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Chapter 7: Problem 77

Use identities to write each expression as a single function of \(x\) or \(\theta\). $$\sin \left(\frac{\pi}{4}+x\right)$$

Short Answer

Expert verified
\[ \sin \left(\frac{\pi}{4}+x\right) = \frac{\sqrt{2}}{2}( \cos x + \sin x) \]

Step by step solution

01

Identify the Trigonometric Identity

Recognize that the given expression \( \sin \left(\frac{\pi}{4}+x\right) \) can be rewritten using the angle addition identity for sine. The identity is: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
02

Apply the Angle Addition Identity

Rewrite the expression using the identified trigonometric identity: \[ \sin \left(\frac{\pi}{4}+x\right) = \sin \frac{\pi}{4} \cos x + \cos \frac{\pi}{4} \sin x \]
03

Substitute Known Values

Recall that \(\sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). Substitute these values into the expression: \[ \sin \left(\frac{\pi}{4}+x\right) = \frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x \]
04

Factor Out the Common Term

Factor out the common term \( \frac{\sqrt{2}}{2} \) from the expression: \[ \sin \left(\frac{\pi}{4}+x\right) = \frac{\sqrt{2}}{2}( \cos x + \sin x) \]

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Key Concepts

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

Angle Addition Identity
The angle addition identity is a fundamental tool in trigonometry. It allows you to express the sine or cosine of a sum or difference of angles in terms of the sines and cosines of the individual angles. For sine, the angle addition identity is:

$$ \sin(a + b) = \sin a \cos b + \cos a \sin b $$

This identity is extremely useful in simplifying expressions involving trigonometric functions, such as transforming \( \sin \left(\frac{\pi}{4}+x\right) \) into a more manageable form. In the given exercise, identify the structure of the sine function and use the angle addition identity to break it into parts that can be more easily handled.
Sine Function
The sine function, represented as \( \sin \theta \), describes the y-coordinate of a point on the unit circle for a given angle \( \theta \). In trigonometry, the sine function has several key properties:

  • The function is periodic with a period of \( 2\pi \).
  • It is symmetric about the origin, meaning \( \sin(-\theta) = -\sin \theta \).
  • It has specific values for key angles, such as \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

Using these values can simplify complex trigonometric expressions. For the problem at hand, knowing that \( \sin \frac{\pi}{4} \) and \( \cos \frac{\pi}{4} \) equal \( \frac{\sqrt{2}}{2} \) allows us to simplify the expression involving \( \sin \left(\frac{\pi}{4}+x\right) \).
Factoring
Factoring is a mathematical process of breaking down expressions into simpler components, often used to simplify algebraic and trigonometric expressions. In trigonometry, you often factor out common terms to streamline expressions. For example:

$$ \frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x $$Both terms are multiplied by \( \frac{\sqrt{2}}{2} \). Factoring out \( \frac{\sqrt{2}}{2} \) gives you:

$$ \frac{\sqrt{2}}{2}( \cos x + \sin x ) $$

This simplifies the expression, making it easier to analyze or compute. Factoring is a powerful strategy when dealing with trigonometric identities and helps in achieving the final simplified form of the expression.

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Most popular questions from this chapter

Amperage, Wattage, and Voltage Amperage is a measure of the amount of electricity that is moving through a circuit, whereas voltage is a measure of the force pushing the electricity. The wattage \(W\) consumed by an electrical device can be determined by calculating the product of the amperage \(I\) and voltage \(V .\) (Source: Wilcox, G. and C. Hesselberth, Electricity for Engineering Technology, Allyn \& Bacon.)(a) A household circuit has voltage $$ V=163 \sin 120 \pi t $$ when an incandescent light bulb is turned on with amperage $$ I=1.23 \sin 120 \pi t $$ Graph the wattage \(W=V I\) consumed by the light bulb in the window \([0,0.05]\) by $$ [-50,300] $$ (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to determine values for \(a, c,\) and \(\omega\) so that \(W=a \cos (\omega t)+c\) (d) Check your answer by graphing both expressions for \(W\) on the same coordinate axes. (e) Use the graph to estimate the average wattage used by the light. For how many watts (to the nearest integer) do you think this incandescent light bulb is rated?

Verify that each trigonometric equation is an identity. $$\sin ^{3} \theta+\cos ^{3} \theta=(\cos \theta+\sin \theta)(1-\cos \theta \sin \theta)$$

Verify that each trigonometric equation is an identity. $$\frac{1-\cos \theta}{1+\cos \theta}=2 \csc ^{2} \theta-2 \csc \theta \cot \theta-1$$

Use a calculator to find each value. Give answers as real numbers. $$\cos \left(\tan ^{-1} 0.5\right)$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \([0,2 \pi)\) and your work leads to \(2 x=\frac{2 \pi}{3}, 2 \pi, \frac{8 \pi}{3} .\) What are the corresponding values of \(x ?\)
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