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Chapter 5: Problem 66

Find the exact value of each expression without using a calculator. Check your answer with a calculator. $$\frac{\sin (-3 \pi / 4)}{\cos (-3 \pi / 4)}$$

Short Answer

Expert verified
The exact value is 1.

Step by step solution

01

Understand the given expression

The given expression is \(\frac{\sin(-3 \pi / 4)}{\cos(-3 \pi / 4)}\). Recall that \(\sin\) and \(\cos\) are trigonometric functions that we can associate with angles on the unit circle.
02

Convert to positive angle

Convert the negative angle \(-3\pi/4\) to a positive angle by adding \(2\pi\): \[ -3\pi/4 + 2\pi = -3\pi/4 + 8\pi/4 = 5\pi/4 \].
03

Analyze the unit circle

The angle \(5\pi / 4\) is in the third quadrant, where \(\sin\) and \(\cos\) are both negative. This can be seen by referencing the unit circle.
04

Evaluate \(\sin(5\pi / 4)\)

\( \sin(5\pi / 4) = -\sin(\pi /4) \). Since \(\sin(\pi / 4) = \frac{\sqrt{2}}{2}\), we have \(\sin(5\pi / 4) = -\frac{\sqrt{2}}{2} \).
05

Evaluate \(\cos(5\pi / 4)\)

\( \cos(5\pi / 4) = -\cos(\pi /4) \). Since \(\cos(\pi / 4) = \frac{\sqrt{2}}{2}\), we have \(\cos(5\pi / 4) = -\frac{\sqrt{2}}{2} \).
06

Substitute and simplify

Substitute the values found into the original expression: \[ \frac{\sin(5\pi / 4)}{\cos(5\pi / 4)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \].

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Key Concepts

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

Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of the coordinate plane. Each point on the unit circle corresponds to an angle measured from the positive x-axis. This makes it easy to understand sine and cosine, which are the y and x coordinates of these points respectively.
Let's explore more:
  • The angle in radians is the length of the arc formed by the angle on the unit circle.
  • Important angles to remember are \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), \( \pi \), and their multiples.
  • In each quadrant of the circle, sine and cosine take on different signs.
Using the unit circle correctly allows us to determine exact values of trigonometric functions without needing a calculator.
Angle Conversion
Sometimes, angles are presented in a way that might not be immediately recognizable. Converting between different representations helps in solving problems.
For instance, converting a negative angle to its positive counterpart:
  • If we have a negative angle like \( -3\frac{\pi}{4} \), we can add \ 2\pi \ (one full rotation) to get a positive angle:
  • \begin{align*} -3\frac{\pi}{4} + 2\pi &= -3\frac{\pi}{4} + 8\frac{\pi}{4} \ & = \5\frac{\pi}{4} \end{align*}
Converting angles ensures they fall within the standard range of 0 to \ 2\pi \, making it easier to reference on the unit circle.
Simplification
Simplification is about reducing expressions to their simplest form. For trigonometric functions, it often involves using known values and properties.
Consider the exercise: \sin(5\frac{\pi}{4}) \ and \cos(5\frac{\pi}{4})\:
  • First, identify the quadrant. \( \5\frac{\pi}{4} \) falls in the third quadrant.
  • In the third quadrant, both sine and cosine are negative.
  • \begin{align*} \sin(5\frac{\pi}{4}) & = -\frac{\sqrt{2}}{2} \ \cos(5\frac{\pi}{4}) & = -\frac{\sqrt{2}}{2} \end{align*}
  • Therefore, \( \frac{\sin(5\frac{\pi}{4})}{\cos(5\frac{\pi}{4})} = 1 \).
Using simplification, complex expressions can be made more manageable, ensuring answers are clear and concise.

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