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Chapter 6: Problem 53

Find the reference angle and the exact function value if they exist. $$\tan 330^{\circ}$$

Short Answer

Expert verified
Reference angle: 30掳, \( \tan 330^{\circ} = -\frac{\sqrt{3}}{3} \)

Step by step solution

01

Calculate the Reference Angle

The reference angle is the positive acute angle that the terminal side of the given angle makes with the x-axis. For an angle in the fourth quadrant like 330掳, the reference angle is calculated by subtracting 330掳 from 360掳:\[ 360^{\text{掳}} - 330^{\text{掳}} = 30^{\text{掳}} \]
02

Determine the Tangent Function Value

Next, find the exact value of the tangent for the reference angle 30掳. The tangent of 30掳 is a known value:\[ \tan 30^{\text{掳}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]
03

Apply the Sign Based on the Quadrant

The angle 330掳 is in the fourth quadrant, where the tangent function is negative. Therefore, apply the negative sign to the tangent value found:\[ \tan 330^{\text{掳}} = -\frac{\sqrt{3}}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reference angle
A reference angle is an important concept in trigonometry. It is the smallest angle that you can make between the terminal side of your given angle and the x-axis. This angle is always positive and acute, meaning it's between 0掳 and 90掳.

For example, in the problem where you have to find the reference angle for 330掳, since 330掳 is in the fourth quadrant, you subtract it from 360掳:
\[ 360^{\text{掳}} - 330^{\text{掳}} = 30^{\text{掳}} \]
So, the reference angle is 30掳.
tangent function
The tangent function, often written as \(\tan\), relates an angle in a right triangle to the ratio of the opposite side to the adjacent side:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In trigonometry, we often use the tangent function with common angles such as 30掳, 45掳, and 60掳. These values are commonly found using the unit circle or trigonometric tables.
For instance, the tangent of 30掳 is:
\[ \tan 30^{\text{掳}} = \frac{1}{\frac{\text{3}}{1}} = \frac{\text{3}}{3} \]
This value is helpful when working through related problems.
angle quadrants
Understanding angle quadrants is crucial for determining the signs of trigonometric functions. The x-y coordinate plane divides into four quadrants:
  • First Quadrant (0掳 to 90掳)
  • Second Quadrant (90掳 to 180掳)
  • Third Quadrant (180掳 to 270掳)
  • Fourth Quadrant (270掳 to 360掳)

The given angle 330掳 is in the fourth quadrant. In this quadrant, the tangent function is negative. So, for our angle:
\[ \tan 330^{\text{掳}} \]
we first find the tangent of the reference angle (30掳) and then apply the negative sign:
\[ \tan 330^{\text{掳}} = -\frac{\text{3}}{3} \]
exact function values
Exact function values are necessary in trigonometry. They are usually found for specific angles like 30掳, 45掳, and 60掳. These values are commonly used in higher-level math problems.

In our example:
\[ \tan 30^{\text{掳}} = \frac{1}{\frac{\text{3}}{3}} = \frac{\text{3}}{3} \]
We then apply the correct sign based on the quadrant to get the exact value:
\[ \tan 330^{\text{掳}} = -\frac{\text{3}}{3} \]

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Most popular questions from this chapter

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Given that $$\begin{array}{ll}\sin 8^{\circ} \approx 0.1392, & \cos 8^{\circ} \approx 0.9903 \\\\\tan 8^{\circ} \approx 0.1405, & \cot 8^{\circ} \approx 7.1154 \\\\\sec 8^{\circ} \approx 1.0098, & \csc 8^{\circ} \approx 7.1853 \end{array}$$ find the six function values of \(82^{\circ} .\)
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