Problem 90 - A Graphical Approach to Precalculus with Limits

Question

For each of the following, find tan \(s\), cot \(s\), sec \(s\), and csc \(s\). Do not use a calculator. $$\sin s=\frac{12}{13}, \cos s=\frac{5}{13}$$

Step-by-Step Solution

Answer

\(\tan s = \frac{12}{5}, \cot s = \frac{5}{12}, \sec s = \frac{13}{5}, \csc s = \frac{13}{12}\).

Step 1: Recall Trigonometric Identities

The trigonometric functions can be defined in terms of sine and cosine:- \( \tan s = \frac{\sin s}{\cos s} \)- \( \cot s = \frac{1}{\tan s} = \frac{\cos s}{\sin s} \)- \( \sec s = \frac{1}{\cos s} \)- \( \csc s = \frac{1}{\sin s} \)

Step 2: Calculate \( \tan s \)

Using the values \( \sin s = \frac{12}{13} \) and \( \cos s = \frac{5}{13} \), we find:\[\tan s = \frac{\sin s}{\cos s} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}\]

Step 3: Calculate \( \cot s \)

By using the reciprocal identity of tangent:\[\cot s = \frac{1}{\tan s} = \frac{5}{12}\]

Step 4: Calculate \( \sec s \)

The secant function is the reciprocal of cosine:\[\sec s = \frac{1}{\cos s} = \frac{13}{5}\]

Step 5: Calculate \( \csc s \)

The cosecant function is the reciprocal of sine:\[\csc s = \frac{1}{\sin s} = \frac{13}{12}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. These identities help in simplifying expressions and solving problems in trigonometry. Some of the most common trigonometric identities include:

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Quotient Identities: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and \( \cot\theta = \frac{\cos\theta}{\sin\theta} \)
• Reciprocal Identities: \( \sec\theta = \frac{1}{\cos\theta} \) and \( \csc\theta = \frac{1}{\sin\theta} \)

These identities are fundamental tools in trigonometry, enabling us to express and manipulate functions in various forms. For the given exercise, using these identities allows you to determine the values of tangent, secant, and cosecant. Understanding these relationships is crucial when dealing with more complex trigonometric problems.

Tangent

The tangent function, denoted as \( \tan \) in trigonometry, represents the ratio of the sine of an angle to the cosine of the same angle. Mathematically, this is expressed as:\[\tan s = \frac{\sin s}{\cos s}\]In the given exercise with \( \sin s = \frac{12}{13} \) and \( \cos s = \frac{5}{13} \), calculating the tangent is straightforward:\[\tan s = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}\]This result signifies how steep the angle \( s \) rises from the baseline in a right triangle context. The tangent function is especially useful in defining the slope of a line and in problems involving rate of change.

Secant

The secant function, symbolized as \( \sec \), is related to the cosine function as its reciprocal. For any angle \( \theta \), the secant can be defined as:\[\sec \theta = \frac{1}{\cos \theta}\]For the exercise, with \( \cos s = \frac{5}{13} \), the secant is found by taking the reciprocal:\[\sec s = \frac{1}{\cos s} = \frac{13}{5}\]The secant function is important when examining lengths in triangles, particularly in situations involving the hypotenuse relative to the adjacent side. Understanding secant helps in solving problems that require finding distances or angles when traditional methods are cumbersome.

Cosecant

Cosecant, noted as \( \csc \), is the reciprocal function of sine. Mathematically, for any angle \( \theta \), it is given by:\[\csc \theta = \frac{1}{\sin \theta}\]In the exercise, with \( \sin s = \frac{12}{13} \), finding cosecant involves:\[\csc s = \frac{1}{\sin s} = \frac{13}{12}\]The cosecant function is particularly useful in problems involving the radius or hypotenuse of a circle or right triangle. Using cosecant can simplify calculations when the direct path is cumbersome. It is also pivotal in solving equations and simplifying expressions in advanced trigonometric problems.