Problem 22 - Precalculus with Limits

Question

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \csc \theta = -5 \), \( \cos \theta < 0 \)

Step-by-Step Solution

Answer

The trigonometric functions evaluated are \(\sin \theta = -1/5\), \(\cos \theta = -24/5\), \(\tan \theta = 1/24\), \(\cot \theta = 24\), \(\sec \theta = -5/24\), and \(\csc \theta = -5\).

Step 1: Calculate \(\sin \theta\)

The sin of an angle is defined as the reciprocal of its cosecant. Thus \(\sin \theta = 1/ \csc \theta = -1/5\) because \(\csc \theta = -5\).

Step 2: Calculate \(\cos \theta\)

Using the Pythagorean Identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can solve for \(\cos \theta\): Subtract \(\sin^2 \theta\) from both sides: \(\cos ^2 \theta = 1 - (-1/5)^2 = 1 - 1/25 = 24/25\). Now take the square root of both sides. There are two possible results, \(\pm \sqrt{24/25}\). However, based on the condition \(\cos \theta < 0\), choose \(- \sqrt{24/25}\) so \(\cos \theta = -24/5\).

Step 3: Calculate \(\tan \theta\)

The tangent of an angle is the sin divided by the cosine. So \(\tan \theta = \sin \theta / \cos \theta = (-1/5) / (- 24/5) = 1/24\).

Step 4: Calculate \(\cot \theta\)

The cotangent of an angle is the reciprocal of its tangent. So \(\cot \theta = 1/ \tan \theta = 24\).

Step 5: Calculate \(\sec \theta\)

The secant of an angle is defined as the reciprocal of its cosine so \(\sec \theta = 1/ \cos \theta = -5/24\).

Key Concepts

Cosecant

The cosecant function is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. That means if you have the sine of an angle, you can find the cosecant by taking the reciprocal of the sine value. For instance, given that \( \csc \theta = -5 \), this means \( \sin \theta = \frac{1}{\csc \theta} = -\frac{1}{5} \).
This concept is especially useful in solving trigonometric problems, where if you know one function value, you can find the related function value of an angle through reciprocal identities. The sine and cosecant functions help simplify complex trigonometric calculations and find missing components.
When using the cosecant, remember it is undefined for angles where the sine is zero, specifically at multiples of \(180^\circ\).

Pythagorean Identity

The Pythagorean identity is a critical tool in trigonometry. It states that for any angle \(\theta\), the square of the sine plus the square of the cosine equals one: \( \sin^2 \theta + \cos^2 \theta = 1 \).
This identity helps find unknown trigonometric function values when one value is known. In our example, after calculating \( \sin \theta = -\frac{1}{5} \), you can find \( \cos \theta \) using this identity:

• First, square the sine: \( (-\frac{1}{5})^2 = \frac{1}{25} \).
• Then subtract from one: \( 1 - \frac{1}{25} = \frac{24}{25} \).
• The remaining value is \( \cos^2 \theta \).
• Since \( \cos \theta < 0 \), \( \cos \theta = -\sqrt{\frac{24}{25}} \).
• Simplify to \( \cos \theta = -\frac{\sqrt{24}}{5} \).

Understanding and using this identity will make solving trigonometric equations more manageble and systematic.

Reciprocal Identities

Reciprocal identities establish relationships between pairs of trigonometric functions. The identities are fundamental for simplifying expressions and solving equations:

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

For example, in the exercise, we've used these identities to solve for \( \sin \theta \) when given \( \csc \theta \). Similarly, knowing \( \cos \theta \), we can find \( \sec \theta \).
This process of flipping the functions is vital in rearranging equations and solving for unknowns. When practicing, ensure you’re comfortable and familiar with these interchangeable expressions to easily navigate problems.

Tangent and Cotangent

Tangent and cotangent are two connected trigonometric functions defined in terms of sine and cosine. The tangent of an angle is the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). In our problem, it simplifies to \( \tan \theta = \frac{-1/5}{-24/5} = \frac{1}{24} \).

The cotangent is the reciprocal of the tangent, hence \( \cot \theta = \frac{1}{\tan \theta} \). For the problem, \( \cot \theta = 24 \). These functions are crucial when analyzing angle measures and their relationships. By knowing \( \tan \theta \), you can automatically determine \( \cot \theta \) and vice versa, supporting the calculation of other trigonometric values.

Secant

The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. It is calculated as \( \sec \theta = \frac{1}{\cos \theta} \). In our example, where \( \cos \theta = -\frac{\sqrt{24}}{5} \), the secant becomes \( \sec \theta = -\frac{5}{\sqrt{24}} \).
Using secant can simplify expressions involving trigonometric functions. It also helps you solve problems that ask for multiple function values like in the exercise. Being familiar with secant ensures you can handle questions involving all six trigonometric functions with ease.
Always remember, like the cosecant, the secant is undefined for angles where its reciprocal, cosine, equals zero.