}{4} \), this identity allows us to find \( \sin \theta \).

By substituting the value of \( \cos \theta \) into our identity, we can solve for \( \sin^2 \theta \), and thus \( \sin \theta \):

• Find \( \sin^2 \theta = 1 - \frac{1}{16} = \frac{15}{16} \)
• Since \( \csc \theta > 0 \), \( \sin \theta = \frac{\sqrt{15}}{4} \)

This step enables further calculations for other trigonometric functions, highlighting how interconnected they are.

Reciprocal Identities

Reciprocal identities are crucial in trigonometry for transforming between different trigonometric functions. These identities take the form:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

Understanding these identities assists in deriving other function values from known ones. In the exercise, we use these identities:

• To find cosine from secant: \( \cos \theta = -\frac{1}{4} \)
• Calculate cosecant from sine: \[ \csc \theta = \frac{4\sqrt{15}}{15} \]
• Derive cotangent from tangent: \[ \cot \theta = -\frac{\sqrt{15}}{15} \]

These identities simplify the process, linking different parts of trigonometric relationships, allowing smooth transitions from one function to another.

Trigonometric Quadrants

Trigonometric quadrants divide the coordinate plane into four sections, helping to determine the sign of trigonometric functions based on the angle's position. These quadrants are as follows:

• Quadrant I: All trigonometric functions positive.
• Quadrant II: Sine and cosecant positive, cosine and secant negative.
• Quadrant III: Tangent and cotangent positive, others negative.
• Quadrant IV: Cosine and secant positive, others negative.

In the exercise, \( \sec \theta = -4 \) and \( \csc \theta > 0 \) indicates \( \theta \) is in Quadrant II. This tells us:

• Cosine is negative.
• Sine is positive.

Having the angle in Quadrant II determines the signs for corresponding trigonometric functions, which is essential for correct calculations. Grasping the concept of trigonometric quadrants is vital for students to solve problems involving angle-based evaluations correctly, ensuring accurate function determination.