Problem 33 - 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. $$ \cos \frac{8 \pi}{3} $$

Step-by-Step Solution

Answer

The exact value of \(\cos \frac{8\pi}{3}\) is \(-\frac{1}{2}\).

Step 1: Cofunction Identity

Recall the cofunction identity for cosine and sine: \[\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)\]Our task is to express \(\cos \frac{8\pi}{3}\) in terms of its cofunction using this identity. However, because angles are typically converted to be within the standard interval \([0, 2\pi])\), we might face angle transformations to fit this identity.

Step 2: Find the Reference Angle

Calculate how many radians \(\frac{8\pi}{3}\) is over \(2\pi\) by writing it as a sum: \[\frac{8\pi}{3} = 2\pi + \frac{2\pi}{3}\]Thus, \(\frac{8\pi}{3}\) is a full circle \(2\pi\) plus an additional \(\frac{2\pi}{3}\). Therefore, \(\frac{8\pi}{3}\) and \(\frac{2\pi}{3}\) are coterminal.

Step 3: Apply Cofunction to Reference Angle

To apply the cofunction identity, we need \(\frac{2\pi}{3}\) instead of \(\frac{8\pi}{3}\) as it's the same position on the unit circle. Therefore: \[\cos \frac{2\pi}{3} = \sin\left(\frac{\pi}{2} - \frac{2\pi}{3}\right)\]Simplify the angle in sine: \[\frac{\pi}{2} - \frac{2\pi}{3} = \frac{3\pi}{6} - \frac{4\pi}{6} = -\frac{\pi}{6}\]Hence: \[\cos \frac{2\pi}{3} = \sin \left(-\frac{\pi}{6}\right)\]

Step 4: Determine the Exact Value

Knowing that \(\sin(-\theta) = -\sin(\theta)\), find: \[\sin \left(-\frac{\pi}{6}\right) = -\sin \frac{\pi}{6}\]Recall \(\sin \frac{\pi}{6} = \frac{1}{2}\), so \[\sin \left(-\frac{\pi}{6}\right) = -\frac{1}{2}\]Thus, \[\cos \frac{8\pi}{3} = -\frac{1}{2}\]

Step 5: Conclusion: Final Answer

By using a combination of angle identity transformations and cofunction identities, we've determined the exact value of \(\cos \frac{8\pi}{3}\) is \(-\frac{1}{2}\). This means the equivalent cofunction equation originally found also gives the same result as the calculated value.

Key Concepts

Cofunction Identities

Cofunction identities are a key concept in trigonometry. They allow us to express trigonometric functions in terms of their complementary angles. The idea behind this is that some trigonometric functions, like sine and cosine, are cofunctions of each other. So, the function of an angle can often be rewritten using this special relationship. For example, the cosine function is related to the sine function through the identity: \[ \cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \] This essentially means, you can transform the cosine of any angle into the sine of its complement. This identity works because in a right triangle, the sine of one angle is equal to the cosine of the other. Applying cofunction identities is helpful in simplifying problems and finding exact values of trigonometric expressions.

Angle Transformation

Angle transformation is the process of converting an angle into a different form while maintaining its trigonometric properties. This often involves expressing angles in terms of common intervals. Angles are typically expressed between \(0\) to \(2\pi\) or \(0\) to \(360\) degrees. Sometimes, angles outside these intervals are transformed by adding or subtracting full rotations. For example, if you have \(\frac{8\pi}{3}\), this is equivalent to having a full circle \(2\pi\) plus \(\frac{2\pi}{3}\). Therefore, \(\frac{8\pi}{3}\) and \(\frac{2\pi}{3}\) are coterminal, meaning they share the same position on the unit circle. By doing an angle transformation, we simplify the problem into a more manageable form.

Unit Circle

The unit circle is a central concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate plane. The unit circle allows us to derive the values of trigonometric functions for any given angle. When an angle is measured starting from the positive x-axis, its sine is the y-coordinate, and its cosine is the x-coordinate of where the angle's terminal side intersects the unit circle. For example, if we look at an angle \(\frac{2\pi}{3}\), this angle falls in the second quadrant, where cosine values are negative and sine values are positive: - At \(\frac{2\pi}{3}\), \(\cos\) is \(-\frac{1}{2}\) and \(\sin\) is \(\frac{\sqrt{3}}{2}\). Understanding how angles relate on the unit circle is fundamental for solving trigonometry problems, especially using cofunction identities and transformations.

Reference Angle

A reference angle is the acute angle that a given angle forms with the x-axis. Trigonometric functions of an angle can be directly determined by knowing its reference angle, as it shares the same sine and cosine values (up to a sign) as the actual angle. To find the reference angle, we typically reduce the original angle to an equivalent angle within the first quadrant: - For positive angles, this is done by wrapping it around into the \([0, \pi/2]\) range using subtraction. - For example, \(\frac{8\pi}{3}\) is reduced to \(\frac{2\pi}{3}\), with its reference angle being \(\frac{\pi}{3}\). Utilizing the reference angle allows us to leverage the positive trigonometric values and apply transformations and identities, essential for finding exact trigonometric function values.