Chapter 1

In Exercises \(39-42,\) express the given quantity in terms of \(\sin x\) and \(\cos x .\) $$ \sin \left(\frac{3 \pi}{2}-x\right) $$

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(0,2), \quad a=2 $$

Solve the inequalities in Exercises \(35-42 .\) Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result \(\sqrt{a^{2}}=|a|\) as appropriate. $$ x^{2}-x<0 $$

A magic trick You may have heard of a magic trick that goes like this: Take any number. Add \(5 .\) Double the result. Subtract \(6 .\) Divide by \(2 .\) Subtract \(2 .\) Now tell me your answer, and I'll tell you what you started with. Pick a number and try it. You can see what is going on if you let \(x\) be your original number and follow the steps to make a formula \(f(x)\) for the number you end up with.

Graph the functions in Exercises \(29-48\) $$ y=\frac{1}{x}-2 $$

In Exercises \(39-42,\) express the given quantity in terms of \(\sin x\) and \(\cos x .\) $$ \cos \left(\frac{3 \pi}{2}+x\right) $$

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(-3,0), \quad a=3 $$

Solve the inequalities in Exercises \(35-42 .\) Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result \(\sqrt{a^{2}}=|a|\) as appropriate. $$ x^{2}-x-2 \geq 0 $$

Graph the functions in Exercises \(29-48\) $$ y=\frac{1}{x}+2 $$

Evaluate \(\sin \frac{7 \pi}{12}\) as \(\sin \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\)

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(-1,5), \quad a=\sqrt{10} $$

Do not fall into the trap \(|-a|=a .\) For what real numbers \(a\) is this equation true? For what real numbers is it false?

Evaluate \(\cos \frac{11 \pi}{12}\) as \(\cos \left(\frac{\pi}{4}+\frac{2 \pi}{3}\right)\)

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(1,1), \quad a=\sqrt{2} $$

Solve the equation \(|x-1|=1-x\)

Evaluate \(\cos \frac{\pi}{12}\)

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(-\sqrt{3},-2), \quad a=2 $$

A proof of the triangle inequality Give the reason justifying each of the numbered steps in the following proof of the triangle inequality. $$ \begin{aligned}|a+b|^{2} &=(a+b)^{2} \\ &=a^{2}+2 a b+b^{2} \\ & \leq a^{2}+2|a||b|+b^{2} \\ &=|a|^{2}+2|a||b|+|b|^{2} \\ &=(|a|+|b|)^{2} \\\|a+b| & \leq|a|+|b| \end{aligned} $$

Graph the functions in Exercises \(29-48\) $$ y=\frac{1}{x^{2}}-1 $$

Evaluate \(\sin \frac{5 \pi}{12}\)

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(3,1 / 2), \quad a=5 $$

Prove that \(|a b|=|a||b|\) for any numbers \(a\) and \(b\)

Graph the functions in Exercises \(29-48\) $$ y=\frac{1}{x^{2}}+1 $$

Find the function values in Exercises \(47-50\) . $$ \cos ^{2} \frac{\pi}{8} $$

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}+4 x-4 y+4=0 $$

If \(|x| \leq 3\) and \(x>-1 / 2,\) what can you say about \(x ?\)

Find the function values in Exercises \(47-50\) . $$ \cos ^{2} \frac{\pi}{12} $$

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}-8 x+4 y+16=0 $$

Graph the inequality \(|x|+|y| \leq 1\)

Find the function values in Exercises \(47-50\) . $$ \sin ^{2} \frac{\pi}{12} $$

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}-3 y-4=0 $$

Let \(f(x)=2 x+1\) and let \(\delta>0\) be any positive number. Prove that \(|x-1|<\delta\) implies \(|f(x)-f(1)|<2 \delta .\) Here the notation \(f(a)\) means the value of the expression \(2 x+1\) when \(x=a\) . This function notation is explained in Section \(1.3 .\)

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}-4 x-(9 / 4)=0 $$

Let \(f(x)=2 x+3\) and let \(\epsilon>0\) be any positive number. Prove that \(|f(x)-f(0)|<\epsilon\) whenever \(|x-0|<\frac{\epsilon}{2} .\) Here the notation \(f(a)\) means the value of the expression \(2 x+3\) when \(x=a\) . (See Section \(1.3 . )\)

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=x^{2}-1, \text { stretched vertically by a factor of } 3 $$

The tangent sum formula The standard formula for the tangent of the sum of two angles is $$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$$ Derive the formula.

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}-4 x+4 y=0 $$

For any number \(a,\) prove that \(|-a|=|a|\)

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=x^{2}-1, \quad \text { compressed horizontally by a factor of } 2 $$

Graph the circles whose equations are given in Exercises 47–52. Label each circle’s center and intercepts (if any) with their coordinate pairs. $$ x^{2}+y^{2}+2 x=3 $$

Let \(a\) be any positive number. Prove that \(|x|>a\) if and only if \(x>a\) or \(x<-a\).

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=1+\frac{1}{x^{2}}, \quad \text { compressed vertically by a factor of } 2 $$

a. If \(b\) is any nonzero real number, prove that \(|1 / b|=1 /|b|\) b. Prove that \(\left|\frac{a}{b}\right|=\frac{|a|}{|b|}\) for any numbers \(a\) and \(b \neq 0\)

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=1+\frac{1}{x^{2}}, \quad \text { stretched horizontally by a factor of } 3 $$

Graph the parabolas in Exercises 53–60. Label the vertex, axis, and intercepts in each case. $$ y=x^{2}+4 x+3 $$

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=\sqrt{x+1}, \quad \text { compressed horizontally by a factor of } 4 $$

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=60^{\circ} .\) Find the length of side \(c .\)

Graph the parabolas in Exercises 53–60. Label the vertex, axis, and intercepts in each case. $$ y=-x^{2}+4 x $$

Exercises \(51-60\) tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph . $$ y=\sqrt{x+1}, \quad \text { stretched vertically by a factor of } 3 $$

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=40^{\circ} .\) Find the length of side \(c .\)