Problem 86 - 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. $$\sin s=\frac{3}{4}, \cos s=\frac{\sqrt{7}}{4}$$

Step-by-Step Solution

Answer

\( \tan s = \frac{3\sqrt{7}}{7} \), \( \cot s = \frac{\sqrt{7}}{3} \), \( \sec s = \frac{4\sqrt{7}}{7} \), \( \csc s = \frac{4}{3} \).

Step 1: Understand the Given Values

We are given \( \sin s = \frac{3}{4} \) and \( \cos s = \frac{\sqrt{7}}{4} \). These values represent the sine and cosine of the angle \( s \). Let's use these to find other trigonometric functions: \( \tan s \), \( \cot s \), \( \sec s \), and \( \csc s \).

Step 2: Calculate Tangent

The tangent of an angle \( s \) is the ratio of the sine to the cosine: \( \tan s = \frac{\sin s}{\cos s} \). Substitute the given values: \[ \tan s = \frac{3/4}{\sqrt{7}/4} = \frac{3}{\sqrt{7}}.\] To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{7} \): \[ \tan s = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{3\sqrt{7}}{7}.\]

Step 3: Calculate Cotangent

The cotangent is the reciprocal of tangent: \( \cot s = \frac{1}{\tan s} \). So, \[ \cot s = \frac{1}{\frac{3\sqrt{7}}{7}} = \frac{7}{3\sqrt{7}} = \frac{7\sqrt{7}}{21}.\] Simplify this to \( \cot s = \frac{\sqrt{7}}{3} \).

Step 4: Calculate Secant

The secant is the reciprocal of cosine: \( \sec s = \frac{1}{\cos s} \). Substitute the given value: \[ \sec s = \frac{1}{\sqrt{7}/4} = \frac{4}{\sqrt{7}}.\] Rationalize the denominator: \[ \sec s = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} = \frac{4\sqrt{7}}{7}.\]

Step 5: Calculate Cosecant

The cosecant is the reciprocal of sine: \( \csc s = \frac{1}{\sin s} \). Substitute the given value: \[ \csc s = \frac{1}{3/4} = \frac{4}{3}.\]

Key Concepts

Tangent

The tangent of an angle, often abbreviated as "tan," is one of the fundamental trigonometric functions. It's defined as the ratio of the sine and cosine of an angle. Specifically, for an angle \( s \), the tangent is given by:

• \[ \tan s = \frac{\sin s}{\cos s} \]

In our given problem, \( \sin s = \frac{3}{4} \) and \( \cos s = \frac{\sqrt{7}}{4} \). Using the tangent formula, we get:

• \[ \tan s = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} \]

To make this expression more conventional in mathematics, we rationalize the denominator by multiplying both the numerator and denominator by \( \sqrt{7} \), resulting in:

• \[ \tan s = \frac{3\sqrt{7}}{7} \]

Rationalizing the denominator is a common practice because it eliminates any square roots or irrational numbers from the denominator, making it simpler and more acceptable for further mathematical operations.

Cotangent

Cotangent, abbreviated as "cot," is another trigonometric function that complements the tangent. It is defined as the reciprocal of the tangent. Thus, for an angle \( s \), it is given by:

• \[ \cot s = \frac{1}{\tan s} \]

This means that if you know tangent, you can easily find cotangent by taking its reciprocal. From the previous section, we found that \( \tan s = \frac{3\sqrt{7}}{7} \). Therefore, the cotangent is:

• \[ \cot s = \frac{1}{\frac{3\sqrt{7}}{7}} = \frac{7}{3\sqrt{7}} \]

To simplify this expression, we once more rationalize the denominator by multiplying by \( \sqrt{7} \), leading us to:

• \[ \cot s = \frac{7\sqrt{7}}{21} = \frac{\sqrt{7}}{3} \]

Cotangent can be a helpful function particularly in solving problems where the tangent is inconvenient, offering an alternative route in trigonometric identities and equations.

Secant

Secant, abbreviated as "sec," is a trigonometric function closely related to cosine. It's defined as the reciprocal of the cosine of an angle. For angle \( s \), the secant is given by:

• \[ \sec s = \frac{1}{\cos s} \]

Given that \( \cos s = \frac{\sqrt{7}}{4} \), the secant function is calculated as follows:

• \[ \sec s = \frac{1}{\frac{\sqrt{7}}{4}} = \frac{4}{\sqrt{7}} \]

Once again, we rationalize the denominator:

• \[ \sec s = \frac{4}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{4\sqrt{7}}{7} \]

Understanding secant is critical when dealing with reciprocal identities in trigonometry. It also appears in various calculus contexts, such as integration and differentiation of trigonometric functions.

Cosecant

Cosecant, abbreviated as "csc," is the reciprocal function of sine. Often used in trigonometric equations and identities, it provides an alternative perspective on trigonometric computations. For angle \( s \), the cosecant is defined as:

• \[ \csc s = \frac{1}{\sin s} \]

Given \( \sin s = \frac{3}{4} \), the cosecant can be calculated simply:

• \[ \csc s = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

Cosecant is useful particularly in problems involving angles and triangles, especially when dealing with right triangles and unit circle methods. Understanding its relationship with sine can allow for easier manipulation and solution of trigonometric equations, giving you a broader toolkit for tackling math problems.