/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 Write each expression as a funct... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 74

Write each expression as a function of \(\alpha\) alone. $$\sin (\alpha-\pi)$$

Short Answer

Expert verified
\sin \alpha (\pi \ ) \ = command.

Step by step solution

01

Use the angle subtraction formula

The sine of a difference of angles can be rewritten using the angle subtraction formula: \ \ \( \theta-\beta \sin = \sin(\theta-)= \sin(\theta) \cos(\beta) - \cos(\theta) \sin(\beta) \).
02

Apply the specific values for sin and cos of \pi

Substitute \theta = \alpha and \beta = \pi into the angle subtraction formula. \ \(\sin(\theta-\beta) = \sin(\theta) \cos(\beta) - \cos(\theta) \sin(\beta) \theta = \alpha, \beta = \pi \)
03

Simplify using known trigonometric identities

We know that \( \cos(\beta) = \cos(\pi) = -1 \) and \( \sin(\beta) = \sin(\pi) = 0 \). Substitute these values into our equation: \ \(\sin(\alpha - \pi) = \sin(\alpha) \cos(\beta) - \cos(\beta) \sin(\beta) = \sin(\beta) (-1) - \cos(\beta) (0) \)

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Key Concepts

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

Angle Subtraction Formula: Understanding the Basics
The angle subtraction formula helps us simplify trigonometric expressions involving the difference of two angles. This formula is crucial in trigonometry and is expressed as: \ \(\sin(\theta - \beta) = \sin(\theta) \cos(\beta) - \cos(\theta) \sin(\beta)\). \ When you see a sine or cosine function with an angle subtraction, you can use this formula to break down and simplify the expression. \ By substituting the components with known values, you can achieve a much simpler form.
Sine Function and Its Properties
The sine function, written as \(\sin(\theta)\), defines the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. \ It鈥檚 a vital function in trigonometry because it describes wave patterns and oscillations. \ Here are some key properties:
  • \(\sin(0) = 0\)
  • \(\sin(\pi/2) = 1\)
  • \(\sin(\pi) = 0\)
  • \(\sin(3\pi/2) = -1\)
  • \(\sin(2\pi) = 0\)
\ These values repeat in cycles, which is known as periodicity.
Cosine Function and Its Role
The cosine function, denoted as \(\cos(\theta)\), represents the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. \ It is often used alongside the sine function to describe angles and lengths in various applications. \ Some important values are:
  • \(\cos(0) = 1\)
  • \(\cos(\pi/2) = 0\)
  • \(\cos(\pi) = -1\)
  • \(\cos(3\pi/2) = 0\)
  • \(\cos(2\pi) = 1\)
\ The cosine function also exhibits periodic behavior, repeating every \(2\pi\) radians.
Simplifying Trigonometric Expressions
When simplifying trigonometric expressions, the key is to use known formulas and identities, like the angle subtraction formula. \ For the given problem: \ \(\sin (\alpha - \pi)\), we apply the angle subtraction formula: \ \(\sin(\alpha - \pi) = \sin(\alpha) \cos(\pi) - \cos(\alpha) \sin(\pi)\).\ We know that \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), which simplifies our expression to: \(\sin(\alpha - \pi) = \sin(\alpha) \cdot (-1) - \cos(\alpha) \cdot 0\). \ Finally, we get: \(\sin(\alpha - \pi) = -\sin(\alpha)\). \ Using these identities and simplifications is essential for solving trigonometric problems effectively.

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Most popular questions from this chapter

Find all values of \(\theta\) in the interval \(0^{\circ}, 360^{\circ}\) ) that satisfy each \right. equation. Round approximate answers to the nearest tenth of a degree. $$\cot ^{4} \theta-4 \cot ^{2} \theta+3=0$$

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