6.4 Chapter 6 Summary and Review

Key Concepts

The distance we travel around a circle of radius is proportional to the angle of displacement.
$Arclength = (fraction of one revolution) \cdot (2 π r)$
We measure angles in radians when we work with arclength.
Note
Radians
The radian measure of an angle is given by
$open parenthesis fraction of one revolution times 2 π close parenthesis$
An arclength equal to one radius determines a central angle of one radian.

Radian measure can be expressed as multiples of

π

or as decimals.

Degrees	$\frac{Radians:}{Exact Values}$	$\frac{Radians: Decimal}{Approximations}$
$0 degrees$	$0$	$0$
$90 degrees$	$\frac{π}{2}$	$1.57$
$180 degrees$	$π$	$3.14$
$270 degrees$	$\frac{3 π}{2}$	$4.71$
$360 degrees$	$2 π$	$6.28$

Quadrantal angles on unit circle radian approximation

We multiply by the appropriate conversion factor to convert between degrees and radians.
Note
Unit Conversion for Angles
$\frac{180^{\circ}}{π radians} = 1$

To convert from radians to degrees we multiply the radian measure by $\frac{180^{\circ}}{π}$ .
To convert from degrees to radians we multiply the degree measure by $\frac{π}{180}$ .
Note
Arclength Formula
On a circle of radius $r$ , the length $s$ of an arc spanned by an angle $θ$ in radians is
$s = r θ$
On a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.
The sine, cosine, or tangent of a particular angle is the same whether the angle is measured in radians or in degrees.

You should be familiar with the trig values of the special angles in radians.

Degrees	Radians	Sine	Cosine	Tangent
$0 degrees$	$0$	$0$	$1$	$0$
$30 degrees$	$\frac{π}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45 degrees$	$\frac{π}{4}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
$60 degrees$	$\frac{π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$the square root of 3$
$90 degrees$	$\frac{π}{2}$	$1$	$0$	undefined

To find the sine or cosine of a real number $t$ , we draw an arc of length $t$ on a unit circle, and then find the sine or cosine of the angle $θ$ determined by the arc.
Note
Coordinates on a Unit Circle
The coordinates of the point $P$ determined by an arc of length $t$ in standard position on a unit circle are
$(x, y) = (\cos (t), \sin (t))$
Note
The Circular Functions
Let $P$ be the terminal point of an arc of length $t$ in standard position on a unit circle. The circular functions of $t$ are defined by
$\begin{aligned} \cos (t) & = x \\ \sin (t) & = y \\ \tan (t) & = \frac{y}{x}, x \neq 0 \end{aligned}$
The domain of a function is the set of all possible input values. The range of a function is the set of all output values for the function.
Note
$f (x) = \sin (x)$
Domain: all real numbers
Range: $[ minus 1, 1 ]$
Period: $2 π$
Note
$g (x) = \cos (x)$
Domain: all real numbers
Range: $[ minus 1, 1 ]$
Period: $2 π$
Note
$h (x) = \tan (x)$
Domain: all real numbers except $\dots, \frac{- 3 π}{2}, \frac{- π}{2}, \frac{π}{2}, \frac{3 π}{2}, \dots$
Range: all real numbers
Period: $π$

Chapter 6 Review Problems

For Problems 1–2, convert from degrees to radians. Give exact answers.

$75 degrees$
$210 degrees$
$340 degrees$

$\frac{5 π}{12}$
$\frac{7 π}{6}$
$\frac{17 π}{9}$

$130 degrees$
$300 degrees$
$12 degrees$

For Problems 3–4, convert from degrees to radians. Round to two decimal places.

$27 degrees$
$142 degrees$
$218 degrees$

$0.47$
$2.48$
$3.80$

$76 degrees$
$328 degrees$
$111 degrees$

For Problems 5–8, convert from radians to degrees. Round to hundredths if necessary.

$\frac{5 π}{6}$
$\frac{3 π}{10}$
$\frac{23 π}{18}$

$150 degrees$
$54 degrees$
$230 degrees$

$\frac{7 π}{4}$
$\frac{8 π}{15}$
$\frac{35 π}{20}$

$2$
$3.6$
$0.8$

$114.59 degrees$
$206.26 degrees$
$45.84 degrees$

$4$
$1.2$
$5.3$

For Problems 9–10, express each fraction of one revolution as an angle in radians.

$\frac{2}{3}$
$\frac{7}{12}$
$\frac{9}{8}$

$\frac{4 π}{3}$
$\frac{7 π}{6}$
$\frac{9 π}{4}$

$\frac{5}{4}$
$\frac{4}{6}$
$\frac{2}{5}$

For Problems 11–12, express each angle in radians as a fraction of one revolution.

$\frac{π}{4}$
$\frac{5 π}{8}$
$\frac{7 π}{3}$

$\frac{1}{8}$
$\frac{5}{16}$
$\frac{7}{6}$

$\frac{11 π}{6}$
$\frac{13 π}{4}$
$\frac{π}{9}$

For Problems 13–14, in which quadrant on a unit circle does an arc with given length lie?

$2.15$
$1.5$
$6.0$

$5.4$
$4.32$
$3.1$

The Earth's radius at the equator is 3960 miles. If you travel 150 miles along the equator, what fraction of its circumference have you covered? How many degrees of longitude have you crossed? Convert your answer to radians.
Use the arclength formula to calculate the answer to part (a). Do your answers agree?

$0.006, 2.17, 0.0379$
$0.0379$

A lawn sprinkler has a range of 15 feet, and waters a porion of a circle whose curved edge is 39.27 feet long. Through what angle does the sprinkler turn?

Many telecommunications satellites are put into geostationary orbits, so that they have the same period as the rotation of the earth, and hence stay in the same relative position seen from earth. Hundreds of these satellites orbit 22,300 miles above the equator in what is called the Clarke belt, named after Arthur C. Clarke. What is the speed of the satellites? (The radius of the earth is about 4000 miles.)

$6885$ mph

The planet Neptune is 4504 million kilometers from the Sun. In one Earth year (365 days), it travels a distance of 171.58 million kilometers around its orbit.

What fraction of its orbit does Neptune travel in one Earth year? What angle in radians does it traverse in that time?
How many days does it take Neptune to complete one orbit around the Sun? What is its speed, in kilometers per day?
Earth is 150 million kilometers from the Sun. What is Earth's orbital speed?

For Problems 19–20, evaluate exactly.

$\cos (\frac{2 π}{3}) + \sin (\frac{π}{6})$
$\tan (\frac{π}{6}) - 3 \tan (\frac{4 π}{3})$
$\sin^{2} (\frac{5 π}{4}) \cos (π)$

$0$
$\frac{- 8}{\sqrt{3}}$
$\frac{- 1}{2}$

$\sin (\frac{7 π}{6}) \cos (\frac{3 π}{4})$
$4 \cos (\frac{3 π}{2}) + \tan (\frac{7 π}{4})$
$2 \tan^{2} (\frac{5 π}{3}) \sin (\frac{4 π}{3})$

For Problems 21–22, sketch an arc with the given length in standard position on a unit circle. Find the coordinates of the terminal point. Round to tenths.

$open parenthesis 0.5, 0.8 close parenthesis$
$open parenthesis minus 0.4, 0.9 close parenthesis$
$open parenthesis minus 1.0, 0.1 close parenthesis$

In Problems 23–24, the circle has radius $r$ , and its center is the point $open parenthesis 0, 0 close parenthesis$ .

Find the coordinates of each point in terms of $α$ .

$(r \cos (α), r \sin (α))$
$(- r \cos (α), r \sin (α))$
$(- r \cos (α), - r \sin (α))$
$(r \cos (α), - r \sin (α))$

Find the length of each arc in terms of $α$ .

$O P$
$O Q$
$P Q$
$Q R$

For Problems 25–26, find an exact value for the area of the sector.

With a central angle of $135 degrees$ in a circle of radius 4 inches.

$6 π$

With a central angle of $240 degrees$ in a circle of radius 12 centimeters.

For Problems 27–30, fill in the correct inequality symbol.

If $\frac{π}{2} < α < β < π$ , then $\cos (α) \underset{―}{} \cos (β)$ .

$>$

If $π < θ < ϕ < \frac{3 π}{2}$ , then $\sin (θ) \underset{―}{} \sin (ϕ)$ .

If $\frac{3 π}{2} < s < t < 2 π$ , then $\tan (s) \underset{―}{} \tan (t)$ .

$<$

If $\frac{π}{2} < x < y < \frac{3 π}{2}$ , then $\cos (x) \underset{―}{} \cos (y)$ .

For Problems 31–34, evaluate the function

$f (t) = 12 - 2.8 \sin (3.5 t - 2)$ for $t equals 8$

$9.86$

$h (x) = 2.4 + 6 \tan (\frac{3 x - 5}{4})$ for $x equals 1.8$

$g (z) = 0.07 \tan (0.4 z + 0.2) - 1.3$ for $z equals 22$

$minus 1.33$

$F (s) = - 1.5 \cos (\frac{s}{8} - 3) + 5$ for $s equals 6.2$

For Problems 35–38, find the reference angle in radians.

$\frac{5 π}{6}$
$\frac{5 π}{4}$
$\frac{3 π}{8}$
$\frac{7 π}{12}$

$\frac{π}{6}$
$\frac{π}{4}$
$\frac{3 π}{8}$
$\frac{5 π}{12}$

$\frac{5 π}{3}$
$\frac{9 π}{8}$
$\frac{17 π}{12}$
$\frac{7 π}{6}$

$2.8$
$3.9$
$5.03$
$1.5$

$0.34$
$0.76$
$1.25$
$1.5$

$1.2$
$6.2$
$2.36$
$4.15$

For Problems 39–40, find the angle of inclination of the line.

$2 x + 5 y = - 3$

$158.2 degrees$

$\frac{x}{8} - \frac{y}{11} = 1$

Prepare a Cartesian coordinate system with the $x$ -axis scaled from 0 to 10 and the $y$ -axis scaled from $minus 2$ to $2$ . Label multiples of $\frac{π}{4}$ on the $x$ -axis.
Sketch an accurate graph of $y = \cos (x)$ on the grid. Sketch an accurate graph of $y = \sin (x)$ on the same grid.

Use your calculator to make a table of values for $Y_{1} = \sin (x), Y_{2} = \cos (x),$ and $Y_{s} = \frac{Y_{1}}{Y_{2}}$ , for $0 less than or equal to x less than or equal to 3,$ and $Δ x equals 0.1$ .
Plot the points $open parenthesis x, Y sub 3 close parenthesis$ from the table. Identify the graph.

For Problems 43–46,

Graph the function for $0 less than or equal to s less than 2 π$ . State the amplitude, period, and midline of the graph.
Use the graph to solve the equation for $0 less than or equal to s less than 2 π$ .

$h (s) = 5 + 3 \cos (2 s)$
$5 + 3 \cos (2 s) = 4.56$

mid: $y equals 5$ , amp: $3$ , period: $π$
$0.86, 2.28, 4.00, 5.42$

$f (s) = 4 - 2 \sin (3 s)$
$4 - 2 \sin (3 s) = 2$

$g (s) = 10 + 4.8 \sin (s - 1.5)$
$10 + 4.8 \sin (s - 1.5) = 12$

mid: $y equals 10$ , amp: $4.8$ , period: $2 π$
$1.93, 4.2$

$j (s) = 1.5 + 0.25 \cos (s + 0.5)$
$1.5 + 0.25 \cos (s + 0.5) = 1.4$

For Problems 47–48, solve the equation graphically for $0 less than or equal to x less than 2 π$ .

$6 + \tan (x - \frac{π}{6}) = 7$

$\frac{5 π}{12}, \frac{17 π}{12}$

$3 - \tan (x + \frac{3 π}{4}) = 4$

For Problems 49–52, solve the equation exactly for $0 less than or equal to x less than 2 π$ .

$\sin (θ) = \frac{\sqrt{3}}{2}$

$\frac{π}{3}, \frac{2 π}{3}$

$\sin (θ) = - \frac{1}{2}$

$\cos (θ) = - 1$

$π$

$\tan (θ) = \sqrt{3}$

For Problems 53–58, find all solutions between $0$ and $2 π$ . Round to two decimal places.

$tan ⁡ open parenthesis t close parenthesis equals 5$

$1.37, 4.51$

$\cos (x) = - 0.63$

$\sin (h) = - 0.26$

$6.02, 3.40$

$\tan (ϕ) = - 2.5$

$cos ⁡ open parenthesis β close parenthesis equals 0.95$

$0.32, 5.97$

$sin ⁡ open parenthesis α close parenthesis equals 0.1$

For Problems 59–62, solve for $x$ .

$cos ⁡ open parenthesis x close parenthesis equals 0.35$
$cos ⁡ open parenthesis 0.35 close parenthesis equals x$

$1.21, 5.07$
$0.9394$

$sin ⁡ open parenthesis x close parenthesis equals 0.84$
$sin ⁡ open parenthesis 0.84 close parenthesis equals x$

$\sin (x) = \frac{π}{8}$
$sin ⁡ open parenthesis π 8 close parenthesis equals x$

$0.40, 2.74$
$0.3827$

$tan ⁡ x equals 1.7 π$
$tan ⁡ open parenthesis 1.7 π close parenthesis equals x$

For Problems 63–66, sketch the graph. State its domain and range.

$g (x) = 2 x^{2} + 4$

Dom: all real numbers, Rge: $y greater than or equal to 4$

$h (w) = 1 - \frac{1}{w^{2}}$

$F (s) = - \sqrt{16 - s^{2}}$

Dom: $minus 4 less than or equal to s less than or equal to 4$ , Rge: $minus 4 less than or equal to y less than or equal to 0$

$G (t) = 4 + \sqrt{4 - t}$

Prove the Pythagorean identity $\cos^{2} (t) + \sin^{2} (t) = 1$ by carrying out the following steps. Sketch a unit circle, and an arc $t$ in standard position.

Write the equation of the unit circle.
Use trig ratios to write the coordinates of the terminal point $P$ of the arc.
Substitute the coordinates of point $P$ into your equation from part (a).
Does the identity hold for all values of $t$ ?

$x^{2} + y^{2} = 1$
$(\cos (t), \sin (t))$
$\cos^{2} (t) + \sin^{2} (t) = 1$
Yes

Prove the tangent identity $\tan (t) = \frac{\sin (t)}{\cos (t)}$ by carrying out the following steps. Sketch an arc $t$ in standard position on a unit circle, and label its terminal point $(x, y)$ .

Write $\sin (t)$ and $\cos (t)$ in terms of $x$ and $y$ .
Write the definition of $\tan (t)$ .
Substitute your results from part (a) into your expression for (b).
Does the identity hold for all values of $t$ ?

Trigonometry by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.