Precalculus

David Cohen, Theodore B. Lee, David Sklar

Chapter 8

Analytical Trigonometry - all with Video Answers

Educators

Section 1

The Addition Formulas

Problem 1

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin \theta \cos 2 \theta+\cos \theta \sin 2 \theta$$

Jose Hannan

Jose Hannan

Numerade Educator

Problem 2

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin \frac{\pi}{6} \cos \frac{\pi}{3}+\cos \frac{\pi}{6} \sin \frac{\pi}{3}$$

Jose Hannan

Numerade Educator

Problem 3

Use the addition formulas for sine and cosine to simplify the expression.

$$\sin 3 \theta \cos \theta-\cos 3 \theta \sin \theta$$

Jose Hannan

Numerade Educator

Problem 4

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin 110^{\circ} \cos 20^{\circ}-\cos 110^{\circ} \sin 20^{\circ}$$

Jose Hannan

Numerade Educator

Problem 5

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos 2 u \cos 3 u-\sin 2 u \sin 3 u$$

Jose Hannan

Numerade Educator

Problem 6

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos 2 u \cos 3 u+\sin 2 u \sin 3 u$$

Jose Hannan

Numerade Educator

Problem 7

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos \frac{2 \pi}{9} \cos \frac{\pi}{18}+\sin \frac{2 \pi}{9} \sin \frac{\pi}{18}$$

Jose Hannan

Numerade Educator

Problem 8

Use the addition formulas for sine and cosine to simplify the expression.
$$\cos \frac{3 \pi}{10} \cos \frac{\pi}{5}-\sin \frac{3 \pi}{10} \sin \frac{\pi}{5}$$

Jose Hannan

Numerade Educator

Problem 9

Use the addition formulas for sine and cosine to simplify the expression.
$$\sin (A+B) \cos A-\cos (A+B) \sin A$$

Jose Hannan

Numerade Educator

Problem 10

Simplify each expression (as in Example 2).
$$\sin \left(\theta-\frac{3 \pi}{2}\right)$$

Jose Hannan

Numerade Educator

Problem 11

Simplify each expression (as in Example 2).
$$\cos \left(\frac{3 \pi}{2}+\theta\right)$$

Jose Hannan

Numerade Educator

Problem 12

Simplify each expression (as in Example 2).
$$\sin \left(\theta-\frac{3 \pi}{2}\right)$$

Jose Hannan

Numerade Educator

Problem 13

Simplify each expression (as in Example 2).
$$\cos (\theta+\pi)$$

Jose Hannan

Numerade Educator

Problem 14

Simplify each expression (as in Example 2).
$$\sin (\theta-\pi)$$

Jose Hannan

Numerade Educator

Problem 15

Expand $\sin (t+2 \pi)$ using the appropriate addition formula, and check to see that your answer agrees with the formula in the first box on page 504

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 16

Follow the directions in Exercise $15,$ but use $\cos (t+2 \pi)$

Jose Hannan

Numerade Educator

Problem 17

Use the formula for $\cos (s+t)$ to compute the exact value of $\cos 75^{\circ}$

Jose Hannan

Numerade Educator

Problem 18

Use the formula for $\sin (s-t)$ to compute the exact value of $\sin \frac{\pi}{12}$

Jose Hannan

Numerade Educator

Problem 19

Use the formula for $\sin (s+t)$ to find $\sin \frac{7 \pi}{12}$

Jose Hannan

Numerade Educator

Problem 20

Determine the exact value of (a) $\sin 105^{\circ}$ and $(\mathrm{b}) \cos 105^{\circ}$

Jose Hannan

Numerade Educator

Problem 21

Use the addition formulas for sine and cosine to simplify each expression.
$$\sin \left(\frac{\pi}{4}+s\right)-\sin \left(\frac{\pi}{4}-s\right)$$

Jose Hannan

Numerade Educator

Problem 22

Use the addition formulas for sine and cosine to simplify each expression.
$$\sin \left(t+\frac{\pi}{6}\right)-\sin \left(t-\frac{\pi}{6}\right)$$

Jose Hannan

Numerade Educator

Problem 23

Use the addition formulas for sine and cosine to simplify each expression.
$$\cos \left(\frac{\pi}{3}-\theta\right)-\cos \left(\frac{\pi}{3}+\theta\right)$$

Jose Hannan

Numerade Educator

Problem 24

Use the addition formulas for sine and cosine to simplify each expression.
$$\cos \left(\theta-\frac{\pi}{4}\right)+\cos \left(\theta+\frac{\pi}{4}\right)$$

Jose Hannan

Numerade Educator

Problem 25

Compute the indicated quantity using the following data.
$$\begin{aligned}
&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2} < \alpha < \pi\\
&\cos \beta=-\frac{3}{5} \quad \text { where } \pi < \beta < \frac{3 \pi}{2}\\
&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta < -\frac{3 \pi}{2}
\end{aligned}$$.
(a) $\sin (\alpha+\beta)$
(b) $\cos (\alpha+\beta)$

Jose Hannan

Numerade Educator

Problem 26

Compute the indicated quantity using the following data.
$$\begin{aligned}
&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2} < \alpha < \pi\\
&\cos \beta=-\frac{3}{5} \quad \text { where } \pi < \beta < \frac{3 \pi}{2}\\
&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta < -\frac{3 \pi}{2}
\end{aligned}$$.
(a) $\sin (\alpha-\beta)$
(b) $\cos (\alpha-\beta)$

Jose Hannan

Numerade Educator

Problem 27

Compute the indicated quantity using the following data.
$$\begin{aligned}
&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2} < \alpha < \pi\\
&\cos \beta=-\frac{3}{5} \quad \text { where } \pi < \beta < \frac{3 \pi}{2}\\
&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta < -\frac{3 \pi}{2}
\end{aligned}$$.
(a) $\sin (\theta-\beta)$
(b) $\sin (\theta+\beta)$

Jose Hannan

Numerade Educator

Problem 28

Compute the indicated quantity using the following data.
$$\begin{aligned}
&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2} < \alpha < \pi\\
&\cos \beta=-\frac{3}{5} \quad \text { where } \pi < \beta < \frac{3 \pi}{2}\\
&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta < -\frac{3 \pi}{2}
\end{aligned}$$.
(a) $\cos (\alpha+\theta)$
(b) $\cos (\alpha-\theta)$

Jose Hannan

Numerade Educator

Problem 29

Suppose that $\sin \theta=1 / 5$ and $0 < \theta < \pi / 2$
(a) Compute $\cos \theta$
(b) Compute $\sin 2 \theta .$ Hint: $\sin 2 \theta=\sin (\theta+\theta)$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 30

Suppose that $\cos \theta=12 / 13$ and $3 \pi / 2 < \theta < 2 \pi$
(a) Compute $\sin \theta$
(b) Compute cos $2 \theta .$ Hint: $\cos 2 \theta=\cos (\theta+\theta)$

Jose Hannan

Numerade Educator

Problem 31

Given $\tan \theta=-2 / 3,$ where $\pi / 2 < \theta < \pi,$ and $\csc \beta=2$
where $0 < \beta < \pi / 2,$ find $\sin (\theta+\beta)$ and $\cos (\beta-\theta)$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 32

Given sec $s=5 / 4,$ where $\sin s<0,$ and $\cot t=-1,$ where $\pi / 2 < t < \pi,$ find $\sin (s-t)$ and $\cos (s+t)$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 33

Prove that each equation is an identity.
$$\sin \left(t+\frac{\pi}{4}\right)=(\sin t+\cos t) / \sqrt{2}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 34

Prove that each equation is an identity.
$$\cos \left(t+\frac{\pi}{4}\right)=(\cos t-\sin t) / \sqrt{2}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 35

Prove that each equation is an identity.
$$\sin \left(t+\frac{\pi}{4}\right)+\cos \left(t+\frac{\pi}{4}\right)=\sqrt{2} \cos t$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 36

Prove that each equation is an identity.
$$\sec (\alpha+\beta)=\frac{\sec \alpha \sec \beta}{1-\tan \alpha \tan \beta}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 37

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$\tan s=2 \quad \text { and } \quad \tan t=3$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 38

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$\tan s=1 / 2 \text { and } \tan t=1 / 3$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 39

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$s=3 \pi / 4 \text { and } \tan t=-4$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 40

Use the given information to compute $\tan (s+t)$ and $\tan (s-t)$.
$$s=7 \pi / 4 \text { and } \tan t=-2$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 41

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan t+\tan 2 t}{1-\tan t \tan 2 t}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 42

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan \frac{\pi}{5}-\tan \frac{\pi}{30}}{1+\tan \frac{\pi}{3} \tan \frac{\pi}{20}}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 43

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan 70_{i}-\tan 10_{i}}{1+\tan 70_{i} \tan 10_{i}}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 44

Use the addition formulas for tangent to simplify each expression.
$$\frac{2 \tan \frac{\pi}{12}}{1-\tan ^{2} \frac{\pi}{12}}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 45

Use the addition formulas for tangent to simplify each expression.
$$\frac{\tan (x-y)+\tan y}{1-\tan (x-y) \tan y}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 46

Use the addition formulas for tangent to simplify each expression.
$$[\tan (\theta+\pi)][\tan (\theta-\pi)]+1$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 47

Compute tan $\frac{7 \pi}{12}$ and rationalize the answer. Hint: $\frac{7 \pi}{22}=\frac{\pi}{3}+\frac{\pi}{4}$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 48

Compute tan $15^{\circ}$ using the fact that $15^{\circ}=45^{\circ}-30^{\circ} .$ Then check that your answer is consistent with the result in Example 7

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 49

Prove that each equation is an identity.
$$\frac{\sin (s+t)}{\cos s \cos t}=\tan s+\tan t$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 50

Prove that each equation is an identity.
$$\frac{\cos (s-t)}{\cos s \sin t}=\cot t+\tan s$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 51

Prove that each equation is an identity.
$$\cos (A-B)-\cos (A+B)=2 \sin A \sin B$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 52

Prove that each equation is an identity.
$$\sin (A-B)+\sin (A+B)=2 \sin A \cos B$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 53

Prove that each equation is an identity.
$$\cos (A+B) \cos (A-B)=\cos ^{2} A-\sin ^{2} B$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 54

Prove that each equation is an identity.
$$\sin (A+B) \sin (A-B)=\cos ^{2} B-\cos ^{2} A$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 55

Prove that each equation is an identity.
$$\cos (\alpha+\beta) \cos \beta+\sin (\alpha+\beta) \sin \beta=\cos \alpha$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 56

Prove that each equation is an identity.
$$\cos \left(\theta+\frac{\pi}{4}\right)+\sin \left(\theta-\frac{\pi}{4}\right)=0$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 57

Prove that each equation is an identity.
$$\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 58

Prove that each equation is an identity.
$\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta$
Hint: Use the addition formulas for tangent and the result in Exercise 57

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 59

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$
In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $f(x)=\sin x .$ Show that
$$
\frac{\Delta f}{\Delta x}=(\sin x)\left(\frac{\cos h-1}{h}\right)+(\cos x)\left(\frac{\sin h}{h}\right)
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 60

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$
In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $g(x)=\cos x .$ Show that
$$
\frac{\Delta g}{\Delta x}=(\cos x)\left(\frac{\cos h-1}{h}\right)-(\sin x)\left(\frac{\sin h}{h}\right)
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 61

You are asked to derive expressions for the average rates of change of the functions $\sin x, \cos x,$ and $\tan x$
In each case, assume that the interval is $[x, x+h] .$ (The results are used in calculus in the study of derivatives.)
Let $T(x)=\tan x .$ Show that
$$
\frac{\Delta T}{\Delta x}=\frac{\tan h}{h} \cdot \frac{\sec ^{2} x}{1-\tan h \tan x}
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 62

Let $\theta$ be the acute angle defined by the following figure.
(FIGURE CAN'T COPY)
Use an addition formula and the figure to show that $5 \sin (x+\theta)=4 \sin x+3 \cos x$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 63

Let $a$ and $b$ be positive constants, and let $\theta$ be the acute angle (in radian measure) defined by the following figure.(a) Use an addition formula and the figure to show that $\sqrt{a^{2}+b^{2}} \sin (x+\theta)=a \sin x+b \cos x$
(b) Use the result in part (a) to specify the maximum value of the function $f(x)=a \sin x+b \cos x$
(FIGURE CAN'T COPY)

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 64

(a) Use an addition formula to show that $2 \sin \left(x+\frac{\pi}{6}\right)=\cos x+\sqrt{3} \sin x$
(b) Use the result in part (a) to graph the function $f(x)=\cos x+\sqrt{3} \sin x$ for one period.

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 65

(a) Use an addition formula to show that $\sqrt{2} \cos \left(x-\frac{\pi}{4}\right)=\cos x+\sin x$
(b) Use the result in part (a) to graph the function $f(x)=\cos x+\sin x$ for one period.

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 66

Let $A, B,$ and $C$ be the angles of a triangle, so that $A+B+C=\pi$
(a) Show that $\sin (A+B)=\sin C$
(b) Show that $\cos (A+B)=-\cos C$
(c) Show that $\tan (A+B)=-\tan C$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 67

Suppose that $A, B,$ and $C$ are the angles of a triangle, so that $A+B+C=\pi .$ Show that
$\cos ^{2} A+\cos ^{2} B+\cos ^{2} C+2 \cos A \cos B \cos C=1$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 68

Prove that
$$
\frac{\sin (\alpha-\beta)}{\cos \alpha \cos \beta}+\frac{\sin (\beta-\gamma)}{\cos \beta \cos \gamma}+\frac{\sin (\gamma-\alpha)}{\cos \gamma \cos \alpha}=0
$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 69

Suppose that $a^{2}+b^{2}=1$ and $c^{2}+d^{2}=1 .$ Prove that $|a c+b d| \leq 1 .$ Hint: Let $a=\cos \theta, b=\sin \theta, c=\cos \phi$
and $d=\sin \phi$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 70

Simplify the expression.
$\cos \left(\frac{\pi}{6}+t\right) \cos \left(\frac{\pi}{6}-t\right)-\sin \left(\frac{\pi}{6}+t\right) \sin \left(\frac{\pi}{6}-t\right)$
Hint: If your solution relies on four separate addition formulas, then you are doing this the hard way.

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 71

Simplify the expression.
$$\begin{array}{l}
\sin \left(\frac{\pi}{3}-t\right) \cos \left(\frac{\pi}{3}+t\right)+\cos \left(\frac{\pi}{3}-t\right) \sin \left(\frac{\pi}{3}+t\right) \\
\tan (A+2 B)-\tan (A-2 B)
\end{array}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 72

Simplify the expression.
$$\frac{\tan (A+2 B)-\tan (A-2 B)}{1+\tan (A+2 B) \tan (A-2 B)}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 73

$$\text { If } \alpha+\beta=\pi / 4, \text { show that }(1+\tan \alpha)(1+\tan \beta)=2$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 74

Outline simple geometric derivations of the
formulas for $\sin (\alpha+\beta)$ and $\cos (\alpha+\beta)$ in the case in which $\alpha$ and $\beta$ are acute angles, with $\alpha+\beta < 90^{\circ} .$ The exercises rely on the accompanying figures, which are constructed as follows. Begin, in Figure $A,$ with $\alpha=\angle G A D, \beta=\angle H A G,$ and $A H=1$ Then, from $H,$ draw perpendiculars to $\overline{A D}$ and to $\overline{A G},$ as shown in Figure $B .$ Finally, draw $\overline{F E} \perp \overline{B H}$ and $\overline{F C} \perp \overline{A D}$
(FIGURES CAN'T COPY)

Formula for $\sin (\alpha+\beta) .$ Supply the reasons or steps behind each statement.
(a) $B H=\sin (\alpha+\beta)$
(b) $F H=\sin \beta$
(c) $\angle B H F=\alpha$
(d) $E H=\cos \alpha \sin \beta \quad$ Hint: Use $\triangle E F H$ and the result in part (b).
(e) $A F=\cos \beta$
(f) $C F=\sin \alpha \cos \beta$
(g) $\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$
Hint: $\sin (\alpha+\beta)=B H=E H+C F$

Anurag Kumar

Numerade Educator

Problem 75

Outline simple geometric derivations of the
formulas for $\sin (\alpha+\beta)$ and $\cos (\alpha+\beta)$ in the case in which $\alpha$ and $\beta$ are acute angles, with $\alpha+\beta < 90^{\circ} .$ The exercises rely on the accompanying figures, which are constructed as follows. Begin, in Figure $A,$ with $\alpha=\angle G A D, \beta=\angle H A G,$ and $A H=1$ Then, from $H,$ draw perpendiculars to $\overline{A D}$ and to $\overline{A G},$ as shown in Figure $B .$ Finally, draw $\overline{F E} \perp \overline{B H}$ and $\overline{F C} \perp \overline{A D}$
(FIGURES CAN'T COPY)

Formula for $\cos (\alpha+\beta) .$ Supply the reasons or steps behind each statement.
(a) $\cos (\alpha+\beta)=A B$
(b) $A C=\cos \alpha \cos \beta$
Hint: Use $\triangle A C F$ and the result in Exercise $74(\mathrm{e})$
(c) $E F=\sin \alpha \sin \beta$
(d) $\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$
Hint: $A B=A C-B C$

Anurag Kumar

Numerade Educator

Problem 76

Let $S$ and $C$ be two functions. Assume that the domain for both $S$ and $C$ is the set of all real numbers and that $S$ and C satisfy the following two identities.
$$
\begin{aligned}
S(x-y) &=S(x) C(y)-C(x) S(y) \\
C(x-y) &=C(x) C(y)+S(x) S(y)
\end{aligned}
$$
Also, suppose that the function $S$ is not identically zero. That is,
$S(x) \neq 0 \quad$ for at least one real number $x$
(a) Show that $S(0)=0 .$ Hint: In identity (1), let $x=y$
(b) Show that $C(0)=1 .$ Hint: In identity (1), let $y=0$
(c) Explain (in complete sentences) why it was necessary to use condition (3) in the work for part (b).
(d) Prove the identity $[C(x)]^{2}+[S(x)]^{2}=1$ Hint: In identity (2), let $y=x$
(e) Show that $C$ is an even function and $S$ is an odd function. That is, prove the identities $C(-x)=C(x)$ and $S(-x)=-S(x)$
Hint: Write $-x$ as $0-x$

Carson Merrill

Numerade Educator

Problem 77

Prove the identities.
$$\frac{\sin (A+B)}{\sin (A-B)}=\frac{\tan A+\tan B}{\tan A-\tan B}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 78

Prove the identities.
$$\frac{\cos (A+B)}{\cos (A-B)}=\frac{1-\tan A \tan B}{1+\tan A \tan B}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 79

Prove the identities.
$$\cot (A+B)=\frac{\cot A \cot B-1}{\cot A+\cot B}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 80

Prove the identities.
$$\cot (A-B)=\frac{\cot A \cot B+1}{\cot B-\cot A}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 81

Let $f(t)=\cos ^{2} t+\cos ^{2}\left(t+\frac{2 \pi}{3}\right)+\cos ^{2}\left(t-\frac{2 \pi}{3}\right)$
(a) Complete the table. (Use a calculator.)
$$\begin{array}{lllll}
\hline t & 1 & 2 & 3 & 4 \\
\hline f(t) & & & & \\
\hline
\end{array}$$
(b) On the basis of your results in part (a), make a conjecture about the function $f .$ Prove that your conjecture is correct.

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 82

(a) Use your calculator to check that $\tan 50^{\circ}-\tan 40^{\circ}=2 \tan 10^{\circ}$
(b) Part (a) is a specific example of a more general identity. State the identity and prove it.

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 83

$$\begin{aligned}
&\text { If } \tan B=\frac{n \sin A \cos A}{1-n \sin ^{2} A}, \text { show that }\\
&\tan (A-B)=(1-n) \tan A
\end{aligned}$$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 84

If triangle $A B C$ is not a right triangle, and $\cos A=\cos B \cos C,$ show that $\tan B \tan C=2$

Subhadeepta Sahoo

Subhadeepta Sahoo

Numerade Educator

Problem 85

(a) The angles of a triangle are $A=20^{\circ}, B=50^{\circ},$ and $C=110^{\circ} .$ Use your calculator to compute the sum $\tan A+\tan B+\tan C$ and then the product
$\tan A \tan B \tan C .$ What do you observe?
(b) The angles of a triangle are $\alpha=\pi / 10, \beta=3 \pi / 10$ and $\gamma=3 \pi / 5 .$ Use your calculator to compute $\tan \alpha+\tan \beta+\tan \gamma$ and $\tan \alpha \tan \beta \tan \gamma$
(c) If triangle $A B C$ is not a right triangle, prove that $\tan A+\tan B+\tan C=\tan A \tan B \tan C$

Aman Gupta

Numerade Educator