True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { The function } y=\frac{1}{2} \sin 2 x \text { has an amplitude that is twice that of }} \\ {\text { the function } y=\sin x .}\end{array}$$

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## Recommended Questions

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { The function } y=\frac{1}{2} \sin 2 x \text { has an amplitude that is twice that of }} \\ {\text { the function } y=\sin x .}\end{array}$$

TRUE OR FALSE? In Exercises 121-123, determine whether the statement is true or false. Justify your answer.

A measurement of 4 radians corresponds to two complete revolutions from the initial side to the terminal side of an angle.

TRUE OR FALSE? In Exercises 71 and 72, determine whether the statement is true or false. Justify your answer.

A line that has an inclination greater than $\pi/2$ radians has a negative slope.

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { The function } y=3 \cos (x / 3) \text { has a period that is three times }} \\ {\text { that of the function } y=\cos x .}\end{array}$$

True or False? In Exercises $77-80$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

Amplitude is always positive

True or False? In Exercises $99-102,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=\ln \pi, \text { then } y^{\prime}=1 / \pi$

TRUE OR FALSE? In Exercises 77-82, determine whether the statement is true or false. Justify your answer.

$\frac{sin\ 60^\circ}{sin\ 30^\circ} = sin\ 2^\circ$

True or False? In Exercises $63-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

\begin{equation}

\begin{array}{l}{\text { The value of }} \\ {\quad\int_{2}^{2} \sin x^{2} d x} \\ {\text { is } 0}.\end{array}

\end{equation}

TRUE OR FALSE? In Exercises 61-64, determine whether the statement is true or false. Justify your answer.

Because sin$-t = -sin t$, it can be said that the sine of a negative angle is a negative number.

TRUE OR FALSE? In Exercises 121-123, determine whether the statement is true or false. Justify your answer.

The difference between the measures of two coterminal angles is always a multiple of $360^{\circ}$ if expressed in degrees and is always a multiple of $2\pi$ radians if expressed in radians.

True or False? In Exercises 85 and $86,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The lines represented by

If a line contains points in both the first and third quadrants, then its slope must be positive.

True or False? In Exercises $93-98$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$4 \int \sin x \cos x d x=-\cos 2 x+C$$

TRUE OR FALSE? In Exercises 73 and 74, determine whether the statement is true or false. Justify your answer.

The graph of $x^2+4y^4-4=0$ is an ellipse.

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

In the polar coordinate system, if a graph that has symmetry with respect to the polar axis were folded on the line $\theta=0$, the portion of the graph above the polar axis would coincide with the portion of the graph below the polar axis.

TRUE OR FALSE? In Exercises 77-82, determine whether the statement is true or false. Justify your answer.

$sin\ 60^\circ$ $csc\ 60^\circ$ $= 1$

TRUE OR FALSE? In Exercises 67-70, determine whether the statement is true or false. Justify your answer.

The conic represented by the following equation is an ellipse.

$r=\dfrac{16}{9-4\ \cos \left(\theta + \dfrac{\pi}{4} \right)}$

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=\pi^{2},$ then $d y / d x=2 \pi$

True or False? In Exercises $83 - 86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array} { l } { \text { If the graphs of } f \text { and } g \text { intersect midway between } x = a \text { and } } \\ { x = b , \text { then } } \\ { \int _ { a } ^ { b } [ f ( x ) - g ( x ) ] d x = 0 } \end{array}$$

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c>0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

The Leaning Tower of Pisa is not vertical, but if you know the angle of elevation $\theta$ to the top of the tower when you stand $d$ feet away from it, you can find its height $h$ using the formula $h=d\ \tan\ \theta$.

True or False? In Exercises $91-96,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$0.75=0.749999 \ldots$$

TRUE OR FALSE? In Exercises 77-82, determine whether the statement is true or false. Justify your answer.

$tan[(5^\circ)^2] = tan^2\ 5^\circ$

True or False? In Exercises $47-50$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $x=\tan \theta,$ then

$$\int_{0}^{\sqrt{3}} \frac{d x}{\left(1+x^{2}\right)^{3 / 2}}=\int_{0}^{4 \pi / 3} \cos \theta d \theta$$

True or False? In Exercises $93-98$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\int_{a}^{b} \sin x d x=\int_{a}^{b+2 \pi} \sin x d x$$

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\begin{array}{l}{\text { If the graph of } f \text { is symmetric with respect to the origin or the }} \\ {y \text { -axis, then } \int_{0}^{\infty} f(x) d x \text { converges if and only if } \int_{-\infty}^{\infty} f(x) d x} \\ {\text { converges. }}\end{array}$$

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

N $24^{\circ}$E means 24 degrees north of east.

TRUE OR FALSE? In Exercises 73 and 74, determine whether the statement is true or false. Justify your answer.

It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricity of the ellipse is large (close to 1).

True or False? In Exercises $103-108$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$f(a)=f(b), \text { then } a=b$$

TRUE OR FALSE? In Exercises 67-70, determine whether the statement is true or false. Justify your answer.

The conic represented by the following equation is a parabola.

$r=\dfrac{6}{3-2\ \cos\ \theta}$

TRUE OR FALSE? In Exercises 112-114, determine whether the statement is true or false. Justify your answer.

$\sin \dfrac{5 \pi}{6}\ = \dfrac{1}{2}$ $\Rightarrow$ $\arcsin \dfrac{1}{2}\ = \dfrac{5 \pi}{6}$

$sec\ 30^\circ\ =\ csc\ 60^\circ$

TRUE OR FALSE? In Exercises 71 and 72, determine whether the statement is true or false. Justify your answer.

To find the angle between two lines whose angles of inclination $\theta_1$ and $\theta_2$ are known, substitute $\theta_1$ and $\theta_2$ for $m_1$ and $m_2$, respectively, in the formula for the angle between two lines.

TRUE OR FALSE? In Exercises 71 and 72, determine whether the statement is true or false. Justify your answer.

Show that the equation

$x^2+y^2=r^2$

is invariant under rotation of axes.

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f^{\prime}(x)=g^{\prime}(x),$ then $f(x)=g(x)$

True or False? In Exercises $93-98$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$\int \sin ^{2} 2 x \cos 2 x d x=\frac{1}{3} \sin ^{3} 2 x+C$$

TRUE OR FALSE? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer.

To find the reference angle for an angle $\theta$ (given in degrees), find the integer $n$ such that $0\ \leq\ 360^{\circ}n\ -\ \theta\ \leq\ 360^{\circ}$. The difference $360^{\circ} n -\ \theta$ is the reference angle.

True or False? In Exercises 83-86, determine whether the

statement is true or false. If it is false, explain why or give an

example that shows it is false.

$$\begin{array}{l}{\text { If } f^{\prime \prime}(x)>0 \text { for all real numbers } x, \text { then } f \text { increases without }} \\ {\text { bound. }}\end{array}$$

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give anexample that shows it is false.

$$\arcsin ^{2} x+\arccos ^{2} x=1$$

True or False? In Exercises $113-116,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

The polar equations $r=\sin 2 \theta, \quad r=-\sin 2 \theta, \quad$ and $r=\sin (-2 \theta)$ all have the same graph.

TRUE OR FALSE? In Exercises 61-64, determine whether the statement is true or false. Justify your answer.

cos$(-\frac{7\pi}{2})$ = cos$(\pi + \frac{\pi}{2})$

True or False? In Exercises $99-102,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=\ln e, \text { then } y^{\prime}=1$

TRUE OR FALSE? In Exercises 103 and 104, determine whether the statement is true or false. Justify your answer.

In each of the four quadrants, the signs of the secant function and sine function will be the same.

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $y$ -axis, then $(4,-5)$ is also a point on the graph.

TRUE OR FALSE? In Exercises 57-59, determine whether the statement is true or false. Justify your answer.

If a triangle contains an obtuse angle, then it must be oblique.

TRUE OR FALSE? In Exercises 95-97, determine whether the statement is true or false. Justify your answer.

The graph of $y =\ - cos\ 2x$ has an amplitude that is twice that of the function given by $y =\ sin(x + \pi/2)$ in the $x$-axis.

$cot^2\ 10^\circ\ - csc^2\ 10^\circ = -1$

$sin\ 45^\circ$ + $csc\ 45^\circ$ $= 1$

TRUE OR FALSE? In Exercises 95-97, determine whether the statement is true or false. Justify your answer.

The graph of the function given by $f(x) =\ sin(x+2\pi)$ translates the graph of $f(x) =\ sin\ x$ exactly one period to the right so that the two graphs look identical.

True or False? In Exercises $77-79$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y$ is a function of $t$ and $x$ is a function of $t,$ then $y$ is a function of $x .$

True or False? In Exercises $113-116,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $x>0,$ then the point $(x, y)$ on the rectangular coordinate system can be represented by $(r, \theta)$ on the polar coordinate system, where $r=\sqrt{x^{2}+y^{2}}$ and $\theta=\arctan (y / x)$

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $x$ -axis, then $(4,-5)$ is also a point on the graph.

TRUE OR FALSE? In Exercises 67-70, determine whether the statement is true or false. Justify your answer.

For a given value of $e>1$ over the interval $\theta=0$ to $\theta=2\pi$, the graph of

$r=\dfrac{ex}{1\ -\ e\ \cos\ \theta}$

is the same as the graph of

$r=\dfrac{e(-x)}{1\ +\ e\ \cos\ \theta}$.

True or False? In Exercises $91-96,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$|r|<1, then \sum_{n=1}^{\infty} a r^{n}=\frac{a}{1-r}$$

TRUE OR FALSE? In Exercises 61-64, determine whether the statement is true or false. Justify your answer.

tan $a =$ tan $(a-6\pi)$

After a rotation of axes is used to eliminate the $xy$-term from an equation of the form

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

the coefficients of the $x^2$- and $y^2$-terms remain $A$ and $C$, respectively.

If $b^{2}-4 a c=0$ and $a \neq 0,$ then the graph of

$$y=a x^{2}+b x+c$$

has only one $x$ -intercept.

True or False? In Exercises $83 - 86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$$ \begin{array} { l } { \text { The line } } \\ { y = ( 1 - \sqrt [ 3 ] { 0.5 } ) x } \\ { \text { divides the region under the curve } } \\ { f ( x ) = x ( 1 - x ) } \\ { \text { on } [ 0,1 ] \text { into two regions of equal area. } } \end{array} $$

True or False? In Exercises $83-86$ , determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false.

The range of $$y=\arcsin x [0, \pi]$$

True or False? In Exercises 87 and 88, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

A slope field shows one particular solution of a differential equation.

True or False? In Exercises $75-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.$\begin{array}{l}{\text { The graph of every cubic polynomial has precisely one point }} \\ {\text { of inflection. }}\end{array}$

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f(x)=0,$ then $f^{\prime}(x)$ is undefined.

True or False? In Exercises $47-50$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $x=\sin \theta,$ then

$$\quad \int \frac{d x}{\sqrt{1-x^{2}}}=\int d \theta$$

True or False? In Exercises 89 and 90 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

\begin{equation}\begin{array}{l}{\text { If the inverse function of } f \text { exists, then the } y \text { -intercept of } f \text { is }} \\ {\text { an } x \text { -intercept of } f^{-1} \text { . }}\end{array}\end{equation}

TRUE OR FALSE? In Exercises 81-84, determine whether the statement is true or false. Justify your answer.

The graph of $y = -f(x)$ is a reflection of the graph of $y=f(x)$ in the $y$-axis.

TRUE OR FALSE? In Exercises 119 and 120, determine whether the statement is true or false. Justify your answer.

If $|r_1| = |r_2|$, then $(r_1, \theta)$ and $(r_2, \theta)$ represent the same point on the polar coordinate system.

TRUE OR FALSE? In Exercises 73-76, determine whether the statement is true or false. Justify your answer.

If $D\neq0$ and $E\neq0$, then the graph of $x^2-y^2+Dx+Ey=0$ is a hyperbola.

TRUE OR FALSE? In Exercises 96 and 97, determine whether the statement is true or false. Justify your answer.

The graph of $y =\ sec\ x$ can be obtained on a calculator by graphing a translation of the reciprocal of $y =\ sin\ x$.

True or False? In Exercises 89 and 90 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

\begin{equation}\text { If }f \text { is an even function, then } f^{-1}\end{equation}

True or False? In Exercises $49-53,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$y=a x+b, \text { then } \frac{\Delta y}{\Delta x}=\frac{d y}{d x}$$

True or False? In Exercises $87-92,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

It is possible for a parabola to intersect its directrix.