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Chapter 6: Problem 90

Find the exact acute angle \(\theta\) for the given function value. $$\tan \theta=\sqrt{3}$$

Short Answer

Expert verified
\( \theta = 60掳 \) or \( \frac{\pi}{3} \) radians

Step by step solution

01

Identify the function and value

The given trigonometric function is \(\tan \theta\). We are given that \(\tan \theta = \sqrt{3}\).
02

Recall the tangent values for special angles

Recall the values of the tangent function for the special angles in the first quadrant (0掳, 30掳, 45掳, 60掳, and 90掳). Specifically, \(\tan 60掳 = \sqrt{3}\).
03

Determine the acute angle

Since \(\theta\) is acute and \(\tan \theta = \sqrt{3}\), the angle \(\theta\) must be 60掳 or \(\frac{\pi}{3}\) radians.

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Key Concepts

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

acute angle
An acute angle is an angle that is less than 90 degrees. These angles are found in various geometric shapes and are often used in trigonometry.
In trigonometry, when we seek the values of functions like sine, cosine, and tangent, we often deal with angles less than 90 degrees.
This is because within this range, these functions have specific and predictable values.
For instance, when given a problem involving \( \tan \theta \), knowing that \( \theta \) is an acute angle helps us narrow down the possible solutions.
tangent function
The tangent function, denoted \( \tan \theta \), relates an angle in a right triangle to the ratio of the opposite side to the adjacent side. In simpler terms, \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
For specific angles, this function has distinct values. For example, at 0掳, 30掳, 45掳, 60掳, and 90掳:
  • \( \tan 0掳 = 0 \)
  • \( \tan 30掳 = \frac{1}{\sqrt{3}} \)
  • \( \tan 45掳 = 1 \)
  • \( \tan 60掳 = \sqrt{3} \)
  • \( \tan 90掳 \) is undefined.
Knowing these specific values helps solve problems more easily when one of these angles or their function values are given.
special angles
Special angles in trigonometry鈥攍ike 30掳, 45掳, and 60掳鈥攈ave known function values. These angles frequently appear in right triangle problems and are termed 'special' because of these recognisable values.
In particular:
  • \( \tan 30掳 = \frac{1}{\sqrt{3}} \)
  • \( \tan 45掳 = 1 \)
  • \( \tan 60掳 = \sqrt{3} \)
By recognizing these values, we can immediately determine angles like we did with our exercise where \( \tan \theta = \sqrt{3} \), leading us to conclude \( \theta \) must be 60掳.
first quadrant
In trigonometry, the first quadrant is the section of the coordinate plane where both x and y values are positive. Angles in this quadrant range from 0掳 to 90掳.
Important to note, trigonometric functions are positive in this quadrant. Knowing this makes it easier to determine theta when we are given function values.
In our case, knowing the function \( \tan \theta = \sqrt{3} \) and that \( \theta \) must be an acute angle confined our search within the first quadrant.
This knowledge helps us conclude that for \( \tan \theta = \sqrt{3} \), \( \theta \) must be 60掳.

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