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Chapter 7: Problem 32

Verify the identity. $$\cos (-x)-\sin (-x)=\cos x+\sin x$$

Short Answer

Expert verified
The identity is verified by applying trigonometric symmetries.

Step by step solution

01

Recognize Trigonometric Function Symmetries

Start by using the even and odd properties of the trigonometric functions. The cosine function is even, meaning \( \cos(-x) = \cos(x) \). The sine function is odd, meaning \( \sin(-x) = -\sin(x) \).
02

Substitute Symmetries Into Both Sides

Substitute the even and odd properties from Step 1 into the left side of the equation. This gives us \( \cos(-x) - \sin(-x) = \cos(x) + \sin(x) \).
03

Simplify the Left Side Using Substitutions

Replace \( \cos(-x) \) with \( \cos(x) \) and \( \sin(-x) \) with \(-\sin(x) \). This transforms the left side to \( \cos(x) + \sin(x) \), matching the right side.

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Key Concepts

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

Even and Odd Functions
Understanding even and odd functions is key when working with trigonometric identities. It's like knowing the personality of each function. Let's start simple:
  • Even functions have something special. They mirror symmetrically about the y-axis. For example, \( f(x) = f(-x) \). Our superstar here is cosine, because for all values, \( \cos(-x) = \cos(x) \).
  • Odd functions have a different kind of symmetry. They rotate 180 degrees about the origin, so \( f(-x) = -f(x) \). The sine function falls here. For sine, \( \sin(-x) = -\sin(x) \).
Recognizing these properties allows us to simplify expressions and solve equations by rewriting terms in a way that might uncover hidden insights or symmetries. As in our example, applying these properties directly transformed the entire equation.
Symmetry in Trigonometry
Symmetry plays a vital role in trigonometry and helps simplify complex equations. Imagine symmetry as a helpful pattern that repeats in predictable ways. Trigonometric functions are great examples:
  • The cosine function is symmetric around the y-axis. This means any reflection across the y-axis doesn't change the cosine values. It's a hallmark of even functions, helping us predict behavior without complex calculations.
  • The sine function exhibits symmetry about the origin, a feature of odd functions. Reflecting across both axes (think of a 180掳 rotation) yields the negative of the original function, simplifying certain scenarios, as seen in our example identity.
In problems or proofs, recognizing such symmetries offers shortcuts and deeper insight, sometimes making complex problems manageable and engaging patterns easy to identify.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions means making them easier to understand or evaluate. It involves using known identities, properties, or transformations. Here's how we approach simplification:
  • Apply symmetry properties. For instance, knowing cosine is even allows immediate substitution, turning \( \cos(-x) \) into \( \cos(x) \).
  • Use known identities like the Pythagorean identity, angle addition formulas, or co-function identities, which often lead directly to simplification or final solution.
  • Substitute known values or variable transformations that might ease complexity. Our solved identity used substitution from symmetry properties to show that the left side matches the right side.
By simplifying, not only do you make expressions easier to work with, but you also uncover elegant patterns or results, enabling a better understanding of the underlying trigonometric relationships.

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Most popular questions from this chapter

Graph \(f\) and \(g\) in the same viewing rectangle. Do the graphs suggest that the equation \(f(x)=g(x)\) is an identity? Prove your answer. $$f(x)=(\sin x+\cos x)^{2}, \quad g(x)=1$$

Find the exact value of the expression, if it is defined. $$\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$$

Use a graphing device to find the solutions of the equation, correct to two decimal places. $$\cos x=\frac{1}{2}\left(e^{x}+e^{-x}\right)$$

(a) Find all solutions of the equation. (b) Use a calculator to solve the equation in the interval \([0,2 \pi),\) correct to five decimal places. $$\cos x=0.4$$

In Philadelphia the number of hours of daylight on day \(t\) (where \(t\) is the number of days after January 1) is modeled by the function $$L(t)=12+2.83 \sin \left(\frac{2 \pi}{365}(t-80)\right)$$ (a) Which days of the year have about 10 hours of daylight? (b) How many days of the year have more than 10 hours of daylight?
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