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Precalculus 7th

David Cohen, Theodore B. Lee, David Sklar

Chapter 10

Additional Topics in Trigonometry

Educators


Problem 1

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle A=30^{\circ}$ and $A B=60 \mathrm{cm},$ find $A C$ and $B C$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 2

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle A=60^{\circ}$ and $A B=12 \mathrm{cm},$ find $A C$ and $B C$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 3

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle B=60^{\circ}$ and $A C=16 \mathrm{cm},$ find $B C$ and $A B$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 4

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle B=45^{\circ}$ and $A C=9 \mathrm{cm},$ find $B C$ and $A B$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 5

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle B=50^{\circ}$ and $A B=15 \mathrm{cm},$ find $B C$ and $A C .$ (Round your answers to one decimal place.)
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 6

For Exercises $1-6,$ refer to the following figure. (However, each problem is independent of the others.)
If $\angle A=25^{\circ}$ and $A C=100 \mathrm{cm},$ find $B C$ and $A B .$ (Round your answers to one decimal place.)
(GRAPH CANT COPY

Sarah L.
Numerade Educator

Problem 7

A ladder 18 ft long leans against a building. The ladder forms an angle of $60^{\circ}$ with the ground.
(a) How high up the side of the building does the ladder reach? [Give two forms for your answer: one with radicals and one (using a calculator) with decimals, rounded to two places. $]$
(b) Find the horizontal distance from the foot of the ladder to the base of the building.

Sarah L.
Numerade Educator

Problem 8

From a point level with and $1000 \mathrm{ft}$ away from the base of the Washington Monument, the angle of elevation to the top of the monument is $29.05^{\circ} .$ Determine the height of the monument to the nearest half foot.

Sarah L.
Numerade Educator

Problem 9

Refer to the following figure.
(a) Use the inverse sine function, as in Example $4,$ to find $\theta$. Express the answer in degrees, rounded to one decimal place.
(b) Follow part (a), but use the inverse cosine function. Check that your answer agrees with the result in part (a).
(c) Follow part (a), but use the inverse tangent function. Again, check that your answer agrees with the result in part (a).
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 10

In isosceles triangle $A B C$, the sides are of length $A C=B C=8$ and $A B=4 .$ Find the angles of the triangle. Express the answers both in radians, rounded to two decimal places, and in degrees, rounded to one decimal place. Hints: To find $\angle A,$ start by drawing an altitude from $C$ to side $\overline{A B}$. Then for $\angle C$, use the fact that the sum of the angles in a triangle is $\pi$ radians or $180^{\circ} .$

Sarah L.
Numerade Educator

Problem 11

Refer to the following figure. At certain times, the planets Earth and Mercury line up in such a way that $\angle E M S$ is a right angle. At such times, $\angle S E M$ is found to be $21.16^{\circ} .$ Use this information to estimate the distance $M S$ of Mercury from the Sun. Assume that the distance from the Earth to the Sun is 93 million miles. (Round your answer to the nearest million miles. Because Mercury's orbit is not really circular, the actual distance of Mercury from the Sun varies from about 28 million miles to 43 million miles.)
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 12

Determine the distance $A B$ across the lake shown in the figure, using the following data: $A C=400 \mathrm{m}, \angle C=90^{\circ}$ and $\angle C A B=40^{\circ} .$ Round the answer to the nearest meter.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 13

A building contractor wants to put a fence around the perimeter of a flat lot that has the shape of a right triangle. One angle of the triangle is $41.4^{\circ},$ and the length of the hypotenuse is $58.5 \mathrm{m}$. Find the length of fencing required. Round the answer to one decimal place.

Sarah L.
Numerade Educator

Problem 14

Suppose that the contractor in Exercise 13 reviews his notes and finds that it is not the hypotenuse that is $58.5 \mathrm{m}$ but rather the side opposite the $41.4^{\circ}$ angle. Find the length of fencing required in this case. Again, round the answer to one decimal place.

Sarah L.
Numerade Educator

Problem 15

For Exercises 15 and $16,$ refer to the following diagram for the roof of a house. In the figure, $x$ is the length of a rafter measured from the top of a wall to the top of the roof; $\theta$ is the acute angle between a rafter and the horizontal; and h is the vertical distance from the top of the wall to the top of the roof.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 16

Suppose that $\theta=34^{\circ}$ and $h=36.5 \mathrm{ft}$
(a) Determine $x$. Round the answer to one decimal place.
(b) Find the area of the gable. Round the final answer to one decimal place. [See Exercise 15 (b) for the definition of gable. $]$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 17

In Exercises 17 and $18,$ find the area of the triangle. In Exercise $18,$ use a calculator and round the final answer to two decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 18

In Exercises 17 and $18,$ find the area of the triangle. In Exercise $18,$ use a calculator and round the final answer to two decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 19

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 20

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 21

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 22

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 23

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 24

In Exercises $19-24$, determine the area of the shaded region, given that the radius of the circle is 1 unit and the inscribed polygon is a regular polygon. Give two forms for each answer: an expression involving radicals or the trigonometric functions; a calculator approximation rounded to three decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 24

Determine the area of the shaded region,
given that the radius of the circle is 1 unit and the inscribed
polygon is a regular polygon. Give two forms for each
answer: an expression involving radicals or the trigonometric
functions; a calculator approximation rounded to three
decimal places.

Sarah L.
Numerade Educator

Problem 25

In Exercises 25 and $26,$ compute the area of the shaded segment of the circle, as in Example $7 .$ Give two forms for each answer: an exact expression and a calculator approximation rounded to two decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 26

In Exercises 25 and $26,$ compute the area of the shaded segment of the circle, as in Example $7 .$ Give two forms for each answer: an exact expression and a calculator approximation rounded to two decimal places.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 27

Show that the perimeter of the pentagon in Example 8 is $20 \sin 36^{\circ} .$ Hint: In Figure $7,$ draw a perpendicular from $O$ to $\overline{A B}$.

Sarah L.
Numerade Educator

Problem 28

In triangle $O A B,$ lengths $O A=O B=6$ in. and $\angle A O B=72^{\circ} .$ Find $A B . \quad$ Hint: Draw a perpendicular from $O$ to $A B .$ Round the answer to one decimal place.

Sarah L.
Numerade Educator

Problem 29

The accompanying figure shows two ships at points $P$ and $Q .$ which are in the same vertical plane as an airplane at point $R .$ When the height of the airplane is 3500 ft, the angle of depression to $P$ is $48^{\circ},$ and that to $Q$ is $25^{\circ} .$ Find the distance between the two ships. Round the answer to the nearest 10 feet.

Sarah L.
Numerade Educator

Problem 30

An observer in a lighthouse is $66 \mathrm{ft}$ above the surface of the water. The observer sees a ship and finds the angle of depression to be $0.7^{\circ} .$ Estimate the distance of the ship from the base of the lighthouse. Round the answer to the nearest 5 feet.

Sarah L.
Numerade Educator

Problem 30

An observer in a lighthouse is $66 \mathrm{ft}$ above the surface of the water. The observer sees a ship and finds the angle of depression to be $0.7^{\circ} .$ Estimate the distance of the ship from the base of the lighthouse. Round the answer to the nearest 5 feet.

Sarah L.
Numerade Educator

Problem 31

From a point on ground level, you measure the angle of elevation to the top of a mountain to be $38^{\circ} .$ Then you walk $200 \mathrm{m}$ farther away from the mountain and find that the angle of elevation is now $20^{\circ} .$ Find the height of the mountain. Round the answer to the nearest meter.

Sarah L.
Numerade Educator

Problem 32

A surveyor stands 30 yd from the base of a building. On top of the building is a vertical radio antenna. Let $\alpha$ denote the angle of elevation when the surveyor sights to the top of the building. Let $\beta$ denote the angle of elevation when the surveyor sights to the top of the antenna. Express the length of the antenna in terms of the angles $\alpha$ and $\beta .$

Sarah L.
Numerade Educator

Problem 33

In $\triangle A C D,$ you are given $\angle C=90^{\circ}, \angle A=60^{\circ},$ and $A C=18 \mathrm{cm} .$ If $B$ is a point on $C D$ and $\angle B A C=45^{\circ}$
find $B D .$ Express the answer in terms of a radical (rather than using a calculator).

Sarah L.
Numerade Educator

Problem 34

The radius of the circle in the following figure is 1 unit. Express the lengths $O A, A B,$ and $D C$ in terms of $\alpha .$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 35

The arc in the next figure is a portion of the unit circle, $x^{2}+y^{2}=1$
(a) Express the following angles in terms of $\theta: \angle B O A$
$\angle O A B, \angle B A P, \angle B P A$. (Assume that $\theta$ is in degrees.)
(b) Express the following lengths in terms of $\sin \theta$ and $\cos \theta: A O, A P, O B, B P$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 36

The following figure shows $\triangle A B C,$ in which $\angle B C A=\theta$ is an obtuse angle. Complete Steps (a)-(c) to show that the area of the triangle is $\frac{1}{2} a b \sin \theta$
(a) Show that $h=a \sin \left(180^{\circ}-\theta\right)$
(b) Use one of the addition formulas to verify that $\sin \left(180^{\circ}-\theta\right)=\sin \theta$
(c) Use the results in parts (a) and (b) to show that the area of $\triangle A B C$ is given by $\frac{1}{2} a b \sin \theta$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 37

Refer to the figure. Express each of the following lengths as a function of $\theta$
(a) $B C$
(b) $A B$
(c) $A C$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 38

In the following figure, $A B=8$ in. Express $x$ as a function of $\theta . \quad$ Hint: First work Exercise 37
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 39

In the figure, line segment $\overline{B A}$ is tangent to the unit circle at $A$. Also, $\overline{C F}$ is tangent to the circle at $F$. Express the following lengths in terms of $\theta$
(a) $D E$
(c) $C F$
(e) $A B$
(b) $O E$
(d) $O C$
(1) $O B$
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 40

At point $P$ on Earth's surface, the moon is observed to be directly overhead, while at the same time at point $T$, the moon is just visible. See the figure on the next column.
(a) Show that $M P=\frac{O T}{\cos \theta}-O P$
(b) Use a calculator and the following data to estimate the distance $M P$ from the earth to the moon: $\theta=89.05^{\circ}$ and $O T=O P=4000$ miles. Round your answer to the nearest thousand miles. (Because the moon's orbit is not really circular, the actual distance varies from about $216,400$ miles to $247,000$ miles.)

Sarah L.
Numerade Educator

Problem 41

Refer to the figure below. Let $r$ denote the radius of the moon.
(a) Show that $r=\left(\frac{\sin \theta}{1-\sin \theta}\right) P S$
(b) Use a calculator and the following data to estimate the radius $r$ of the moon: $P S=238,857$ miles and $\theta=0.257^{\circ} .$ Round your answer to the nearest 10 miles.
(GRAPH CANT COPY)

Sarah L.
Numerade Educator

Problem 42

Figure A shows a regular hexagon inscribed in a circle of radius 1. Figure B shows a regular heptagon (seven-sided polygon) inscribed in a circle of radius $1 .$ In Figure $A$ a line segment drawn from the center of the circle perpendicular to one of the sides is called an apothem of the polygon.
(a) Show that the length of the apothem in Figure A is $\sqrt{3} / 2$
(b) Show that the length of one side of the heptagon in Figure $\mathrm{B}$ is $2 \sin \left(180^{\circ} / 7\right)$
(c) Use a calculator to evaluate the expressions in parts (a) and (b). Round each answer to four decimal places, and note how close the two values are. Approximately two thousand years ago, Heron of Alexandria made use of this coincidence when he used the length of the apothem of the hexagon to approximate the length of the side of the heptagon. (The apothem of the hexagon can be constructed with ruler and compass; the side of the regular heptagon cannot.)
(GRAPH CANT COPY)

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Problem 43

The following figure shows a regular seven-sided polygon inscribed in a circle of radius 1.
(a) Explain why the area of $\triangle A O B$ is $\frac{1}{2} \sin (2 \pi / 7)$ Hint: Use the area formula from Example 5 .
(b) Explain why the area of the entire polygon is $\frac{7}{2} \sin (2 \pi / 7)$
(c) Let $a_{n}$ denote the area of a regular $n$ -sided polygon inscribed in a circle of radius $1 .$ Use the ideas from parts (a) and (b) to show that $a_{n}=\frac{1}{2} n \sin (2 \pi / n)$
(d) Use the formula from part (c) and a calculator to complete the following table. Round each result to eight decimal places.
\begin{tabular}{lccccccc}
\hline$n$ & 5 & 10 & 50 & 100 & $10^{3}$ & $10^{4}$ & $10^{5}$ \\
$a_{n}$ & & & & & & \\
\hline
\end{tabular}
(e) Explain (in complete sentences) why the values of $a_{n}$ in your table get closer and closer to $\pi$. (The value of $\pi, \text { correct to ten decimal places, is } 3.1415926535 .)$

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Problem 44

(a) The following figure shows a segment with central angle $\alpha$ in a circle of radius $r .$ (Assume $\alpha$ is in radians.) Show that the area $A$ of the segment is given by
$$
A=\frac{1}{2} r^{2}(\alpha-\sin \alpha)
$$
(b) In the following figure, the arc is a semicircle with diameter $A O B$, where radius $O B=1 .$ Use the formula in part (a) to show that the sum of the areas of the two shaded segments is $\pi / 2-\sin \theta,$ while the difference (larger minus smaller) is $\pi / 2-\theta$
(GRAPH CANT COPY)

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Problem 45

In the following figure, arc $A B C$ is a portion of a circle with center $D(0,-1)$ and radius $\overline{D C}$. The shaded crescent-shaped region is called a lune. Verify the following result, which was discovered (and proved) by the Greek mathematician Hippocrates of Chios approximately 2500 years ago: The area of the lune is equal to the area of the square $O C E D$. Hint: In computing the area of the lune, make use of the formula given in Exercise $44(\mathrm{a})$ for the area of a segment of a circle.

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Problem 46

In this exercise, you'll verify another result about lunes that was discovered by Hippocrates of Chios. The figure shows a regular hexagon inscribed in a circle of radius 1 Outward from each side of the hexagon, (congruent) semicircles are constructed with the sides of the hexagon as diameters. Follow steps (a) through (f) to show that the area of the hexagon is equal to the sum of the areas of the six (congruent) lunes plus twice the area of one of the semicircles.
(a) What is the radian measure of $\angle A O B ?$
(b) Show that the area of the hexagon is $3 \sqrt{3} / 2$
(c) Show that the area of the shaded segment is $(2 \pi-3 \sqrt{3}) / 12$
(d) Show that the area of semicircle $A C B$ is $\pi / 8$.
(e) Use the results in parts (c) and (d) to show that the area of lune $A C B$ is $\frac{6 \sqrt{3}-\pi}{24}$
(f) Use the results in parts (d) and (e) to verify Hippocrates' result:
area of hexagon $=6 \times$ (area of lune $A C B$ ) $+2 \times(\text { area of semicircle } A C B)$
(GRAPH CANT COPY)

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Problem 47

In the following figure, $\overline{A B}$ is a chord in a circle of radius 1 The length of $\overline{A B}$ is $d,$ and $\overline{A B}$ subtends an angle $\theta$ at the center of the circle, as shown. In this exercise we derive the following formula for the length $d$ of the chord in terms of the angle $\theta:$
$$
d=\sqrt{2-2 \cos \theta}
$$
(The derivation of this formula does not require any new material from this section. It is developed here for use in subsequent exercises.)
(a) We place the figure in an $x$ -y coordinate system and orient it so that the angle $\theta$ is in standard position and the point $B$ is located at $(1,0) .$ (See the following figure.) What are the coordinates of the point $A$ (in terms of $\theta$ )?
(b) Use the formula for the distance between two points to show that $d=\sqrt{2-2 \cos \theta}$
(GRAPH CANT COPY)

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Problem 48

This exercise provides practice in using the chord-length formula developed in Exercise 47
(a) Use the formula from Exercise 47 to compute the length $d$ in the following figure. Round the answer to one decimal place.
(b) In the following figure, $A B=1.2 .$ Use the formula developed in Exercise 47 to compute the angle $\theta$. Express the answer in radians, rounded to one decimal place.
(GRAPH CANT COPY)

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Problem 49

In the following figure, arc $A B C$ is a semicircle with diameter $\overline{A C}$, and arc $C D E$ is a semicircle with diameter $\overline{C E}$
(a) Show that the area of semicircle $C D E$ is $\pi(1-\cos \theta) / 4$ Hint: Use the chord-length formula in Exercise 47
(b) Express the area of lune $C D E$ in terms of $\theta$. Hint: Use the result in part (a) along with the formula in Exercise $44(\mathrm{a})$ for the area of a segment.
(c) Express the area of lune $A B C$ in terms of $\theta$.
(d) Express the area of $\triangle A C E$ in terms of $\theta .$
(e) Use the results in parts (b), (c), and (d) to verify that the area of $\triangle A C E$ is equal to the sum of the areas of the two lunes $C D E$ and $A B C$. Remark: As with the results in Exercises 45 and $46,$ this result about lunes was discovered and proved by the ancient Greek mathematician Hippocrates of Chios. According to Professor George F. Simmons in his book Calculus Gems (New York:
McGraw-Hill Book Co., 1992 ), these results appear "to be the earliest precise determination of the area of a region bounded by curves."
(GRAPH CANT COPY)

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Problem 50

Using the ruler-and-compass constructions of elementary geometry, there is a well known method for bisecting any angle. (Do you remember this from a geometry class?) However, there is no similar method for trisecting an angle. This exercise demonstrates a geometric method for the approximate trisection of small acute angles. IThe origins of the method can be traced back to the German cleric Nicolaus Cusanus $(1401-1464)$ and the Dutch physicist Willebrord Snell ( $1580-1626$ ). ]

In the following figure, $O$ is the center of unit circle and
$\angle D \overline{O C}=\theta$ is the angle to be trisected. Radius $\overline{O B}$ is extended to a point $A$ so that $A B=O B=1 .$ Then line segment $\overline{A D}$ is drawn, creating $\angle D A C=\beta$
(a) Draw a perpendicular from $D$ to $\overline{A C}$, meeting $\overline{A C}$ at $E$ Express the two lengths $D E$ and $O E$ in terms of $\theta$.
(b) Show that $\tan \beta=(\sin \theta) /(2+\cos \theta)$
(c) From part (b) it follows that
$$
\beta=\tan ^{-1}[(\sin \theta) /(2+\cos \theta)]
$$
Use this formula to complete the following table. As you will see by completing the table, $\beta \approx \theta / 3 .$ In the table, express $\beta$ in degrees, rounded to four decimal places. For the percentage error in the approximation $\beta \approx \theta / 3,$ use the formula
percentage error $=\frac{\theta / 3-\beta}{\theta / 3} \times 100$
Round the percentage error to two decimal places. \begin{tabular}{lcc}
& & Percentage Error in Approximation \\
$\boldsymbol{\theta}$ & $\boldsymbol{\theta} / 3$ & $\boldsymbol{\beta}$ & $\boldsymbol{\beta} \approx \boldsymbol{\theta} / 3$ \\
\hline $30^{\circ}$ & & & \\
$15^{\circ}$ & & & \\
$9^{\circ}$ & & & \\
$6^{\circ}$ & & & \\
$3^{\circ}$ & & &
\end{tabular}
(GRAPH CANT COPY)

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Problem 51

In this exercise we prove the following trigonometric identity:
$$
\sin 3 \theta=\sin \theta+2 \sin \theta \cos 2 \theta
$$
[This identity is valid for all angles $\theta$, but in this exercise we use right-triangle trigonometry and the resulting proof is valid only when $0^{\circ}<3 \theta<90^{\circ} .$ The idea for the proof is due to Professors J. Chris Fisher and E. L. Koh in Mathematics Magazine, vol. $65 \text { no. } 2 \text { (April } 1992 \text { ). }]$
(a) In the following figure, $O$ is the center of the circle and the radius is 1. Show that $A B=2 \sin \theta .$ Hint: Draw
a perpendicular from $O$ to $\overline{A B}$.
(b) For the remainder of this exercise refer to the following figure in which the arc is a portion of the unit circle, lines $\overline{A B}$ and $D C$ are parallel to the $y$ -axis, and $\overline{D B}$ is parallel to the $x$ -axis. Why does $A B=\sin \theta ?$ Why does $C B=2 \sin \theta ?$
(c) Use the fact that $\triangle O B C$ is isosceles to show that
$$
\angle O B C=90^{\circ}-\theta
$$
(d) From elementary geometry we know that alternate interior angles are equal. Consequently $\angle D B O=\angle B O A=\theta .$ Use this observation and the result in part (c) to show that $\angle D B C=90^{\circ}-2 \theta$
(e) By referring to $\triangle C D B$ and using two of the previous results, show that $C D=2 \sin \theta \cos 2 \theta$
(f) From the figure, you can see that the $y$ -coordinate of point $C$ is equal to $A B+C D .$ But independent of that fact, why is the $y$ -coordinate of point $C$ also equal to sin $3 \theta ?$ After you've answered this, use these observations to conclude that $\sin 3 \theta=\sin \theta+2 \sin \theta \cos 2 \theta$
as required.
(g) Substituting $\theta=10^{\circ}$ in the identity $\sin 3 \theta=\sin \theta+$
$2 \sin \theta \cos 2 \theta$ yields the statement $\frac{1}{2}=\sin 10^{\circ}+$
$2 \sin 10^{\circ} \cos 20^{\circ} .$ Use a calculator to check this last
equation.
(h) Substituting $\theta=\pi / 9$ in the identity $\sin 3 \theta=\sin \theta+$
$2 \sin \theta \cos 2 \theta$ yields the statement
$$
\frac{\sqrt{3}}{2}=\sin \frac{\pi}{9}+2 \sin \frac{\pi}{9} \cos \frac{2 \pi}{9}
$$
Use a calculator to check this last equation.
(GRAPH CANT COPY)

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Problem 52

In the accompanying figure, the smaller circle is tangent to the larger circle. Ray $P Q$ is a common tangent and ray $P R$ passes through the centers of both circles. If the radius of the smaller circle is $a$ and the radius of the larger circle is $b,$ show that $\sin \theta=(b-a) /(a+b)$ Then, using the identity $\sin ^{2} \theta+\cos ^{2} \theta=1,$ show that $\cos \theta=2 \sqrt{a b} /(a+b)$
(GRAPH CANT COPY)

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Problem 53

A vertical tower of height $h$ stands on level ground. From a point $P$ at ground level and due south of the tower, the angle of elevation to the top of the tower is $\theta .$ From a point $Q$ at ground level and due west of the tower, the angle of elevation to the top of the tower is $\beta .$ If $d$ is the distance between $P$ and $Q$, show that
$$
h=\frac{d}{\sqrt{\cot ^{2} \theta+\cot ^{2} \beta}}
$$

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Problem 54

The following problem is taken from An Elementary Treatise on Plane Trigonometry, 8 th ed., by R. D. Beasley (London: Macmillan and Co.; first published in 1884 ):
The [angle of] elevation of a tower standing on a horizontal plane is observed; a feet nearer it is found to be $45^{\circ}$ b feet nearer still it is the complement of what it was at the first station; show that the height of the tower is $a b /(a-b)$ feet.

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Problem 55

(a) The following problem is taken from Plane Trigonometry, 5 th ed., by Isaac Todhunter (London:
Macmillan and Co., 1874 ).
$A B$ is the diameter of a circle, $C$ its centre; a straight line $\overline{A P}$ is drawn dividing the [area of the] semicircle into two equal parts; $\theta$ is the circular [radian] measure of the complement of $\angle P C B$ : shew that $\cos \theta=\theta$ Hint: Use the figure below. Let $r$ denote the radius of the circle and note that $\angle P C B=\frac{\pi}{2}-\theta$
(b) Use Figure 14 in Section 8.2 to estimate, to the nearest
$0.05,$ the value for $\theta$ for which $\cos \theta=\theta$
(c) Use a graphing calculator to show that the actual value for the root in part (b), rounded to three decimal places, is $\theta=0.739 .$ Use this result to compute the percentage error for the estimate in part (b). Also, use the value $\theta=0.739$ to compute $\angle P C B$. Express that answer in degrees, rounded to the nearest one degree.
(GRAPH CANT COPY)

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