Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(15,-8)$$

Step-by-Step Solution

Answer

The point (15, -8) is in the fourth quadrant. For this, \(\sin\theta = \frac{-8}{17}, \cos\theta = \frac{15}{17}, \tan\theta = \frac{-8}{15}, \csc\theta = -\frac{17}{8}, \sec\theta = \frac{17}{15}, \cot\theta = -\frac{15}{8}.\)

Step 1: Determine the Quadrant

First, we need to determine in which quadrant the point \((15, -8)\) is located. Since the x-coordinate is positive and the y-coordinate is negative, the point is in the fourth quadrant.

Step 2: Find the Radius

The radius (or the hypotenuse of the right triangle) can be found using the Pythagorean theorem. \[ r = \sqrt{x^2 + y^2} = \sqrt{15^2 + (-8)^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \]

Step 3: Identify the Angle

We are looking for the angle \(\theta\) in standard position. Since the point is in the fourth quadrant, \(\theta\) is negative when measuring clockwise from the positive x-axis. Therefore, the least positive angle equivalent to \(\theta\) will be in the range of \(0^\circ\) to \(360^\circ\).

Step 4: Define Trigonometric Functions

Now calculate each of the six trigonometric functions. - \(\sin\theta = \frac{y}{r} = \frac{-8}{17}\)- \(\cos\theta = \frac{x}{r} = \frac{15}{17}\)- \(\tan\theta = \frac{y}{x} = \frac{-8}{15}\) Reciprocals:- \(\csc\theta = \frac{r}{y} = \frac{17}{-8} = -\frac{17}{8}\)- \(\sec\theta = \frac{r}{x} = \frac{17}{15}\)- \(\cot\theta = \frac{x}{y} = \frac{15}{-8} = -\frac{15}{8}\)

Key Concepts

Angles in Standard Position

Understanding angles in standard position is essential in trigonometry. An angle is said to be in standard position when its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. From this starting point, the angle can open in the counter-clockwise direction for positive angles or clockwise for negative angles.

In this context, knowing the quadrant in which an angle's terminal side lies helps us understand the properties of the angle. The quadrants are numbered counter-clockwise, starting from the positive x-axis. For example:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, and y is negative.

For the point (15, -8), the angle is in the fourth quadrant. This means the angle in standard position will measure at a negative degree when drawn clockwise from the positive x-axis if not adjusted to fit within 0° to 360°.

Trigonometric Functions

Trigonometric functions are a core concept in trigonometry associated with angles. These functions take an angle as input and relate to the ratios of the sides of right triangles.

Let's break down the primary functions using our example angle with the point (15, -8):

• **Sine (\( ext{sin} heta\))**: This is defined as the ratio of the length of the opposite side to the hypotenuse. For our example angle, \( ext{sin} heta = \frac{-8}{17}\).
• **Cosine (\( ext{cos} heta\))**: This is the ratio of the length of the adjacent side to the hypotenuse. Hence, \( ext{cos} heta = \frac{15}{17}\).
• **Tangent (\( ext{tan} heta\))**: The ratio of the opposite side to the adjacent side, giving us \( ext{tan} heta = \frac{-8}{15}\).

The reciprocal functions provide further relationships:

• **Cosecant (\( ext{csc} heta\))**: \( ext{csc} heta = \frac{1}{ ext{sin} heta} = -\frac{17}{8}\).
• **Secant (\( ext{sec} heta\))**: \( ext{sec} heta = \frac{1}{ ext{cos} heta} = \frac{17}{15}\).
• **Cotangent (\( ext{cot} heta\))**: \( ext{cot} heta = \frac{1}{ ext{tan} heta} = -\frac{15}{8}\).

Understanding these functions is crucial as they appear in various mathematical contexts and applications.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry and trigonometry, which relates the sides of a right triangle. This theorem states: in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Using this theorem, you can find the radius or hypotenuse, \(r\), of a right triangle. Let's see how it's applied:

• Given a point (15, -8), associate these as x and y coordinates of a right triangle.
• We find the hypotenuse using \( r = \sqrt{x^2 + y^2} \) which becomes \( r = \sqrt{15^2 + (-8)^2} \).
• This simplifies to \( r = \sqrt{225 + 64} = \sqrt{289} = 17 \).

The hypotenuse here is crucial for determining the values of trigonometric functions as it helps in establishing the ratios necessary for sine, cosine, and tangent.