Problem 52 - Algebra 2

Question

Writing Explain how to find the sine and cosine of angles with measures of \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ},\) and \(360^{\circ}\) without using a calculator.

Step-by-Step Solution

Answer

The cosine values for the angles \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, 360^{\circ}\) are 1, 0, -1, 0, 1 respectively and the sine values are 0, 1, 0, -1, 0 respectively.

Step 1: Understanding the unit circle

Draw a unit circle. This is a circle with radius 1 and center at the origin of the coordinate plane. The x-coordinate of a point on the unit circle gives the cosine of the angle from the positive x-axis to the point, and the y-coordinate gives the sine.

Step 2: Locating the angles

Plot points corresponding to the angles of \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}, and 360^{\circ}\). Note that the angles are measured counter-clockwise from the positive x-axis.

Step 3: Evaluate the cosine and sine values

At each point, note the (x, y) coordinates and state them. For \(0^{\circ}\) and \(360^{\circ}\), the point is (1, 0), so \(\cos(0^{\circ}) = \cos(360^{\circ}) = 1\) and \(\sin(0^{\circ}) = \sin(360^{\circ}) = 0\). For \(90^{\circ}\), the point is (0, 1), so \(\cos(90^{\circ}) = 0\) and \(\sin(90^{\circ}) = 1\). For \(180^{\circ}\), the point is (-1, 0), so \(\cos(180^{\circ}) = -1\) and \(\sin(180^{\circ}) = 0\). Lastly, for \(270^{\circ}\), the point is (0, -1), so \(\cos(270^{\circ}) = 0\) and \(\sin(270^{\circ}) = -1\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry that helps us understand the trigonometric functions of angles in a simple and visual way. Picture a circle centered at the origin on a coordinate plane with a radius of 1. This circle is special because every point on its circumference has

• The x-coordinate representing the cosine of an angle
• The y-coordinate representing the sine of that angle.

The unit circle effectively translates the trigonometric ratios into coordinates, making it a powerful tool for finding sine and cosine values of common angles like \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ},\) and \(360^{\circ}\). By visually mapping these angles onto the unit circle, you can easily determine their corresponding sine and cosine values.

Angle Measurement

Angles in trigonometry are usually measured using degrees or radians. For simplicity, we'll focus on degrees in this context. Understanding how angles relate to the unit circle requires knowing that:

• Angles are measured starting from the positive x-axis.
• They increase in a counter-clockwise direction.

When looking at angles like \(0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ},\) and \(360^{\circ},\) each has a distinct place on the unit circle. For instance:- \(0^{\circ}\) and \(360^{\circ}\) lie along the positive x-axis.- \(90^{\circ}\) points straight up along the positive y-axis.- \(180^{\circ}\) extends along the negative x-axis.- \(270^{\circ}\) points downward along the negative y-axis.

Sine and Cosine Values

Sine and cosine are essential trigonometric functions that describe the relationships of the unit circle. For any angle, these functions are derived from the coordinates of the equivalent point on the unit circle. Let's look at the key angles:

• For \(0^{\circ}\) and \(360^{\circ}\), the point is (1, 0) on the unit circle. Therefore, \(\cos(0^{\circ}) = \cos(360^{\circ}) = 1\), and \(\sin(0^{\circ}) = \sin(360^{\circ}) = 0\).
• At \(90^{\circ}\), the point is (0, 1). Thus, \(\cos(90^{\circ}) = 0\), and \(\sin(90^{\circ}) = 1\).
• For \(180^{\circ}\), you find the point (-1, 0), making \(\cos(180^{\circ}) = -1\), and \(\sin(180^{\circ}) = 0\).
• Finally, \(270^{\circ}\) corresponds to the point (0, -1), where \(\cos(270^{\circ}) = 0\), and \(\sin(270^{\circ}) = -1\).

By knowing these points, you can quickly find sine and cosine for these fundamental angles without a calculator.