Problem 82 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

Use the unit circle to find values: (a) at \(\pi\), (b) at \(-\frac{\pi}{2}\), (c) at \(0\), (d) at \(\frac{\pi}{2}\). Many result in undefined due to division by zero.

Step 1: Convert Degrees to Radians (if applicable)

For angle (a) and (b), first convert the degree measures to radians because it's often easier to work with radians in trigonometry. - (a): An angle of \( 180^{\circ} \) corresponds to \( \pi \) radians since \( 180^{\circ} = \pi \) radians.- (b): An angle of \( -90^{\circ} \) corresponds to \( -\frac{\pi}{2} \) radians since \( -90^{\circ} = -\frac{\pi}{2} \).

Step 2: Calculate Trig Functions for Known Angles

Use the unit circle to find the trigonometric values for the known angles.- (a): \( \pi \) radians. - Sine: \( \sin \pi = 0 \) - Cosine: \( \cos \pi = -1 \) - Tangent: \( \tan \pi = 0 \) - Cosecant: \( \csc \pi = \text{undefined} \) (since 1/0 is undefined) - Secant: \( \sec \pi = -1 \) - Cotangent: \( \cot \pi = \text{undefined} \)- (b): \( -\frac{\pi}{2} \) radians. - Sine: \( \sin(-\frac{\pi}{2}) = -1 \) - Cosine: \( \cos(-\frac{\pi}{2}) = 0 \) - Tangent: \( \tan(-\frac{\pi}{2}) = \text{undefined} \) - Cosecant: \( \csc(-\frac{\pi}{2}) = -1 \) - Secant: \( \sec(-\frac{\pi}{2}) = \text{undefined} \) - Cotangent: \( \cot(-\frac{\pi}{2}) = 0 \)

Step 3: Simplify Angles Greater than \(2\pi\) or Less than Zero

For angles such as (c) and (d), simplify them to angles between 0 and \(2\pi\) by subtracting or adding \(2\pi\) multiples.- (c): \(2\pi\) - This angle corresponds to \(0\) because full rotations return to the starting point.- (d): \( \frac{5\pi}{2} \) - Simplify by subtracting \(2\pi\) to get \( \frac{\pi}{2} \) since \( \frac{5\pi}{2} - 2\pi = \frac{\pi}{2} \).

Step 4: Calculate Trig Functions for Simplified Angles

Now using the unit circle, calculate the trigonometric functions for these simplified angles.- (c): \(0\) radians (or \(2\pi\)). - Sine: \( \sin 0 = 0 \) - Cosine: \( \cos 0 = 1 \) - Tangent: \( \tan 0 = 0 \) - Cosecant: \( \csc 0 = \text{undefined} \) - Secant: \( \sec 0 = 1 \) - Cotangent: \( \cot 0 = \text{undefined} \)- (d): \( \frac{\pi}{2} \) radians. - Sine: \( \sin \frac{\pi}{2} = 1 \) - Cosine: \( \cos \frac{\pi}{2} = 0 \) - Tangent: \( \tan \frac{\pi}{2} = \text{undefined} \) - Cosecant: \( \csc \frac{\pi}{2} = 1 \) - Secant: \( \sec \frac{\pi}{2} = \text{undefined} \) - Cotangent: \( \cot \frac{\pi}{2} = 0 \)

Key Concepts

Unit Circle

The unit circle is your friend when it comes to understanding trigonometric functions. It is a circle with a radius of 1, centered at the origin of the coordinate plane. Every point on the unit circle corresponds to an angle measured from the positive x-axis. Here, both the x-coordinate and y-coordinate have significant meanings related to trigonometry.

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

For angles in radians, the full circumference of the circle is \( 2\pi \), which makes it extremely easy to visualize and calculate the values of trigonometric functions. As you move around the circle, you're essentially rotating through different angles, and each angle will align with specific points on this circle, offering immediate access to their sine and cosine values. The unit circle becomes especially handy when working with specific known angles like 0, \( \pi/2 \), \( \pi \), and \( 2\pi \), or similarly, \( \frac{3\pi}{2} \), where values are straightforward to find. This one-tool arms you with almost everything you need to solve basic trig problems efficiently.

Radian Conversion

Converting degrees into radians is essential for working in trigonometry because radian is the standard unit for measuring angles in mathematics. To convert an angle from degrees to radians, use the formula \[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]This formula stems from the fact that a full circle is \( 360^{\circ} \) and that corresponds to \( 2 \pi \) radians. By understanding this relationship, you can easily switch between degrees and radians.

For example:

• \( 180^{\circ} \) is equivalent to \( \pi \) radians.
• \( -90^{\circ} \) equates to \( -\frac{\pi}{2} \) radians.

Mastering radian conversion is crucial because it simplifies calculations and makes it easier to use other mathematical tools, such as the unit circle, for determining trigonometric function values. It's always good practice to start any problem by converting angles into radians when necessary, to ensure consistency and accuracy in your work.

Exact Values of Trigonometric Functions

Grasping the exact values of trigonometric functions for specific angles allows you to solve trigonometry problems effectively. These values can usually be derived straightforwardly from the unit circle for angles like \( 0 \), \( \pi/2 \), \( \pi \), \( 3\pi/2 \), and \( 2\pi \). Each of these angles provides a unique set of sine, cosine, and tangent values, often memorized through practice:

• At \( 0 \) radians:
  • Sine: \(0\)
  • Cosine: \(1\)
  • Tangent: \(0\)
• At \( \pi/2 \) radians:
  • Sine: \(1\)
  • Cosine: \(0\)
  • Tangent: undefined
• At \( \pi \) radians:
  • Sine: \(0\)
  • Cosine: \(-1\)
  • Tangent: \(0\)
• At \( 3\pi/2 \) radians:
  • Sine: \(-1\)
  • Cosine: \(0\)
  • Tangent: undefined
• At \( 2\pi \) radians:
  • Sine: \(0\)
  • Cosine: \(1\)
  • Tangent: \(0\)

These values are pivotal in solving trigonometric equations and analyzing graphs. Recognizing these key values can help predict the behavior of trigonometric functions as angles change, leading to a deeper understanding and smoother problem-solving experience.