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Chapter 5: Problem 52

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

Short Answer

Expert verified
The equation \( \sin^{2} x + \cos^{2} x = 1 \) is an identity because it holds true for all angles \( x \).

Step by step solution

01

Define the types of equations

Before solving the equation, let's recall the definitions: An identity is an equation that is true for every value of the variable. A conditional equation is true for certain values of the variable. A contradiction is never true, regardless of the value of the variable.
02

Recognize the trigonometric identity

The equation \( \sin^{2} x + \cos^{2} x = 1 \) is a well-known trigonometric identity. This identity states that for any angle \( x \), the square of the sine of the angle plus the square of the cosine of the angle equals 1.
03

Test the equation for all possible values

Since \( \sin^{2} x + \cos^{2} x = 1 \) holds true for all angles \( x \), for instance: when \( x = 0 \), \( \sin^{2} 0 + \cos^{2} 0 = 0 + 1 = 1 \) and when \( x = \pi/2 \), \( \sin^{2}(\pi/2) + \cos^{2}(\pi/2) = 1 + 0 = 1 \).
04

Conclusion

Since the equation holds true for any angle \( x \), it satisfies the condition to be an identity. Thus, \( \sin^{2} x + \cos^{2} x = 1 \) is an identity.

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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 a type of mathematical equation that holds true only for certain values of the variable involved. This concept is essential because it highlights the difference between equations that apply universally and those that apply conditionally. When tackling problems in mathematics, particularly in trigonometry, knowing whether an equation is conditional is invaluable.

Imagine an equation as a key that fits only specific locks 鈥� these locks represent the particular values where the equation is valid. Let's consider the example of solving a simple trigonometric equation like \(\sin(x) = \frac{1}{2}\). This equation is conditional because it only holds true for values such as \(x = \frac{\pi}{6}\) or \(x = \frac{5\pi}{6}\), among other periodic solutions.
  • Conditional equations can have a finite or infinite set of solutions.
  • These solutions depend on the equation and the constraints applied, such as domain restrictions.
  • Understanding these conditions helps solve mathematical problems accurately.
Encountering conditional equations in math helps cultivate skills in critical thinking and problem-solving. Analyzing the conditions that make equations true sharpens our ability to approach problems analytically, leading to better insights and solutions.
Trigonometric Functions
Trigonometric functions are a fundamental part of mathematics, particularly in the study of triangles and modeling periodic phenomena. These functions include sine, cosine, and others like tangent. With roots in geometry, trigonometric functions relate the angles of a triangle to the lengths of its sides.

In the context of the provided exercise, we deal with the sine and cosine functions, which are pivotal in forming the Pythagorean identity: \(\sin^2x + \cos^2x = 1\). This identity is a cornerstone of trigonometry.
  • \(\sin(x)\) represents the ratio of the opposite side to the hypotenuse in a right triangle.
  • \(\cos(x)\) represents the ratio of the adjacent side to the hypotenuse.
  • The identity \(\sin^2x + \cos^2x = 1\) shows the intrinsic relationship between these two functions.
Understanding trigonometric functions is key to mastering advanced math topics, such as calculus and continuous periodic signals. It forms the basis for exploring deeper mathematical theories, helping in both academic and real-world applications such as engineering and physics.
Contradiction in Equations
A contradiction in equations arises when no possible value of the variable makes the equation true. In mathematical terms, contradictions highlight inconsistencies within a proposed equation. Unlike identities or conditional equations, contradictions signal an equation that cannot be satisfied.

A simple example of a contradiction would be \(x + 2 = x - 3\). No value of \(x\) can satisfy this because adding something and subtracting from the same doesn't equal the same expression 鈥� it's inherently inconsistent.
  • Contradictions are crucial in verifying propositions and assumptions.
  • Recognizing contradictions quickly can confirm that an equation or problem setup is flawed.
  • In mathematical proofs, showing a contradiction can discredit an assumed statement.
Being able to spot contradictions helps improve logical reasoning, both in problem-solving and proof-writing. This skill is not only applicable in mathematics but extends to fields such as computer science and logic-based disciplines, honing an individual's ability to assess and analyze systematically.

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Most popular questions from this chapter

In Exercises 45-48, graph the functions. $$ y=4 \cos ^{2}\left(\frac{x}{2}\right) $$

A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the ground is \(\theta\) such that \(\sin \theta=\frac{2 \sqrt{2}}{3}\). For the novice division, the angle \(\theta\) is cut in half to lower the ramp. What is \(\tan \left(\frac{\theta}{2}\right)\), the steepness of the ramp?

Consider the triangle below, where the vertex angle measures \(\theta\), the equal sides measure \(a\), the height is \(h\), and half the base is \(b\). (In an isosceles triangle, the perpendicular dropped from the vertex angle divides the triangle into two congruent triangles.) The two triangles formed are right triangles. In the right triangles, \(\sin \left(\frac{\theta}{2}\right)=\frac{b}{a}\) and \(\cos \left(\frac{\theta}{2}\right)=\frac{h}{a}\). Multiply each side of each equation by \(a\) to get \(b=a \sin \left(\frac{\theta}{2}\right), h=a \cos \left(\frac{\theta}{2}\right)\). The area of the entire isosceles triangle is \(A=\frac{1}{2}(2 b) h=b h\). Substitute the values for \(b\) and \(h\) into the area formula. Show that the area is equivalent to \(\frac{a^{2}}{2} \sin \theta\).

An ore crusher wheel consists of a heavy disk spinning on its axle. Its normal (crushing) force \(F\) in pounds between the wheel and the inclined track is determined by $$ F=W \sin \theta+\frac{1}{2} \psi^{2}\left[\frac{C}{R}(1-\cos 2 \theta)+\frac{A}{l} \sin 2 \theta\right] $$ where \(W\) is the weight of the wheel, \(\theta\) is the angle of the axis, \(C\) and \(A\) are moments of inertia, \(R\) is the radius of the wheel, \(l\) is the distance from the wheel to the pin where the axle is attached, and \(\psi\) is the speed in rpm that the wheel is spinning. The optimum crushing force occurs when the angle \(\theta\) is between \(45^{\circ}\) and \(90^{\circ}\). Find \(F\) if the angle is \(75^{\circ}\), \(W\) is 500 pounds, and \(\psi\) is \(200 \mathrm{rpm}, \frac{C}{R}=750\), and \(\frac{A}{l}=3.75\)

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