Problem 15 - Algebra 2

Question

The measure \(\boldsymbol{\theta}\) of an angle in standard position is given. Find the exact values of \(\cos \theta\) and \(\sin \theta\) for each angle measure. \(\frac{\pi}{3}\) radians

Step-by-Step Solution

Answer

For the angle measure \(\frac{\pi}{3}\) radians, the cosine value (cos θ) is \(\frac{1}{2}\) and the sine value (sin θ) is \(\frac{\sqrt{3}}{2}\).

Step 1: Converting radians to degrees

First, convert the given radians into degrees. The conversion factor is 180 degrees equals \(\pi\) radians. So, \(\frac{\pi}{3}\) radians would be \(\frac{\pi}{3}* \frac{180}{\pi}\) degrees, which simplifies to 60 degrees.

Step 2: Calculating cosine

Now, calculate the cosine of the angle. The cosine of 60 degrees is \(\frac{1}{2}\), therefore, \(\cos\frac{\pi}{3} = \frac{1}{2}\).

Step 3: Calculating sine

Finally, calculate the sine of the angle. The sine of 60 degrees is \(\frac{\sqrt{3}}{2}\), therefore \(\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}\).

Key Concepts

Radians to Degrees Conversion

When working with trigonometric functions, you often need to convert between radians and degrees. Radians and degrees are two different units used to measure angles. The relationship between them is crucial for solving problems effectively. The formula to convert radians to degrees uses the equivalence: \(180° = \pi \: \text{radians}\). Thus, to convert an angle from radians to degrees, you can multiply the radian measure by \(\frac{180}{\pi}\).

• Example: To convert \(\frac{\pi}{3}\) radians to degrees, calculate \(\frac{\pi}{3} \times \frac{180}{\pi}\).
• Cancel out \(\pi\) to simplify: \(\frac{180}{3}\).
• The result is 60 degrees.

Converting radians to degrees helps to easily interpret and work with angles in trigonometric functions.

Cosine of an Angle

The cosine function is a fundamental concept in trigonometry. It represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle, when referring to an angle in standard position. The cosine function is represented by \(\cos \theta\), where \(\theta\) is the angle.
When dealing with angles measured in degrees or radians, having standard values for specific angles facilitates quick calculations:

• The \(\cos(60°) = \frac{1}{2}\), and since \(60°\) is equivalent to \(\frac{\pi}{3}\) radians, then \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

These standard values are often memorized to simplify work with trigonometric problems.

Sine of an Angle

The sine function is another essential trigonometric function. It represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle, again considering the angle in standard position. The sine function is noted as \(\sin \theta\), with \(\theta\) denoting the angle.
Certain angles have significant sine values that reoccur in trigonometric calculations:

• For example, \(\sin(60°) = \frac{\sqrt{3}}{2}\).
• Knowing that \(60°\) is \(\frac{\pi}{3}\) radians, \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).

Recognizing these set values simplifies the process, making calculations quicker and easeful.

Standard Position Angles

An angle is said to be in standard position when its vertex is located at the origin of a coordinate plane, and its initial side lies along the positive x-axis. Standard position angles provide a useful framework for exploring trigonometric functions and analyzing their behaviors.

Understanding standard position is important because:

• It simplifies the process of determining values for trigonometric functions like sine and cosine.
• Provides a base to visualize unit circle principles where angles and their function values are mapped conveniently.
• Informs the understanding of periodicity and symmetry properties of trigonometric functions.

Analyzing angles in standard position builds a foundation for learning more complex aspects of trigonometry.

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