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Chapter 14: Problem 17

Find the exact value of each expression. \(\cos 225^{\circ}\)

Short Answer

Expert verified
\( \cos 225^\circ = -\frac{\sqrt{2}}{2} \)

Step by step solution

01

Identify Reference Angle and Quadrant

First, note that the angle 225掳 is in the third quadrant. To find the reference angle, subtract from the nearest x-axis angle, which is 180掳: 225掳 - 180掳 = 45掳. Thus, the reference angle is 45掳.
02

Determine Cosine Value in Third Quadrant

In the third quadrant, the cosine value is negative because the x-coordinates are negative. Since the reference angle is 45掳, first consider the cosine of 45掳, which is \( \cos 45^\circ = \frac{\sqrt{2}}{2} \). Thus, \( \cos 225^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2} \).

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Key Concepts

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

Reference Angle
The reference angle is instrumental when dealing with trigonometric functions, as it helps simplify the process of finding angles that are not primary angles like 0掳, 30掳, 45掳, 60掳, and 90掳. The reference angle is the smallest angle that the terminal side of an angle makes with the x-axis.
To find the reference angle for any angle, you need to determine the angle's location within one of the four quadrants of the Cartesian plane. When an angle, like 225掳, falls beyond the first quadrant, finding the reference angle involves:
  • Identifying the angle's quadrant: 225掳 is in the third quadrant because it is between 180掳 and 270掳.
  • Calculating its reference angle: Subtract the nearest x-axis angle. For the third quadrant, you subtract 180掳, so 225掳 - 180掳 = 45掳.
With the reference angle of 45掳, you can easily determine trigonometric values using known values of primary angles.
Trigonometric Functions
Trigonometric functions relate the angles of a triangle to the ratios of its sides. The primary functions are sine (\(\sin\theta\)), cosine (\(\cos\theta\)), and tangent (\(\tan\theta\)). Each function represents a specific ratio:
  • Cosine, or \(\cos\theta\), is the ratio of the adjacent side to the hypotenuse in a right triangle.
  • Sine, or \(\sin\theta\), is the ratio of the opposite side to the hypotenuse.
  • Tangent, or \(\tan\theta\), is the ratio of the opposite side to the adjacent side.
The value of these functions changes depending on the quadrant in which the angle resides. This property of trigonometric functions is crucial for determining the signs of function values; in the third quadrant, the cosine function is intrinsically negative. Therefore, knowing your reference angle, like 45掳, assists in using these functions effectively to find exact angle values.
Third Quadrant
The third quadrant of the unit circle is where both x and y coordinates are negative. Understanding this quadrant's properties is crucial, as they impact the trigonometric function values:
  • In the third quadrant, angles range between 180掳 and 270掳.
  • The cosine of any angle here is negative, while sine is also negative because both the x- and y- coordinates on the unit circle are negative.
  • Tangent, however, is positive in this quadrant since it is the ratio of sine to cosine, i.e., \(\tan\theta = \frac{-y}{-x} = \frac{y}{x}\).
For 225掳, which lies in the third quadrant, you apply these sign rules. Since the cosine of 45掳, the reference angle, is \(\frac{\sqrt{2}}{2}\), the cosine of 225掳, incorporating the negative sign from the third quadrant, results in \(-\frac{\sqrt{2}}{2}\). Understanding the quadrant's influence is vital to finding precise trigonometric values and avoiding common mistakes.

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Most popular questions from this chapter

Find the exact value of each expression. \(\cos \left(-120^{\circ}\right)\)

Find the amplitude, if it exists, and period of each function. Then graph each function. \(y=3 \sin \theta\)

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Write an example of a trigonometric equation that has no solution.

Find the exact values of \(\sin 2 \theta, \cos 2 \theta, \sin \frac{\theta}{2},\) and \(\cos \frac{\theta}{2}\) for each of the following. $$ \sin \theta=-\frac{3}{4} ; 270^{\circ}<\theta<360^{\circ} $$
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