Problem 6 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\tan \frac{5 \pi}{3}\)

Step-by-Step Solution

Answer

\(-\sqrt{3}\).

Step 1: Convert the Angle to Degrees

First, convert the angle from radians to degrees. Multiply by \(180\degree/\pi\):\[\frac{5\pi}{3} \times \frac{180\degree}{\pi} = 300\degree.\]This means \(\frac{5\pi}{3}\) radians is equivalent to \(300\degree\).

Step 2: Identify the Reference Angle

Determine the reference angle by subtracting \(360\degree - 300\degree = 60\degree\). The reference angle for \(300\degree\) is \(60\degree\), since \(300\degree\) is located in the fourth quadrant where the reference angle is \(360\degree - \theta\).

Step 3: Determine Function Sign in Fourth Quadrant

In the fourth quadrant, the tangent function is negative. This is because tangent is the ratio of sine to cosine, and sine is negative while cosine is positive in this quadrant, resulting in \(\tan(\theta) = \frac{-\sin(\theta)}{\cos(\theta)}\).

Step 4: Find Exact Value of \(\tan 60\degree\)

Using known values for the reference angle, \(\tan 60\degree = \sqrt{3}\).

Step 5: Apply Sign to Result

Given that tangent is negative in the fourth quadrant, the value of \(\tan 300\degree\) is \(-\tan 60\degree = -\sqrt{3}\). Therefore, \(\tan \frac{5\pi}{3} = -\sqrt{3}\).

Key Concepts

Radians to Degrees Conversion

Understanding the relationship between radians and degrees is fundamental in trigonometry. A radian is a measure based on the radius of a circle. To convert radians to degrees, we use the fact that a full circle is both 2\(\pi\) radians and 360 degrees. This gives us the conversion factor:

• 1 radian = \( \frac{180\degree}{\pi} \).

To convert an angle from radians to degrees, multiply the radian measure by \( \frac{180\degree}{\pi} \). For example, to convert \( \frac{5\pi}{3} \) radians to degrees, we calculate:\[\frac{5\pi}{3} \times \frac{180\degree}{\pi} = 300\degree.\]This calculation shows that \( \frac{5\pi}{3} \) radians equals 300 degrees. Converting radians to degrees can make it easier to understand angles, especially if you are more familiar with degrees.

Reference Angle

Reference angles help simplify the process of finding trigonometric values for any given angle. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always between 0 degrees and 90 degrees, or 0 and \( \frac{\pi}{2} \) radians.For example, if we take the angle 300 degrees, which is in the fourth quadrant, the reference angle can be found using:

• Reference angle = 360 degrees - 300 degrees = 60 degrees.

This means that the reference angle for 300 degrees is 60 degrees. Knowing the reference angle allows you to calculate trigonometric functions using familiar acute angles. This is useful in the context of special angles, where certain trigonometric values are known by heart.

Tangent Function

The tangent function is a crucial part of trigonometry, representing the ratio of the sine and cosine of an angle:\[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.\]In different quadrants of the unit circle, the sine and cosine values can change from positive to negative, affecting the sign of the tangent. Observing the angle 300 degrees, it lies in the fourth quadrant where:

• Sine is negative.
• Cosine is positive.

Thus, the tangent in this quadrant becomes negative since a negative divided by a positive is a negative result. For example, with a reference angle of 60 degrees, and knowing that \( \tan(60\degree) = \sqrt{3} \), the tangent function produces a value of:\[\tan(300\degree) = -\sqrt{3}.\]Understanding which sign the trigonometric function takes on in each quadrant will make it easier to compute exact values for any angle.