/*! 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 78 Write each expression as a funct... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 78

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

Short Answer

Expert verified
\( \cos(\alpha - \pi) = -\cos \alpha \)

Step by step solution

01

Recall the trigonometric identity

Use the trigonometric identity for the cosine of a difference: \[ \ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
02

Substitute the values into the identity

Substitute \( a = \alpha \) and \( b = \pi \) into the identity: \[ \cos(\alpha - \pi) = \cos \alpha \cos \pi + \sin \alpha \sin \pi \]
03

Use known values of trigonometric functions

Recall the trigonometric values for \( \pi \): \[ \cos \pi = -1 \] and \[ \sin \pi = 0 \]. Substitute these values into the equation: \[ \cos(\alpha - \pi) = \cos \alpha(-1) + \sin \alpha(0) \] This simplifies to: \[ \cos(\alpha - \pi) = -\cos \alpha \]

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Key Concepts

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

Cosine Function
The cosine function, often abbreviated as cos, is one of the primary trigonometric functions. It relates an angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, the cosine of an angle \( \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the circle.

Some important properties of the cosine function include:
  • It is an even function: \( \text{cos}(-\theta) = \text{cos}(\theta) \).
  • The range of the cosine function is \( [-1, 1] \).
  • It is periodic with period \( 2\text{蟺} \), meaning \( \text{cos}(\theta + 2\text{蟺}) = \text{cos}(\theta) \).
  • The graph of the cosine function is a wave that starts at 1 when \( \theta = 0 \), decreases to -1, and then returns to 1.

Understanding these properties can help in solving trigonometric problems and recognizing patterns in trigonometric graphs.
Trigonometric Values
Trigonometric values are specific values of the trigonometric functions (sine, cosine, tangent, etc.) for common angles like \( 0, \text{蟺}/2, \text{蟺}, 3\text{蟺}/2, \text{蟺}/4, \text{蟺}/6, \text{蟺}/3 \) and others. These values often need to be memorized or quickly referenced because they frequently appear in trigonometric identities and equations.

For example:
  • \( \text{cos}(\text{蟺}) = -1 \)
  • \( \text{sin}(\text{蟺}) = 0 \)
  • \( \text{cos}(0) = 1 \)
  • \( \text{sin}(0) = 0 \)

In the original exercise, these values were used to simplify the expression. Knowing these exact values can make solving trigonometric problems much easier since they often simplify complex expressions.
Angle Subtraction
Angle subtraction involves finding the trigonometric function values for the difference of two angles. A key identity often used is the cosine of a difference formula:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
This identity can simplify the analysis and calculations involving trigonometric expressions with angle subtraction.

In the given problem, the angles were \( \text{伪} \) and \( \text{蟺} \). Substituting these into the identity we get:
\[ \cos(\text{伪} - \text{蟺}) = \cos \text{伪} \cos \text{蟺} + \sin \text{伪} \sin \text{蟺} \]
Knowing the trigonometric values:
  • \( \cos \text{蟺} = -1 \)
  • \( \sin \text{蟺} = 0 \)

We substitute and get:
\[ \cos(\text{伪} - \text{蟺}) = \cos \text{伪}(-1) + \sin \text{伪}(0) \approx -\cos \text{伪} \]
This shows how angle subtraction can be used effectively in trigonometric expressions.

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Most popular questions from this chapter

Solve each problem. Wave Action The vertical position of a floating ball in an experimental wave tank is given by the equation \(x=2 \sin (\pi t / 3)\) where \(x\) is the number of feet above sea level and \(t\) is the time in seconds. For what values of \(t\) is the ball \(\sqrt{3} \mathrm{ft}\) above sea level?

Find all values of \(\alpha\) in degrees that satisfy each equation. Round approximate answers to the nearest tenth of a degree. $$\sec 2 \alpha=4.5$$

Motion of a Spring A block is attached to a spring and set in motion on a frictionless plane. Its location on the surface at any time \(t\) in seconds is given in meters by \(x=\sqrt{3} \sin 2 t+\cos 2 t .\) For what values of \(t\) is the block at its resting position \(x=0 ?\)

Use identities to simplify each expression. Do not use a calculator. $$\cos ^{2}\left(\frac{\pi}{9}\right)-\sin ^{2}\left(\frac{\pi}{9}\right)$$

Verify that each equation is an identity. $$\frac{1-\sin ^{2}\left(\frac{x}{2}\right)}{1+\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1+\cos x}{3-\cos x}$$
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