Problem 26 - Algebra 2 and Trigonometry

Question

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \sin \frac{\pi}{3} $$

Step-by-Step Solution

Answer

\(\sin \frac{\pi}{3} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).

Step 1: Understanding Cofunction Identities

Cofunction identities relate the trigonometric function values of \(\theta\) and \(\frac{\pi}{2} - \theta\). For sine, the cofunction identity is \(\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)\).

Step 2: Applying Cofunction Identity

Apply the cofunction identity for sine: \[ \sin \frac{\pi}{3} = \cos \left(\frac{\pi}{2} - \frac{\pi}{3}\right) \]

Step 3: Simplify the Expression

Calculate the expression inside the cosine function: \[ \frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \] Thus, \[ \sin \frac{\pi}{3} = \cos \frac{\pi}{6} \]

Step 4: Find the Exact Value Using the Unit Circle

The cosine of \(\frac{\pi}{6}\) radians is known from the unit circle or trigonometric tables to be \(\frac{\sqrt{3}}{2}\). Thus, \[ \sin \frac{\pi}{3} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \]

Key Concepts

Cofunction Identities

Cofunction identities are a fundamental part of trigonometry, helping to relate one trigonometric function to another. They are particularly useful in simplifying expressions and solving equations. The core idea is that each trigonometric function has a cofunction that, when calculated at a complementary angle, yields the same value.For sine and cosine, the cofunction identity is defined as follows:

• \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \)
• \( \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \)

This means if you know the sine of an angle, you can find the cosine of its complementary angle, and vice versa. Cofunction identities rely on the complementary relationship of angles, where two angles sum up to \( \frac{\pi}{2} \) radians or 90 degrees. By using these identities, we can express trigonometric functions in different but equivalent forms.

Unit Circle

The unit circle is a basic yet powerful tool in trigonometry. It is a circle of radius 1 centered at the origin (0,0) on the coordinate plane. The significance of the unit circle is that every point on it corresponds to the sine and cosine of angles in standard position.

• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

The unit circle allows us to visualize and determine trigonometric function values at various angles. By traversing the circle from 0 to \(2\pi \) radians, we can observe how sine and cosine values change, periodically cycling through their respective ranges. When working with the unit circle, keep in mind the important angles like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3} \), etc., and their corresponding coordinates. These specific angles yield exact values which are critical for solving trigonometric problems more efficiently.

Exact Values of Trig Functions

Finding exact values of trigonometric functions is a key aspect of trigonometry, especially for angles that appear frequently in applications. The exact values are derived from the properties of the unit circle and the symmetry of trigonometric functions.For certain angles such as \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), etc., the values of sine, cosine, and other trig functions are well-known:

• \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \)
• \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \)
• \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)

Understanding these values makes computations simpler and ensures accuracy without the need for calculators for these specific angles. The symmetry of the unit circle and its inherent properties help in remembering these values and applying them effectively across different problems.