Problem 79 - 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 \(s\). $$\left(\frac{3}{5}, y\right), y>0$$

Step-by-Step Solution

Answer

\( y = \frac{4}{5} \), trigonometric values are: \( \cos s = \frac{3}{5}, \sin s = \frac{4}{5}, \tan s = \frac{4}{3}, \csc s = \frac{5}{4}, \sec s = \frac{5}{3}, \cot s = \frac{3}{4} \).

Step 1: Apply the Pythagorean Identity

Use the identity \( \cos^2 s + \sin^2 s = 1 \) and substitute \( \cos s = \frac{3}{5} \) to find \( \sin s \). The identity becomes \( \left(\frac{3}{5}\right)^2 + y^2 = 1 \).

Step 2: Solve for \( y^2 \)

Calculate \( \left(\frac{3}{5}\right)^2 = \frac{9}{25} \). Substitute this into the equation to get \( \frac{9}{25} + y^2 = 1 \). Subtract \( \frac{9}{25} \) from \( 1 \) to find \( y^2 \): \( y^2 = 1 - \frac{9}{25} = \frac{16}{25} \).

Step 3: Find the Value of \( y \)

Since \( y > 0 \), take the positive square root of \( y^2 \): \( y = \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Step 4: Calculate the Six Trigonometric Functions

Given that \( \cos s = \frac{3}{5} \) and \( \sin s = \frac{4}{5} \), calculate the other four trigonometric functions: \( \tan s = \frac{\sin s}{\cos s} = \frac{4/5}{3/5} = \frac{4}{3} \), \( \csc s = \frac{1}{\sin s} = \frac{5}{4} \), \( \sec s = \frac{1}{\cos s} = \frac{5}{3} \), and \( \cot s = \frac{1}{\tan s} = \frac{3}{4} \).

Key Concepts

Pythagorean Identity in Trigonometry

The Pythagorean Identity is a fundamental concept in trigonometry and it goes as follows: \[\cos^2 s + \sin^2 s = 1\] This identity is derived from the Pythagorean Theorem, which deals with right-angled triangles. It's called a Pythagorean Identity because it relates the squares of the cosine and sine functions, much like how the squares of the sides of a right triangle relate to the square of the hypotenuse. This identity is essential for finding unknown values of trigonometric functions, like when you're given either \(\cos s\) or \(\sin s\) and need to find the other. Let's imagine we know that \(\cos s = \frac{3}{5}\). To find \(\sin s\), we substitute into the identity: \[\left(\frac{3}{5}\right)^2 + \sin^2 s = 1\] From here, solving for \(\sin^2 s\), we get \[\sin^2 s = 1 - \frac{9}{25} = \frac{16}{25}\] Since \(y > 0\), \(\sin s\) becomes \(\frac{4}{5}\). This shows the utility of the Pythagorean Identity in problem-solving.

• It allows for the calculation of unknown trigonometric values.
• It supports calculations on the unit circle and beyond.

Understanding the Unit Circle

The Unit Circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It provides a powerful visual tool to understand trigonometric functions. Every point on this circle corresponds to trigonometric function values. The equation for the unit circle is \[x^2 + y^2 = 1\] This is because the radius is 1. Each point \((x, y)\) on this circle represents \((\cos s, \sin s)\). In essence, the angle \(s\) corresponds to a point on the unit circle where \(x\) is the cosine of \(s\) and \(y\) is the sine of \(s\). When you are given a point like \(\left(\frac{3}{5}, y\right)\), it implies \(\cos s = \frac{3}{5}\) and calls for using the Pythagorean Identity to find \(\sin s\): \[\left(\frac{3}{5}\right)^2 + y^2 = 1\] Solving gives \(y = \frac{4}{5}\). The positive value for \(y\) is chosen since \(y > 0\) indicating the point is in the upper half of the unit circle.

• The unit circle helps in visualizing and calculating tangent, cotangent, secant, and cosecant.
• Angles in radians align with points on the unit circle.

Exploring Trigonometric Functions

Trigonometric Functions are fundamental to understanding the relationships between angles and sides in right triangles and circles. They include sine \(\sin\), cosine \(\cos\), tangent \(\tan\), cosecant \(\csc\), secant \(\sec\), and cotangent \(\cot\). These functions can be visualized and calculated using the unit circle: - **Sine and Cosine:** For an angle \(s\), \(\cos s\) corresponds to the x-coordinate and \(\sin s\) to the y-coordinate on the unit circle.- **Tangent and Cotangent:** \(\tan s = \frac{\sin s}{\cos s}\), \(\cot s = \frac{\cos s}{\sin s}\).- **Secant and Cosecant:** \(\sec s = \frac{1}{\cos s}\), \(\csc s = \frac{1}{\sin s}\). In our earlier example, with \(\cos s = \frac{3}{5}\) and \(\sin s = \frac{4}{5}\), you can calculate:

• \(\tan s = \frac{4}{3}\)
• \(\csc s = \frac{5}{4}\)
• \(\sec s = \frac{5}{3}\)
• \(\cot s = \frac{3}{4}\)

Understanding these functions allows the solving of a variety of mathematical problems by relating angles to side lengths on triangles or positions on circles.