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Chapter 7: Problem 36

Simplify. Check your results using a graphing calculator. $$\frac{1+\sin 2 x+\cos 2 x}{1+\sin 2 x-\cos 2 x}$$

Short Answer

Expert verified
\(\frac{1+\text{sin}\thinspace x \thinspace \text{cos}\thinspace x - \text{sin}^2\thinspace x}{\text{sin}\thinspace x \thinspace \text{cos}\thinspace x + \text{sin}^2\thinspace x}\)

Step by step solution

01

Recognize Trigonometric Identities

Identify the trigonometric identities for \(\frac{1+\text{sin}\thinspace 2x + \text{cos}\thinspace 2x}{1+\text{sin}\thinspace 2x - \text{cos}\thinspace 2x}\).
02

Express Sine and Cosine in Double Angle Form

Recall that \(\text{sin}\thinspace 2x = 2\thinspace \text{sin}\thinspace x \thinspace \text{cos}\thinspace x\) and \(\text{cos}\thinspace 2x = 1-2\thinspace \text{sin}^2\thinspace x\) or \(\text{cos}\thinspace 2x = 2\thinspace \text{cos}^2\thinspace x - 1\).
03

Apply Trigonometric Identities

Substitute \(\text{sin}\thinspace 2x\) and \(\text{cos}\thinspace 2x\) into the original expression: \(\frac{1 + 2\thinspace \text{sin}\thinspace x \thinspace \text{cos}\thinspace x + 1 - 2\thinspace \text{sin}^2\thinspace x}{1 + 2\thinspace \text{sin}\thinspace x \thinspace \text{cos}\thinspace x - 1 + 2\thinspace \text{sin}^2\thinspace x}\).
04

Simplify the Numerator and Denominator

Combine like terms in the numerator and denominator: \(\frac{2 + 2\thinspace \text{sin}\thinspace x \thinspace \text{cos}\thinspace x - 2\thinspace \text{sin}^2\thinspace x}{2\thinspace \text{sin}\thinspace x \thinspace \text{cos}\thinspace x + 2\thinspace \text{sin}^2\thinspace x}\).
05

Factor Out Common Terms

Factor out common terms from the numerator and denominator. Let \(A = \text{sin}\thinspace x \thinspace \text{cos}\thinspace x\) and \(B = \text{sin}^2\thinspace x\), yielding \(\frac{2(1 + A - B)}{2(A + B)} \text{.}\)
06

Simplify Final Expression

Cancel the common factor of 2 in the numerator and the denominator, resulting in \(\frac{1 + \text{sin}\thinspace x \thinspace \text{cos}\thinspace x - \text{sin}^2\thinspace x}{\text{sin}\thinspace x \thinspace \text{cos}\thinspace x + \text{sin}^2\thinspace x}\).
07

Validate Using a Graphing Calculator

Input the original expression and the simplified version into a graphing calculator to check that they yield the same plot, confirming the accuracy of the simplification.

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Key Concepts

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

Double Angle Formulas
Double angle formulas are essential in trigonometry for expressing functions of double angles (like \(2x\)) in terms of single angles (like \(x\)).
These formulas are grounded in the basic sine and cosine identities and become particularly useful when simplifying more complex trigonometric expressions.
Simplification
Simplification is about reducing complex mathematical expressions into simpler forms.
This helps in easier calculations and understanding of relationships between trigonometric functions.
Using a Graphing Calculator
A graphing calculator can be invaluable in verifying trigonometric simplifications.
It provides a powerful way to visualize and validate the equivalence of different expressions.

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Most popular questions from this chapter

Solve, finding all solutions in \([0,2 \pi)\). $$\sin 2 x \cos x+\sin x=0$$

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Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=a \sec \theta\) in \(\sqrt{x^{2}-a^{2}} .\) Then find \(\sin \theta\) and \(\cos \theta\)
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