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Magazine Chapter 7: Problem 99
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. Show that the equation is not an identity, (a) \(\sin 2 x=2 \sin x\) (b) \(\sin (x+y)=\sin x+\sin y\) (c) \(\sec ^{2} x+\csc ^{2} x=1\) (d) \(\frac{1}{\sin x+\cos x}=\csc x+\sec x\)
Short Answer
Expert verified
The equations are not identities as they do not hold universally across all values of x.
Step by step solution
01
Rewrite Each Expression
For each part of the question, we'll begin by rewriting the given equations in recognizable mathematical identities or canonical forms:(a) \(\sin 2x = 2\sin x\) can be expanded using the double angle formula: \(\sin 2x = 2\sin x \cos x\).(b) \(\sin(x+y) = \sin x + \sin y\) can be expanded using sum of angles identity: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\).(c) \(\sec^2 x + \csc^2 x = 1\) will need to be expressed using reciprocal identities: \(\sec^2 x = 1/\cos^2 x\) and \(\csc^2 x = 1/\sin^2 x\).(d) \(1/(\sin x + \cos x) = \csc x + \sec x\) rewrites to: \(\csc x = 1/\sin x\) and \(\sec x = 1/\cos x\).
02
Assess Algebraic Simplifications
Evaluate whether the rewritten forms match or simplify to an identity:(a) \(2\sin x \cos x eq 2\sin x\) unless \(\cos x = 1\), implying it's not universally true.(b) \(\sin x \cos y + \cos x \sin y eq \sin x + \sin y\), no simplification makes this universally true.(c) Adding a common denominator: \(\sin^2 x \cos^2 x (1/\cos^2 x + 1/\sin^2 x) = (\sin^2 x + \cos^2 x)/(\sin^2 x \cos^2 x) eq 1\).(d) Cross multiply to get \(1 = (\sin x + \cos x)(\frac{1}{\sin x} + \frac{1}{\cos x})\), which simplifies to \(1 = 1 + \sin(x)/\cos(x) + \cos(x)/\sin(x) eq 1\).
03
Graphical Evaluation
Graph each pair of functions in each subpart to evaluate intersections:For each equation, graph both sides of the equation over a common domain, such as [0, 2\(\pi\)].Intersections of graphs determine points where both sides of equations equal, which are not consistently true over whole domains, indicating no identity.
04
Conclusion and Proof
Check specific point evaluations to confirm no identity:(a) Pick \(x = \pi/4\), \(\sin(\pi/2) = 2\sin(\pi/4)\cos(\pi/4)\), which holds, but not true for \(x = \pi/6\).(b) For \(x = y = \pi/4\), \(\sin(\pi/2) = \sqrt{2} eq \sqrt{2}/2 + \sqrt{2}/2\), (c) \(x = \pi/4\) and check: \(1/(\cos^2(\pi/4)) + 1/(\sin^2(\pi/4)) eq 1\)(d) Plug in \(x = \pi/4\) shows \(1/(\sin(\pi/4) + \cos(\pi/4)) eq \sqrt{2} + \sqrt{2}\).
05
Final Verification
Each identity is verified by algebraic proof and graphically suggested by multiple graph intersections:(a) As not holding over all points, confirm \(\sin 2xeq 2\sin x\) in all \(x\).(b) No true simplification holds that \(\sin(x+y)\) equals \(\sin x + \sin y\).(c) Reciprocal forms don't allow \(\sec^2 x + \csc^2 x = 1\).(d) The form \(1/(\sin x + \cos x) eq \csc x + \sec x\) throughout.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphical Representation
Graphs provide a powerful visual tool for comparing and analyzing mathematical equations. In this exercise, graphing the provided functions helps determine whether the proposed equations hold true as identities. For each expression given in the exercise, when we graph both sides over a common interval, such as [0, 2\(\pi\)], we can observe whether the two functions are identical across the entire domain.
- If the graphs overlap entirely, the equation is likely an identity.
- If they intersect at certain points but not others, it suggests the equation is not an identity.
Double Angle Formulas
Trigonometric double angle formulas help break down complex expressions into products of simpler sine and cosine functions. For example, the double angle formula for sine is given by:\[\sin(2x) = 2\sin(x)\cos(x)\]In this exercise, this formula is used to analyze the proposed non-identity mathematical statement \(\sin(2x) = 2\sin(x)\).
- The challenge was to examine if both the original and expanded forms result in identical expressions for all values of \(x\).
- Here, it becomes clear that they only match when \(\cos(x) = 1\), meaning \(x\) must equal integer multiples of \(\pi\).
Reciprocal Identities
The reciprocal identities for trigonometric functions express one trigonometric ratio in terms of another. These identities are critical for transforming equations into more comparable forms. Specifically:
These identities become particularly useful when exploring such statements by simplifying or rewriting terms to demonstrate inconsistencies or errors in initial assumptions.
- The cosecant function, \(\csc(x)\), is \(1/\sin(x)\).
- The secant function, \(\sec(x)\), is \(1/\cos(x)\).
These identities become particularly useful when exploring such statements by simplifying or rewriting terms to demonstrate inconsistencies or errors in initial assumptions.
Mathematical Proofs
Mathematical proofs serve as the backbone for validating or refuting equations by methodically demonstrating truth through logical deductions. This exercise leans on proof methods to establish whether given trigonometric equations are identities or not.
- The proof process often begins by rewriting equations using known identities and then simplifying.
- Direct evaluation at specific points, such as \(x = \pi/4\), further supports or refutes the statement.
