Problem 136 - Calculus Volume 1

Question

For the following exercises, \(P\) is a point on the unit circle. a. Find the (exact) missing coordinate value of each point and b. find the values of the six trigonometric functions for the angle \(\theta\) with a terminal side that passes through point \(P .\) Rationalize denominators. $$P\left(\frac{-15}{17}, y\right), y<0$$

Step-by-Step Solution

Answer

y-coordinate is \(\frac{-8}{17}\); \(\sin(\theta) = \frac{-8}{17}\), \(\cos(\theta) = \frac{-15}{17}\), \(\tan(\theta) = \frac{8}{15}\), \(\csc(\theta) = \frac{-17}{8}\), \(\sec(\theta) = \frac{-17}{15}\), \(\cot(\theta) = \frac{15}{8}\).

Step 1: Use the Unit Circle Equation

Since point \(P\left(\frac{-15}{17}, y\right)\) is on the unit circle, it satisfies the equation \(x^2 + y^2 = 1\). Substitute \(x = \frac{-15}{17}\) into the equation: \(\left(\frac{-15}{17}\right)^2 + y^2 = 1\).

Step 2: Solve for y

Calculate \(\left(\frac{-15}{17}\right)^2 = \frac{225}{289}\). Substitute and simplify the equation: \(\frac{225}{289} + y^2 = 1\). This simplifies to \(y^2 = 1 - \frac{225}{289} = \frac{64}{289}\). Taking the square root of both sides gives \(y = \pm \frac{8}{17}\). Since \(y < 0\), \(y = \frac{-8}{17}\).

Step 3: Find Sine and Cosine

For angle \(\theta\), \(\sin(\theta) = y = \frac{-8}{17}\) and \(\cos(\theta) = x = \frac{-15}{17}\).

Step 4: Find the Remaining Trigonometric Functions

Using \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{-8}{17}}{\frac{-15}{17}} = \frac{8}{15}\). Using the reciprocal identities: \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{-17}{8}\), \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{-17}{15}\), \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{15}{8}\). Ensure to rationalize the denominators.

Key Concepts

Trigonometric Functions in the Unit Circle

Trigonometric functions are essential for understanding relationships within circles and solving real-world problems involving rotation and oscillation. When dealing with the unit circle, these functions take on simplified yet powerful forms.
The unit circle has a radius of 1, which means any point on the unit circle can be represented by the coordinates \( (x, y) \). The coordinate \( x \) is equivalent to \( \cos(\theta) \) and \( y \) is \( \sin(\theta) \), where \( \theta \) is the angle formed by the line from the origin to the point and the positive x-axis. This relationship helps to find function values directly.

• For sine, \( \sin(\theta) \), use the y-coordinate.
• For cosine, \( \cos(\theta) \), use the x-coordinate.
• Tangent, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) is the ratio of the y-coordinate to the x-coordinate.

Beyond these primary functions, you have the reciprocals:

• Cosecant, \( \csc(\theta) = \frac{1}{\sin(\theta)} \)
• Secant, \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Cotangent, \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

By mastering these functions, you can solve many problems related to angles and their respective positions on the unit circle.

Rationalizing Denominators

Rationalizing denominators is a useful algebraic technique that makes expressions simpler or more elegant. This process involves eliminating any irrational or radical expressions found in the denominator of a fraction.
Let's see how rationalizing works in the context of trigonometric functions derived from the unit circle:

• Consider the function \( \csc(\theta) = \frac{1}{\sin(\theta)} \) with \( \sin(\theta) = \-\frac{8}{17} \).
• Initially, this becomes \( \csc(\theta) = \,\frac{1}{\frac{-8}{17}} = \frac{-17}{8} \).
• This fraction is already rational, but ensure this process for irrational or complex denominators.

By rationalizing, we simplify computations and interpretations within other mathematical contexts, ensuring clarity. It's crucial for situations where precise numbers are needed, such as in engineering and physics applications.

Angle Calculation on the Unit Circle

Calculating angles in the unit circle involves understanding their coordinates and the relationships they have with trigonometric functions. Point \( P(x, y) \) on the unit circle directly corresponds to an angle \( \theta \), where \( \cos(\theta) = x \) and \( \sin(\theta) = y \).
Every point intersects with a particular angle from the origin, traveling counter-clockwise from the positive x-axis.
Steps to determine the angle include:

• Identify \( x \) and \( y \) coordinates, use them to find \( \sin, \cos, \tan \) from your known point.
• Recognize which quadrant the point resides in to determine the sign of \( \sin \) and \( \cos \).
• Consider full-circle symmetry: for example, if \( \theta \) is negative, the angle's position is measured from clockwise rotation.

Each point on the unit circle helps find critical trigonometric function values at that angle, handy for accurately mapping any rotational pattern or wave function periodicity.