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Chapter 5: Problem 55

Determine whether each equation is an identity, a conditional equation, or a contradiction. $$ \sin x=\sqrt{1-\cos ^{2} x} $$

Short Answer

Expert verified
The equation is an identity.

Step by step solution

01

Understanding the Equation

The given equation is \( \sin x = \sqrt{1 - \cos^2 x} \). This is a trigonometric equation relating sine and cosine, which are fundamental trigonometric functions.
02

Investigate the Identity

We know the Pythagorean identity; \( \sin^2 x + \cos^2 x = 1 \). Rearranging gives \( \sin^2 x = 1 - \cos^2 x \), and taking the square root of both sides results in \( \sin x = \sqrt{1 - \cos^2 x} \) or \( \sin x = -\sqrt{1 - \cos^2 x} \). This simplifies to \( \sin x = \sqrt{1 - \cos^2 x} \) when considering only the positive root in typical functions.
03

Check Conditional Solutions

For \( \sin x = \sqrt{1 - \cos^2 x} \) to be true and not dependent on specific values of \( x \), check if it holds for all values. Rewrite \( \sqrt{1 - \cos^2 x} \) using \( \sin x = |\sin x| \), which is true as \( \sqrt{y^2} = |y| \).
04

Determine Equation Type

The equation holds for all \( x \) in its domain where both sides are defined, which includes all real \( x \) except where the square root is negative. Considerations confirm it holds universally with typical function domain constraints.

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Key Concepts

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

Conditional Equation
A conditional equation is true only for certain values of the variable involved in the equation. These solutions are dependent upon specific conditions being met. For instance, the equation \( x + 1 = 2 \) is satisfied only when \( x = 1 \).

In trigonometric equations, like the one given in the exercise \( \sin x = \sqrt{1 - \cos^2 x} \), conditional solutions mean that the equation holds under the condition that both sides are defined for appropriate values of \( x \).
  • A conditional trigonometric equation will meet the requirements that the trigonometric functions involved are defined and real.
  • It typically narrows down the equation's validity to certain domain constraints, influenced by both algebraic manipulation and the inherent properties of trigonometric functions.
Contradiction in Equations
A contradiction in equations occurs when an equation is false for all possible values of the variable. In simpler terms, no value will satisfy a contradictory equation.

Consider an equation such as \( x + 1 = x + 3 \). Subtracting \( x \) from both sides yields \( 1 = 3 \), which clearly is never true. Hence, this is a classic contradiction.
  • Unlike identities, which are true universally, contradictory equations have no solution.
  • Recognizing a contradiction often involves reducing the equation to an impossible statement (e.g., a nonsensical equality like \( 0 = 4 \)).
Pythagorean Identity
The Pythagorean identity is a cornerstone concept in trigonometry. It expresses the fundamental relationship between the sine and cosine of an angle: \( \sin^2 x + \cos^2 x = 1 \).

This identity is essential for transforming and simplifying trigonometric expressions. Understanding and utilizing this identity can reveal the equivalence of various trigonometric forms.
  • For example, transforming \( \sin^2 x = 1 - \cos^2 x \) using the Pythagorean identity allows us to dive deeper into trigonometric equations and proofs.
  • When you take the square root of \( 1 - \cos^2 x \), two possibilities arise: \( \sin x = \sqrt{1 - \cos^2 x} \) or \( \sin x = -\sqrt{1 - \cos^2 x} \). The choice usually depends on the specific half of the unit circle (positive or negative) under consideration.
  • This flexibility makes the Pythagorean identity a versatile tool in verifying the equality of trigonometric expressions, identifying identities, and assessing conditional equations.

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Most popular questions from this chapter

Verify each identity. $$ \cos (4 x)=[\cos (2 x)-\sin (2 x)][\cos (2 x)+\sin (2 x)] $$

An auger used to deliver grain to a storage bin can be raised and lowered, thus allowing for different size bins. Let \(\alpha\) be the angle formed by the auger and the ground for bin \(\mathrm{A}\) such that \(\sin \alpha=\frac{40}{41}\). The angle formed by the auger and the ground for bin B is half of \(\alpha\). If the height \(h\), in feet, of a bin can be found using the formula \(h=75 \sin \theta\), where \(\theta\) is the angle formed by the ground and the auger, find the height of bin \(\mathrm{B}\).

Simplify each expression using half-angle identities. Do not evaluate. $$ \frac{\sin 150^{\circ}}{1+\cos 150^{\circ}} $$

Use the half-angle identities to find the exact values of the trigonometric expressions. $$ \cos \left(\frac{3 \pi}{8}\right) $$

Verify each identity. $$ \cos (4 x)=1-\sin ^{2}(2 x)-4(\sin x \cos x)^{2} $$
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