Problem 110 - Precalculus Essentials

Question

Will help you prepare for the material covered in the next section.$$\text { Give exact values for } \sin 30^{\circ}, \cos 30^{\circ}, \sin 60^{\circ}, \text { and } \cos 60^{\circ}$$

Step-by-Step Solution

Answer

\(\sin 30^{\circ} = 0.5\), \(\cos 30^{\circ} = √3/2\), \(\sin 60^{\circ} = √3/2\), \(\cos 60^{\circ} = 0.5\).

Step 1: Calculate sin(30°)

In the unit circle, an angle of 30° corresponds to half of one-sixth of the circle. This forms a right-angled triangle where the hypotenuse is 1 (radius of the unit circle), the opposite side is 0.5 and the adjacent side is √3/2. Therefore, \(\sin 30^{\circ}= \frac{{opposite side}}{hypotenuse} = \frac{1/2}{1} = 0.5\)

Step 2: Calculate cos(30 °)

Continuing from the previous step, \(\cos\) is defined as the ratio of the adjacent side to the hypotenuse. So \(\cos 30^{\circ}= \frac{{adjacent side}}{hypotenuse} = \frac{√3/2}{1} = √3/2.\)

Step 3: Calculate sin(60 °)

An angle of 60° corresponds to one-third of a half-circle, forming a right-angled triangle where the opposite side is √3/2 and the adjacent side is 1/2. Therefore, \(\sin 60^{\circ} = \frac{{opposite side}}{hypotenuse} = \frac{√3/2}{1} = √3/2\).

Step 4: Calculate cos(60 °)

Remaining on the triangle from the last step, \(\cos 60^{\circ} = \frac{{adjacent side}}{hypotenuse} = \frac{1/2}{1} = 0.5.\)

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle with a radius of 1, centered at the origin of a coordinate plane. This simple setup helps us to easily calculate trigonometric functions like sine and cosine.
Each angle, when measured from the positive x-axis, corresponds to a point on this circle. The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.
This visual representation allows for a clear understanding of how these trigonometric values are derived. For example, the point on the unit circle for a 30° angle is found in the first quadrant, providing us with the coordinates \((\frac{\sqrt{3}}{2}, \frac{1}{2})\) for cosine and sine, respectively.

Right-angled Triangle

Triangles that include a 90-degree angle offer a straightforward way to understand trigonometric functions. In a right-angled triangle, sides are related to angles using trigonometry.
Typically, the longest side is known as the hypotenuse, the side opposite the angle of interest is the opposite, and the side adjacent to the angle is the adjacent.
Knowing these terms is critical for understanding how to find sine and cosine. For example, with a 30° angle, the opposite side is \(\frac{1}{2}\), the adjacent is \(\frac{\sqrt{3}}{2}\), and the hypotenuse is 1 on the unit circle.

Sine and Cosine Values

Sine and cosine are two of the primary trigonometric functions.

• Sine (\(\sin \theta \)): Calculated as the ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos \theta \)): Calculated as the ratio of the adjacent side to the hypotenuse.

On the unit circle, these functions help us find exact values for angles like 30° and 60°. The sine of 30° is \(0.5\), while the cosine is \(\frac{\sqrt{3}}{2}\). Conversely, the sine of 60° is \(\frac{\sqrt{3}}{2}\) and the cosine is \(0.5\). Each of these values corresponds numerically to the coordinates found around the unit circle.

Exact Trigonometric Values

Exact trigonometric values are not approximations but precise values. These are typically memorized for specific angles such as 30°, 45°, and 60° because they form the basis of much of trigonometry.
For 30°, the values are \(\sin 30^{\circ} = 0.5\) and \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\). For 60°, we have \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\) and \(\cos 60^{\circ} = 0.5\). These are derived using either the unit circle or right-angled triangles.
Understanding these values helps in solving a variety of problems across mathematics and physics.