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Chapter 4: Problem 53

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$225^{\circ}$$

Short Answer

Expert verified
The values are as follows: \( \sin(225^{\circ}) = -\sqrt{2}/2 \), \( \cos(225^{\circ}) = -\sqrt{2}/2 \), and \( \tan(225^{\circ}) = 1 \).

Step by step solution

01

Convert Degrees to Radians

225 degrees is located in the third quadrant of the unit circle. It helps if you convert it to radians using the formula: \( \theta_{rad} = \theta_{degree} \cdot \frac{\pi}{180} \). So, \( \theta_{rad} = 225 \cdot \frac{\pi}{180} = \frac{5\pi}{4} \) radian.
02

Compute the Sine Function

Recall that on the unit circle, the y-coordinate represents the value of the sine function. For \( \theta = \frac{5\pi}{4} \), the y-coordinate is -1/ \(\sqrt{2}\) or -\(\sqrt{2}/2\). Therefore, \( \sin(\frac{5\pi}{4}) = -\sqrt{2}/2 \)
03

Compute the Cosine Function

On the unit circle, the x-coordinate represents the value of the cosine function. For \( \theta = \frac{5\pi}{4} \), the x-coordinate is also -1/ \(\sqrt{2}\) or -\(\sqrt{2}/2\). Therefore, \( \cos(\frac{5\pi}{4}) = -\sqrt{2}/2 \)
04

Compute the Tangent Function

The tangent function is the ratio of the sine divided by the cosine: \( \tan(\theta) = \sin(\theta)/\cos(\theta) \). For \( \theta = \frac{5\pi}{4} \), both sine and cosine have the same absolute value but different signs, which makes \( \tan(\frac{5\pi}{4}) = -\sqrt{2}/2 / -\sqrt{2}/2 = 1 \).

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\) (a) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\tan x$$

Use a graphing utility to graph the function. $$f(x)=\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{\pi}\right)$$

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Is a degree or a radian the greater unit of measure? Explain.
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