Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Use the identity \(\cos ^{2} s+\sin ^{2} s=1\) to find the value of \(x\) or \(y,\) as appropriate. Then, assuming that \(s\) corresponds to the given point on the unit circle, find the six circular function values for \(s\). $$\left(\frac{7}{25}, y\right), y>0$$

Step-by-Step Solution

Answer

\( y = \frac{24}{25} \); six circular functions found.

Step 1: Use Pythagorean Identity

We start by using the identity \( \cos^2 s + \sin^2 s = 1 \). We are given \( \cos s = \frac{7}{25} \), so we plug this into the identity: \[ \left(\frac{7}{25}\right)^2 + \sin^2 s = 1. \]

Step 2: Simplify and Solve for \( \sin^2 s \)

Calculate \( \left(\frac{7}{25}\right)^2 \): \[ \cos^2 s = \frac{49}{625}. \] Substitute back into the identity: \[ \frac{49}{625} + \sin^2 s = 1. \] Subtract \( \frac{49}{625} \) from both sides to isolate \( \sin^2 s \): \[ \sin^2 s = 1 - \frac{49}{625} = \frac{576}{625}. \]

Step 3: Solve for \( \sin s \)

Since \( y > 0 \), \( \sin s \) is positive. Take the square root of \( \sin^2 s = \frac{576}{625} \) to get \( \sin s \): \[ \sin s = \sqrt{\frac{576}{625}} = \frac{24}{25}. \]

Step 4: Find the Tangent Value

Calculate \( \tan s \) using \( \tan s = \frac{\sin s}{\cos s} \): \[ \tan s = \frac{\frac{24}{25}}{\frac{7}{25}} = \frac{24}{7}. \]

Step 5: Find Reciprocal Functions Cosecant, Secant, and Cotangent

Calculate \( \csc s = \frac{1}{\sin s} = \frac{25}{24} \), \( \sec s = \frac{1}{\cos s} = \frac{25}{7} \), and \( \cot s = \frac{1}{\tan s} = \frac{7}{24} \).

Step 6: Summarize All Six Circular Functions

The six circular functions are: \( \cos s = \frac{7}{25} \), \( \sin s = \frac{24}{25} \), \( \tan s = \frac{24}{7} \), \( \csc s = \frac{25}{24} \), \( \sec s = \frac{25}{7} \), and \( \cot s = \frac{7}{24} \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry. It's closely related to the Pythagorean theorem you might have encountered in geometry, expressed as: \( a^2 + b^2 = c^2 \). In the realm of trigonometric functions, this identity becomes \( \cos^2 s + \sin^2 s = 1 \). It's a powerful tool for analyzing circular functions because it relates the sine and cosine of an angle on the unit circle.

Here's what makes the Pythagorean identity so useful:

• It holds true for any angle \( s \), regardless of how large or small.
• It helps in verifying and simplifying expressions involving trigonometric functions.
• If you know either the sine or cosine of an angle, you can easily find the other using this identity, as shown in the original exercise.

In solving trigonometric problems, especially those on the unit circle, the equation \( \cos^2 s + \sin^2 s = 1 \) often serves as a precursor to finding angle measures and ensuring that calculations conform to this inherent truth about circular functions.

Unit Circle

The unit circle is a key concept in understanding trigonometric identities. It is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. Every point \((x, y)\) on the unit circle corresponds to an angle \( s \), and these coordinates represent the values of \( \cos s \) and \( \sin s \) respectively.

Some essential features of the unit circle include:

• The cosine of an angle \( s \) is the x-coordinate of the corresponding point on the unit circle.
• The sine of an angle \( s \) is the y-coordinate.
• Because the hypotenuse is 1, the coordinates also satisfy the equation \( x^2 + y^2 = 1 \), aligning with the Pythagorean identity.

This circular representation allows you to visualize how angles and trigonometric functions are interconnected. Understanding this model is crucial when solving for unknowns and verifying calculations, as you did in the problem by identifying that \( \cos s = \frac{7}{25} \) corresponded to a specific point on the unit circle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. They are pivotal tools for simplifying expressions and solving equations in trigonometry.

Some commonly used trigonometric identities include:

• The Pythagorean identities, which include \( \cos^2 s + \sin^2 s = 1 \).
• The ratio identities such as \( \tan s = \frac{\sin s}{\cos s} \).
• The reciprocal identities which involve \( \csc s = \frac{1}{\sin s} \), \( \sec s = \frac{1}{\cos s} \), and \( \cot s = \frac{1}{\tan s} \).

In the exercise, you applied these identities effectively. After determining \( \sin s \) and \( \cos s \), you calculated the other trigonometric functions: tangent, cosecant, secant, and cotangent through these identities. This showcases how understanding and applying these identities help in solving trigonometry problems, as they reveal relationships between different circular functions.