Problem 82 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact values of the six trigonometric unctions of each angle, whenever possible. (a) \(180^{\circ}\) (b) \(-90^{\circ}\) (c) \(2 \pi\) (d) \(5 \pi / 2\)

Step-by-Step Solution

Answer

The trigonometric values depend on the standard position of the angles on the unit circle.

Step 1: Analyze the Angle 180°

For angle \( 180^{\circ} \), we consider its position on the unit circle. This angle corresponds to the point \((-1, 0)\) on the unit circle. Thus, the trigonometric functions are: - \( \sin(180^{\circ}) = 0 \) - \( \cos(180^{\circ}) = -1 \) - \( \tan(180^{\circ}) = 0 \) - \( \csc(180^{\circ}) = \text{undefined} \) because \( \sin(180^{\circ}) = 0 \)- \( \sec(180^{\circ}) = -1 \)- \( \cot(180^{\circ}) = \text{undefined} \) because \( \tan(180^{\circ}) = 0 \).

Step 2: Analyze the Angle -90°

For angle \( -90^{\circ} \), we convert it to standard position as \( 270^{\circ} \), and it corresponds to the point \((0, -1)\) on the unit circle. Thus, the trigonometric functions are: - \( \sin(-90^{\circ}) = -1 \) - \( \cos(-90^{\circ}) = 0 \) - \( \tan(-90^{\circ}) = \text{undefined} \) - \( \csc(-90^{\circ}) = -1 \)- \( \sec(-90^{\circ}) = \text{undefined} \)- \( \cot(-90^{\circ}) = 0 \).

Step 3: Analyze the Angle 2π

For angle \( 2\pi \), it corresponds to \( 360^{\circ} \), a complete rotation back to the original position at \((1, 0)\) on the unit circle. Thus, the trigonometric functions are: - \( \sin(2\pi) = 0 \) - \( \cos(2\pi) = 1 \) - \( \tan(2\pi) = 0 \) - \( \csc(2\pi) = \text{undefined} \) because \( \sin(2\pi) = 0 \)- \( \sec(2\pi) = 1 \)- \( \cot(2\pi) = \text{undefined} \) because \( \tan(2\pi) = 0 \).

Step 4: Analyze the Angle 5π/2

For angle \( \frac{5\pi}{2} \), first convert it to degrees to get \( 450^{\circ} \). This is equivalent to \( 90^{\circ} \) after subtracting a full rotation \( 360^{\circ} \) twice. This corresponds to the point \((0, 1)\) on the unit circle. Thus, the trigonometric functions are: - \( \sin\left(\frac{5\pi}{2}\right) = 1 \) - \( \cos\left(\frac{5\pi}{2}\right) = 0 \) - \( \tan\left(\frac{5\pi}{2}\right) = \text{undefined} \)- \( \csc\left(\frac{5\pi}{2}\right) = 1 \)- \( \sec\left(\frac{5\pi}{2}\right) = \text{undefined} \)- \( \cot\left(\frac{5\pi}{2}\right) = 0 \).

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry, acting as a circle with a radius of 1 centered at the origin of a coordinate plane. The equation of the unit circle is given by \( x^2 + y^2 = 1 \). Each point on the unit circle represents a specific angle, measured in radians or degrees, starting from the positive x-axis and sweeping counterclockwise.

• The x-coordinate corresponds to the cosine of the angle, \( \cos(\theta) \).
• The y-coordinate corresponds to the sine of the angle, \( \sin(\theta) \).

By understanding the positions of key angles on the unit circle, you can easily determine the sine and cosine values for those angles. For example, at \( 0^{\circ} / 2\pi \), the coordinates are \((1,0)\). At \(90^{\circ} / \pi/2\), the coordinates are \((0,1)\). A thorough grasp of this circle allows you to quickly visualize and solve trigonometric problems.

Exploring Sine and Cosine

Sine and cosine are the primary trigonometric functions that give valuable information about angles and their reference on the unit circle.

• Sine, \( \sin(\theta) \), represents the y-coordinate of an angle's point on the unit circle.
• Cosine, \( \cos(\theta) \), represents the x-coordinate of that same point.

The sine and cosine functions are periodic with a period of \( 2\pi \), indicating their values repeat every full circle rotation. For instance, at \( 180^{\circ} \) or \( \pi \), \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \). At \( 360^{\circ} \) or \( 2\pi \), both \( \sin(\theta) \) and \( \cos(\theta) \) return to their starting values of 0 and 1, respectively. Recognizing these patterns is crucial in solving more advanced trigonometric equations.

Mastering Angle Conversion

Converting angles between degrees and radians is an essential skill in trigonometry. Since radians are more commonly used in calculus and higher mathematics, it's important to be comfortable switching between these units.

• To convert degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
• Conversely, to convert radians to degrees, use: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

As in the example, \( -90^{\circ} \) converts to \( 270^{\circ} \) when considering its counterclockwise position, which simplifies to \( 3\pi/2 \) radians. By regular practice, angle conversion becomes a seamless step in solving trigonometric problems.

Understanding Undefined Function Values

In trigonometry, certain function values can become undefined due to division by zero in the respective formulas. Consider the following functions that often result in undefined values:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) becomes undefined where \( \cos(\theta) = 0 \), for example, at \( \pm 90^{\circ} \) or \( \pi/2 \), \( 3\pi/2 \).
• \( \csc(\theta) = \frac{1}{\sin(\theta)} \) is undefined where \( \sin(\theta) = 0 \), like at \( 0^{\circ} \) or \( \pi \).
• \( \sec(\theta) = \frac{1}{\cos(\theta)} \) is undefined wherever \( \cos(\theta) = 0 \).
• \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) is undefined where \( \sin(\theta) = 0 \).

Knowing how and why these values become undefined results in a deeper understanding of trigonometric functions and prepares you for tackling more complex mathematical analyses.