Problem 18 - 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. $$(-12,-5)$$

Step-by-Step Solution

Answer

Angle \( \theta \) is in the third quadrant, with trigonometric functions: \( \sin = \frac{-5}{13} \), \( \cos = \frac{-12}{13} \), \( \tan = \frac{5}{12} \), \( \csc = -\frac{13}{5} \), \( \sec = -\frac{13}{12} \), \( \cot = \frac{12}{5} \).

Step 1: Understand the Problem

We need to find an angle \( \theta \) in standard position where the terminal side passes through the point \( (-12, -5) \). We will then calculate the six trigonometric functions for this angle.

Step 2: Determine the Quadrant

The point \( (-12, -5) \) is located in the third quadrant of the Cartesian plane, where both x and y values are negative.

Step 3: Calculate the Reference Angle

Find the reference angle by calculating the arctangent of the absolute values of the point: \( \tan^{-1} \left( \frac{5}{12} \right) \). This will give us the reference angle for \( \theta \).

Step 4: Find the Angle \( \theta \)

Since the point is in the third quadrant, add 180 degrees to the reference angle: \( \theta = 180^\circ + \tan^{-1} \left( \frac{5}{12} \right) \) to find the measure of the angle \( \theta \).

Step 5: Calculate the Hypotenuse

Use the Pythagorean theorem to find the hypotenuse \( r \): \( r = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = 13 \).

Step 6: Determine the Trigonometric Functions

Using \( x = -12 \), \( y = -5 \), and \( r = 13 \), calculate:- \( \sin \theta = \frac{y}{r} = \frac{-5}{13} \)- \( \cos \theta = \frac{x}{r} = \frac{-12}{13} \)- \( \tan \theta = \frac{y}{x} = \frac{-5}{-12} = \frac{5}{12} \)- \( \csc \theta = \frac{r}{y} = \frac{13}{-5} = -\frac{13}{5} \)- \( \sec \theta = \frac{r}{x} = \frac{13}{-12} = -\frac{13}{12} \)- \( \cot \theta = \frac{x}{y} = \frac{-12}{-5} = \frac{12}{5} \)

Key Concepts

Angles in Standard Position

When we talk about angles in standard position, we're working with a specific format: the angle is drawn on the Cartesian plane, centered at the origin. The initial side of the angle lies along the positive x-axis. When you draw the terminal side of your angle, it will rotate counterclockwise if the angle is positive and clockwise if it's negative.

For the given problem, the point

• The terminal side passes through (-12, -5), which places our angle in the third quadrant.
• These details help us identify key elements.

Identifying the quadrant is crucial because different trigonometric functions have positive or negative values depending on where the terminal side lands.

Trigonometric Functions

Trigonometric functions relate the angles and sides of a triangle, often helping us understand the position of angles in different quadrants. For this exercise, the six foundational trigonometric functions are: sine, cosine, tangent, cosecant, secant, and cotangent.

Each function typically uses values based on a right triangle made by dropping a perpendicular from the point on the terminal side to the x-axis:

• Sine ( \( \sin \theta \) ): Ratio of the opposite side to the hypotenuse.
• Cosine ( \( \cos \theta \) ): Ratio of the adjacent side to the hypotenuse.
• Tangent ( \( \tan \theta \) ): Ratio of the opposite side to the adjacent side.
• Other functions are reciprocals: cosecant, secant, and cotangent, representing respective reciprocal ratios.

Given point (-12, -5), we calculate:

• \( \sin \theta = -\frac{5}{13} \)
• \( \cos \theta = -\frac{12}{13} \)
• \( \tan \theta = \frac{5}{12} \)

Reference Angle

In trigonometry, the reference angle helps determine the measure of an angle relative to the x-axis. It simplifies calculating the trigonometric functions because these values are often the same for any angle that shares a reference angle.The reference angle is always a non-negative, acute angle. For a point in the third quadrant, as with (-12, -5),

• Compute it by finding \( \tan^{-1} \left( \frac{5}{12} \right) \) using the absolute values.
• Then adjust using the quadrant location to find the full angle measure (by adding 180°).

Using the reference angle, you can correctly determine positive or negative signs for the function values based on whether the point is in the first, second, third, or fourth quadrant.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry, providing connections between the sides of right triangles in the Cartesian plane.

In this exercise, you need to find the hypotenuse of a right triangle formed from point (-12, -5):

• Using the coordinates as the lengths of the two sides (-12 along the x-axis, -5 along the y-axis)
• Apply the theorem: \( r = \sqrt{(-12)^2 + (-5)^2} \), thus \( r = \sqrt{144 + 25} \) leading to a hypotenuse of \( 13 \).

The hypotenuse helps calculate the sine and cosine functions, which play a critical role in finding other trigonometric functions. Understanding how all these elements interact is key to mastering trigonometry.