Problem 82 - 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 \(s\). $$\left(x, \frac{8}{17}\right), x<0$$

Step-by-Step Solution

Answer

The six values for \(s\) are: \(\sin(s) = \frac{8}{17}, \cos(s) = -\frac{15}{17}, \tan(s) = -\frac{8}{15}, \csc(s) = \frac{17}{8}, \sec(s) = -\frac{17}{15}, \cot(s) = -\frac{15}{8}\).

Step 1: Understanding the Trigonometric Identity

The problem states that we need to use the identity \(\cos ^{2} s+\sin ^{2} s=1\). This identity will help us to find \(x\), given that we have \(\sin(s) = \frac{8}{17}\). Since \(x\) corresponds to \(\cos(s)\), we will use this identity to solve for \(x\).

Step 2: Apply the Identity

Using the identity \(\cos ^{2} s+\sin ^{2} s=1\), substitute the known value of \(\sin(s)\): \[ \cos^2(s) = 1 - \sin^2(s) = 1 - \left(\frac{8}{17}\right)^2. \] Calculate \(\sin^2(s)\): \(\sin^2(s) = \left(\frac{8}{17}\right)^2 = \frac{64}{289}\). Hence, \(\cos^2(s) = 1 - \frac{64}{289} = \frac{225}{289}\).

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

With \(\cos^2(s) = \frac{225}{289}\), solve for \(\cos(s)\): \[ \cos(s) = \pm \sqrt{\frac{225}{289}} = \pm \frac{15}{17}. \] Since the problem specifies that \(x < 0\), choose \(\cos(s) = -\frac{15}{17}\). Hence, \(x = \cos(s) = -\frac{15}{17}\).

Step 4: Six Circular Function Values

The six trigonometric functions are: sine \(\sin(s)\), cosine \(\cos(s)\), tangent \(\tan(s)\), cosecant \(\csc(s)\), secant \(\sec(s)\), and cotangent \(\cot(s)\). - \(\sin(s) = \frac{8}{17}\)- \(\cos(s) = -\frac{15}{17}\)- \(\tan(s) = \frac{\sin(s)}{\cos(s)} = \frac{\frac{8}{17}}{-\frac{15}{17}} = -\frac{8}{15}\)- \(\csc(s) = \frac{1}{\sin(s)} = \frac{17}{8}\)- \(\sec(s) = \frac{1}{\cos(s)} = -\frac{17}{15}\)- \(\cot(s) = \frac{1}{\tan(s)} = -\frac{15}{8}\)

Key Concepts

Trigonometric Identity

A trigonometric identity is an equation involving trigonometric functions that is true for every angle. Among these, the Pythagorean identity, \( \cos^2 s + \sin^2 s = 1 \), is fundamental. It is derived from the Pythagorean theorem and applies to any angle on the unit circle.

• This identity helps us find missing values when one of the sine or cosine values is known.
• This is particularly useful for solving equations where either sine or cosine value is provided, as it allows us to find the other.

To solve for cosine or sine, when one is known, rearrange the identity: \[ \cos^2 s = 1 - \sin^2 s \] or \[ \sin^2 s = 1 - \cos^2 s \] Substitute the known values within these equations to find the missing trigonometric function value.

Circular Functions

Circular functions are another name for trigonometric functions, particularly when they are associated with angles on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.

• Each angle in the unit circle relates to a point \((x, y)\).
• The x-coordinate is equivalent to \(\cos(s)\) and the y-coordinate is \(\sin(s)\) for angle \(s\).

All trigonometric functions derive from these primary circular functions. For example, once you know the coordinates of a point on the unit circle, you can calculate other trigonometric function values. This connection establishes circular functions as an essential framework for understanding the behavior of trigonometric functions.

Sine and Cosine

Sine and cosine are the foundational trigonometric functions, vital for describing oscillations and waves. In the context of the unit circle, they are the projections of the radius line onto the x and y axes.

• Sine \((\sin(s))\) represents the y-coordinate of the point on the unit circle.
• Cosine \((\cos(s))\) represents the x-coordinate.

In a given problem where the sine of an angle is known, the Pythagorean identity allows you to find the cosine or vice versa. In our problem, we were given \(\sin(s) = \frac{8}{17}\). To determine the cosine, use \(\cos^2 s = 1 - \sin^2 s\). By substituting the value for sine, we found that \(\cos s = -\frac{15}{17}\). The negative value was chosen according to the condition \(x < 0\).

Trigonometric Functions

Trigonometric functions extend the concept of sine and cosine by introducing additional functions that describe the relationships between different sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

• \(\sin(s)\) and \(\cos(s)\) give the basic circular coordinates.
• \(\tan(s) = \frac{\sin(s)}{\cos(s)}\), hence in our problem, \(\tan(s) = -\frac{8}{15}\).
• \(\csc(s) = \frac{1}{\sin(s)} = \frac{17}{8}\).
• \(\sec(s) = \frac{1}{\cos(s)} = -\frac{17}{15}\).
• \(\cot(s) = \frac{1}{\tan(s)} = -\frac{15}{8}\).

These functions provide a complete view of the angle's properties, allowing us to solve a wide range of geometric and real-world problems. Understanding these functions in the unit circle context makes it easier to visualize how these ratios change with different angles.