Question

Write a brief paragraph that explains how to quickly compute the value of the trigonometric functions of \(0^{\circ}, 90^{\circ}, 180^{\circ},\) and \(270^{\circ} .\)

Short answer

Use the unit circle: \( \sin(0^{\circ})=0 \), \( \sin(90^{\circ})=1 \), \( \sin(180^{\circ})=0 \), \( \sin(270^{\circ})=-1 \); \( \cos(0^{\circ})=1 \), \( \cos(90^{\circ})=0 \), \( \cos(180^{\circ})=-1 \), \( \cos(270^{\circ})=0 \); \( \tan(0^{\circ}) = 0 \), \( \tan(90^{\circ})=\text{undefined} \), \( \tan(180^{\circ}) = 0 \), \( \tan(270^{\circ})=\text{undefined} \).

Step-by-step solution

Understanding the Problem

The task is to find the values of trigonometric functions (sine, cosine, tangent) at angles typically used in trigonometric problems: 0°, 90°, 180°, and 270°.

Recall the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. The angles 0°, 90°, 180°, and 270° correspond to specific points on the unit circle. Knowing these points helps to find the values of the trigonometric functions.

Evaluate the Sine Function

The sine of an angle corresponds to the y-coordinate of the point on the unit circle. Therefore, \( \sin(0^{\circ}) = 0 \), \( \sin(90^{\circ}) = 1 \), \( \sin(180^{\circ}) = 0 \), and \( \sin(270^{\circ}) = -1 \).

Evaluate the Cosine Function

The cosine of an angle corresponds to the x-coordinate of the point on the unit circle. Hence, \( \cos(0^{\circ}) = 1 \), \( \cos(90^{\circ}) = 0 \), \( \cos(180^{\circ}) = -1 \), and \( \cos(270^{\circ}) = 0 \).

Evaluate the Tangent Function

The tangent of an angle is the ratio of the sine to the cosine at that angle. Therefore, \( \tan(0^{\circ}) = \frac{0}{1} = 0 \), \( \tan(90^{\circ}) = \frac{1}{0} = \text{undefined} \), \( \tan(180^{\circ}) = \frac{0}{-1} = 0 \), and \( \tan(270^{\circ}) = \frac{-1}{0} = \text{undefined} \).

Key concepts

unit circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1 unit centered at the origin of a coordinate plane. The key idea is that the circle's radius always equals 1, making calculations simpler.
Each point on the unit circle has coordinates \( (x, y) \) and corresponds to an angle measured from the positive x-axis.
The angles you typically work with, such as 0°, 90°, 180°, and 270°, correspond to specific points on the circle.
This helps in quickly determining the sine, cosine, and tangent values for these angles without complex calculations.

sine function

The sine function relates to the y-coordinate of a point on the unit circle.
To find the sine of an angle, you only need to look at the corresponding angle's y-coordinate.
For example:

• At 0°, the y-coordinate is 0, so \( \sin(0^{\circ}) = 0 \).
• At 90°, the y-coordinate is 1, so \( \sin(90^{\circ}) = 1 \).
• At 180°, the y-coordinate is 0, so \( \sin(180^{\circ}) = 0 \).
• At 270°, the y-coordinate is -1, so \( \sin(270^{\circ}) = -1 \).

As you can see, understanding the unit circle coordinates makes it simple to find sine values quickly.

cosine function

The cosine function is related to the x-coordinate of a point on the unit circle.
You can determine the cosine of an angle by looking at the x-coordinate of that angle's corresponding point.
For example:

• At 0°, the x-coordinate is 1, so \( \cos(0^{\circ}) = 1 \).
• At 90°, the x-coordinate is 0, so \( \cos(90^{\circ}) = 0 \).
• At 180°, the x-coordinate is -1, so \( \cos(180^{\circ}) = -1 \).
• At 270°, the x-coordinate is 0, so \( \cos(270^{\circ}) = 0 \).

By remembering these key points on the unit circle, you can easily compute the cosine values for these important angles.

tangent function

The tangent function is a bit different from sine and cosine because it involves the ratio of these two functions.
The tangent of an angle is the ratio \( \frac{\sin \theta}{\cos \theta} \).
For example:

• At 0°, \( \tan(0^{\circ}) = \frac{0}{1} = 0 \).
• At 90°, \( \tan(90^{\circ}) = \frac{1}{0} \), which is undefined.
• At 180°, \ \( \tan(180^{\circ}) = \frac{0}{-1} = 0 \).
• At 270°, \ \( \tan(270^{\circ}) = \frac{-1}{0} \), which is again undefined.

Understanding the relationships between these functions is crucial for quickly finding their values at key angles.