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Chapter 6: Problem 48

Find the reference angle and the exact function value if they exist. $$\tan 675^{\circ}$$

Short Answer

Expert verified
Reference angle: 45掳, Exact value: -1

Step by step solution

01

Convert the angle to an equivalent angle between 0掳 and 360掳

First, subtract 360掳 repeatedly from the given angle until the result is between 0掳 and 360掳. Here, we have: \[675^{\text{掳}} - 360^{\text{掳}} = 315^{\text{掳}}\]
02

Determine the reference angle

For an angle of 315掳 in standard position, the reference angle is found by subtracting the angle from 360掳 since 315掳 is in the fourth quadrant. \[360^{\text{掳}} - 315^{\text{掳}} = 45^{\text{掳}}\]
03

Find the function value

We need to find the tangent of the reference angle. The tangent of 45掳 is a well-known value: \tan(45掳) = 1. Since 315掳 is in the fourth quadrant, where the tangent is negative, the value is: \[\tan 315^{\text{掳}} = -1\]

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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 the acute angle formed between the terminal side of an angle and the horizontal axis. To find it, you often need to subtract the given angle from 360掳 or 180掳, depending on which quadrant the angle lies in.
The angle 315掳 is situated in the fourth quadrant. Here, the reference angle is found by subtracting the given angle from 360掳: 315掳 - 360掳 = -45掳. Since a reference angle is always positive, we add 45掳 to get 45掳 as the reference angle.
Tangent Function
The tangent function relates the angles of a triangle to the ratio of its opposite side over its adjacent side. It is one of the fundamental trigonometric functions, often abbreviated as tan. For the reference angle 45掳, the function value is easy to remember: \((tan 45掳 = 1)\). In the fourth quadrant, where 315掳 lies, the tangent function is negative. Therefore, tan 315掳 = -1.
Angle Conversion
Converting angles to a standard range is crucial for simplifying trigonometric calculations. Often, you convert an angle beyond 360掳 or less than 0掳 to an equivalent value between 0掳 and 360掳. Here鈥檚 how to do it: To convert 675掳:
  • Subtract 360掳 until you fall within the 0掳-360掳 range:
  • 675掳 - 360掳 = 315掳
  • Thus, 675掳 converts to 315掳.
Converting angles helps in identifying their reference angles and calculating trigonometric functions more efficiently. This conversion simplifies the problem and avoids confusion.
Remember: Repeatedly subtract or add 360掳 until you get an angle within the desired range of 0掳 to 360掳.

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