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Chapter 7: Problem 39

Complete the identity. $$1-\cos ^{2} x=$$

Short Answer

Expert verified
1 - \cos^2 x = \sin^2 x

Step by step solution

01

Recall the Pythagorean Identity

The Pythagorean identity states that \[ \sin^2 x + \cos^2 x = 1 \] This identity is true for all values of x.
02

Rearrange the Identity

We need to isolate \sin^2 x on one side of the equation. To do this, subtract \cos^2 x from both sides: \[ \sin^2 x = 1 - \cos^2 x \]
03

Substitute and Simplify

Substitute the right-hand side of the rearranged identity into the given expression: \[ 1 - \cos^2 x = \sin^2 x \]

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Key Concepts

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

Understanding Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They are fundamental tools in calculus, physics, engineering, and many other fields. These identities can simplify complex trigonometric expressions and solve equations that involve trigonometric functions. A well-known category is the Pythagorean identities, derived from the Pythagorean Theorem. These include \(\sin^2 x + \cos^2 x = 1\), \(1 + \tan^2 x = \sec^2 x\), and \(1 + \cot^2 x = \csc^2 x\). Recognizing these identities is crucial, as they help in transforming and simplifying trigonometric expressions.
Exploring \(\sin^2 x\)
The function \(\sin^2 x\) represents the square of the sine function. Sine is one of the key trigonometric functions, defined in a right-angled triangle as the ratio of the length of the opposite side to the hypotenuse. When squared, \(\sin^2 x\) is always a non-negative value, restricted between 0 and 1. According to the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we can derive that \(\sin^2 x = 1 - \cos^2 x\). This relationship is essential for solving various trigonometric problems. Through this identity, you can convert between sine and cosine functions, making it easier to work with and solve equations.
Understanding \(\cos^2 x\)
The function \(\cos^2 x\) represents the square of the cosine function. Cosine is another fundamental trigonometric function, defined in a right-angled triangle as the ratio of the length of the adjacent side to the hypotenuse. Similar to \(\sin^2 x\), \(\cos^2 x\) ranges between 0 and 1. The Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) allows us to express \(\cos^2 x\) in terms of \(\sin^2 x\): \(\cos^2 x = 1 - \sin^2 x\). This is particularly useful when \(\cos^2 x\) is more straightforward to handle in trigonometric equations. By understanding these fundamental relationships, you'll find it simpler to manipulate and solve trigonometric problems.

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Most popular questions from this chapter

Simplify. Check your results using a graphing calculator. $$\frac{\cos x-\sin \left(\frac{\pi}{2}-x\right) \sin x}{\cos x-\cos (\pi-x) \tan x}$$

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Prove the identity. \(\tan ^{-1} x=\sin ^{-1} \frac{x}{\sqrt{x^{2}+1}}\)

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ Trigonometric functions with domains composed of real numbers are called ____________ functions.

Simplify. Check your results using a graphing calculator. $$\sin \left(\frac{\pi}{2}-x\right)[\sec x-\cos x]$$
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