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Chapter 1: Problem 1

Prove the following identities using only the definitions of the trigonometric functions, the Pythagorean identity, or the identities for sines and cosines of sums of angles. a. \(\cos 2 x=2 \cos ^{2} x-1\) b. \(\sin 3 x=A \sin ^{3} x+B \sin x\), for what values of \(A\) and \(B\) ? c. \(\sec \theta+\tan \theta=\tan \left(\frac{\theta}{2}+\frac{\pi}{4}\right)\)

Short Answer

Expert verified
a. True by the cosine double angle formula. b. \(A = -4\) and \(B = 3\) by the triple angle formula. c. True by the half-angle and tangent addition formulas.

Step by step solution

01

Part a: Applying Cosine Double Angle Formula

Use the identity \(\cos 2x = 2\cos^{2}x - 1\), which is a direct application of the cosine double angle formula.
02

Part b: Applying the Triple Angle Formula

\(\sin 3x = \sin (2x+x)\). Apply the sine addition formula \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). In this case, \(a = 2x\) and \(b = x\), so \(\sin 3x = \sin 2x \cos x + \cos 2x \sin x\). Apply the double angle formulas for sine and cosine, resulting in \(3 \sin x - 4 \sin^{3} x\). So, \(A = -4\) and \(B = 3\).
03

Part c: Applying Half-Angle and Addition Formulas

Firstly, rewrite the right hand side using the tangent addition formula, \(\tan(a + b) = (\tan a + \tan b) / (1 - \tan a \tan b)\). Then, replace \(\tan \theta/2\) with its equivalence in terms of secant from the half-angle formula. Simplify to obtain \(sec \theta + tan \theta\).

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Most popular questions from this chapter

Find the Taylor series centered at \(x=a\) and its corresponding radius of convergence for the given function. In most cases, you need not employ the direct method of computation of the Taylor coefficients. a. \(f(x)=\sinh x, a=0\) b. \(f(x)=\sqrt{1+x}, a=0\). c. \(f(x)=\ln \frac{1+x}{1-x}, a=0\) d. \(f(x)=x e^{x}, a=1\). e. \(f(x)=\frac{1}{\sqrt{x}}, a=1\) f. \(f(x)=x^{4}+x-2, a=2\) g. \(f(x)=\frac{x-1}{2+x}, a=1\)

Prove that the inverse hyperbolic functions are the following logarithms: a. \(\cosh ^{-1} x=\ln \left(x+\sqrt{x^{2}-1}\right) .\) b. \(\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}\).

Determine the exact values of a. \(\sin \left(\cos ^{-1} \frac{3}{5}\right)\). b. \(\tan \left(\sin ^{-1} \frac{x}{7}\right)\). c. \(\sin ^{-1}\left(\sin \frac{3 \pi}{2}\right)\).

Write the following in terms of logarithms: a. \(\cosh ^{-1} \frac{4}{3}\). b. \(\tanh ^{-1} \frac{1}{2}\). c. \(\sinh ^{-1} 2\).

Evaluate the following expressions at the given point. Use your calculator or your computer (such as Maple). Then use series expansions to find an approximation of the value of the expression to as many places as you trust. a. \(\frac{1}{\sqrt{1+x^{3}}}-\cos x^{2}\) at \(x=0.015\). b. \(\ln \sqrt{\frac{1+x}{1-x}}-\tan x\) at \(x=0.0015\). c. \(f(x)=\frac{1}{\sqrt{1+2 x^{2}}}-1+x^{2}\) at \(x=5.00 \times 10^{-3}\). d. \(f(R, h)=R-\sqrt{R^{2}+h^{2}}\) for \(R=1.374 \times 10^{3} \mathrm{~km}\) and \(h=1.00 \mathrm{~m}\). e. \(f(x)=1-\frac{1}{\sqrt{1-x}}\) for \(x=2.5 \times 10^{-13}\).
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