3.E: Identities (Exercises)

3.1 Exercises

3.1.1 We showed that $\;\sin\;\theta ~=~ \pm\,\sqrt{1 ~-~ \cos^2 \;\theta}\;$ for all $\theta$. Give an example of an angle $\theta$ such that $\sin\;\theta ~=~ -\sqrt{1 ~-~ \cos^2 \;\theta}\;$.

3.1.2 We showed that $\;\cos\;\theta ~=~ \pm\,\sqrt{1 ~-~ \sin^2 \;\theta}\;$ for all $\theta$. Give an example of an angle $\theta$ such that $\cos\;\theta ~=~ -\sqrt{1 ~-~ \sin^2 \;\theta}\;$.

3.1.3 Suppose that you are given a system of two equations of the following form:

$$\nonumber \begin{align*} A\,\cos\;\phi ~ &= ~ B\, \nu_1 ~-~ B\nu_2 \;\cos\;\theta\\ \nonumber A\,\sin\;\phi ~ &= ~ B\, \nu_2 \;\sin\;\theta ~. \end{align*} \nonumber$$

Show that $\;A ^2 ~=~ B^2 \left( \nu_1^2 ~+~ \nu_2^2 ~-~ 2\nu_1 \nu_2 \;\cos\theta\ \right)$.

For Exercises 4-16, prove the given identity.

3.1.4 $\cos\;\theta ~ \tan\;\theta ~=~ \sin\;\theta$

3.1.5 $\sin\;\theta ~ \cot\;\theta ~=~ \cos\;\theta$

3.1.6 $\dfrac{\tan\;\theta}{\cot\;\theta} ~=~ \tan^2 \;\theta$

3.1.7 $\dfrac{\csc\;\theta}{\sin\;\theta} ~=~ \csc^2 \;\theta$

3.1.8 $\dfrac{\cos^2 \;\theta}{1 ~+~ \sin\;\theta} ~=~ 1 ~-~ \sin\;\theta$

3.1.9 $\dfrac{1 ~-~ 2\;\cos^2 \;\theta}{\sin\;\theta ~ \cos\;\theta} ~=~ \tan\;\theta ~-~ \cot\;\theta$

3.1.10 $\sin^4 \;\theta ~-~ \cos^4 \;\theta ~=~ \sin^2 \;\theta ~-~ \cos^2 \;\theta$

3.1.11 $\cos^4 \;\theta ~-~ \sin^4 \;\theta ~=~ 1 ~-~ 2\;\sin^2 \;\theta$

3.1.12 $\dfrac{1 ~-~ \tan\;\theta}{1 ~+~ \tan\;\theta} ~=~ \dfrac{\cot\;\theta ~-~ 1}{\cot\;\theta ~+~ 1}$

3.1.13 $\dfrac{\tan\;\theta ~+~ \tan\;\phi}{\cot\;\theta ~+~ \cot\;\phi} ~=~ \tan\;\theta ~ \tan\;\phi$

3.1.14 $\dfrac{\sin^2 \;\theta}{1 ~-~ \sin^2 \;\theta} ~=~ \tan^2 \;\theta$

3.1.15 $\dfrac{1 ~-~ \tan^2 \;\theta}{1 ~-~ \cot^2 \;\theta} ~=~ 1 ~-~ \sec^2 \;\theta$

3.1.16 $\sin\;\theta ~=~ \pm\,\dfrac{\tan\;\theta}{\sqrt{1 ~+~ \tan^2 \;\theta}}\qquad$ (Hint: Solve for $\;\sin^2 \theta\;$ in Exercise 14.)

3.1.17 Sometimes identities can be proved by geometrical methods. For example, to prove the identity in Exercise 16, draw an acute angle $\theta$ in QI and pick the point $(1,y)$ on its terminal side, as in Figure 3.1.2. What must $y$ equal? Use that to prove the identity for acute $\theta$. Explain the adjustment(s) you would need to make in Figure 3.1.2 to prove the identity for $\theta$ in the other quadrants. Does the identity hold if $\theta$ is on either axis?

Figure 3.1.2

3.1.18 Similar to Exercise 16 , find an expression for $\cos\;\theta$ solely in terms of $\tan\;\theta$.

3.1.19 Find an expression for $\tan\;\theta$ solely in terms of $\sin\;\theta$, and one solely in terms of $\cos\;\theta$.

3.1.20 Suppose that a point with coordinates $(x,y)=(a\;(\cos\;\psi\;-\;\epsilon),a\sqrt{1 - \epsilon^2}~\sin\;\psi)$ is a distance $r>0$ from the origin, where $a>0$ and $0 < \epsilon < 1$. Use $\;r^2 = x^2 + y^2$ to show that $\;r = a\;(1 \;-\; \epsilon\;\cos\;\psi)\;$.\\(Note: These coordinates arise in the study of elliptical orbits of planets.)

3.1.21 Show that each trigonometric function can be put in terms of the sine function.

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3.2 Exercises

3.2.1 Verify the addition formulas 3.12 and 3.13 for $A=B=0^\circ$.

For Exercises 2 and 3, find the exact values of $\sin\;(A+B)$, $\cos\;(A+B)$, and $\tan\;(A+B)$.

3.2.2 $\sin\;A = \frac{8}{17}$, $\cos\;A = \frac{15}{17}$, $\sin\;B = \frac{24}{25}$,
$\cos\;B = \frac{7}{25}$

3.2.3 $\sin\;A = \frac{40}{41}$, $\cos\;A = \frac{9}{41}$, $\sin\;B = \frac{20}{29}$,
$\cos\;B = \frac{21}{29}$

3.2.4 Use $75^\circ = 45^\circ + 30^\circ$ to find the exact value of $\;\sin\;75^\circ$.

3.2.5 Use $15^\circ = 45^\circ - 30^\circ$ to find the exact value of $\;\tan\;15^\circ$.

3.2.6 Prove the identity $\;\sin\;\theta + \cos\;\theta = \sqrt{2}\;\sin\;(\theta + 45^\circ)\;$. Explain why this shows that

$$-\sqrt{2} ~\le~ \;\sin\;\theta ~+~ \cos\;\theta ~\le~ \sqrt{2} \nonumber$$

for all angles $\theta$. For which $\theta$ between $0^\circ$ and $360^\circ$ would $\;\sin\;\theta \;+\; \cos\;\theta\;$ be the largest?

For Exercises 7-14, prove the given identity.

3.2.7 $\cos\;(A+B+C) \;=\; \cos\;A~\cos\;B~\cos\;C \;-\; \cos\;A~\sin\;B~\sin\;C \;-\; \sin\;A~\cos\;B~\sin\;C \;-\; \sin\;A~\sin\;B~\cos\;C$

3.2.8 $\tan\;(A+B+C) ~=~ \dfrac{\tan\;A \;+\; \tan\;B \;+\; \tan\;C \;-\; \tan\;A~\tan\;B~\tan\;C}{1 \;-\; \tan\;B~\tan\;C \;-\; \tan\;A~\tan\;C \;-\; \tan\;A~\tan\;B}$

3.2.9 $\cot\;(A+B) ~=~ \dfrac{\cot\;A~\cot\;B \;-\; 1}{\cot\;A \;+\; \cot\;B}$

3.2.10 $\cot\;(A-B) ~=~ \dfrac{\cot\;A~\cot\;B \;+\; 1}{\cot\;B \;-\; \cot\;A}$

3.2.11 $\tan\;(\theta + 45^\circ) ~=~ \dfrac{1 \;+\; \tan\;\theta}{1 \;-\; \tan\;\theta}$

3.2.12 $\dfrac{\cos\;(A+B)}{\sin\;A~\cos\;B} ~=~ \cot\;A \;-\; \tan\;B$

3.2.13 $\cot\;A ~+~ \cot\;B ~=~ \dfrac{\sin\;(A+B)}{\sin\;A~\sin\;B}$

3.2.14 $\dfrac{\sin\;(A-B)}{\sin\;(A+B)} ~=~ \dfrac{\cot\;B \;-\; \cot\;A}{\cot\;B \;+\; \cot\;A}$

3.2.15 Generalize Exercise 6: For any $a$ and $b$, $-\sqrt{a^2 + b^2} \;\le\; a\;\sin\;\theta \;+\; b\;\cos\;\theta \;\le\; \sqrt{a^2 + b^2}\;$ for all $\theta$.

3.2.16 Continuing Example 3.12, use Snell's law to show that the s-polarization transmission Fresnel coefficient

$$\tag{3.22} t_{1\;2\;s} ~=~ \frac{2\;n_1 ~\cos\;\theta_1}{n_1 ~\cos\;\theta_1 ~+~ n_2 ~\cos\;\theta_2} \nonumber$$

can be written as:

$$t_{1\;2\;s} ~=~ \frac{2\;\cos\;\theta_1~\sin\;\theta_2}{\sin\;(\theta_2 + \theta_1)} \nonumber$$

3.2.17 Suppose that two lines with slopes $m_1$ and $m_2$, respectively, intersect at an angle $\theta$ and are not perpendicular (i.e. $\theta \ne 90^\circ$), as in the figure on the right. Show that

$$\tan\;\theta ~=~ \left| \frac{m_1 ~-~ m_2}{1 ~+~ m_1 \; m_2} \right| ~. \nonumber$$

(Hint: Use Example 1.26 from Section 1.5.)

3.2.18 Use Exercise 17 to find the angle between the lines $y=2x+3$ and $y=-5x-4$.

3.2.19 For any triangle $\triangle\,ABC$, show that $\;\cot\;A~\cot\;B ~+~ \cot\;B~\cot\;C ~+~ \cot\;C~\cot\;A ~=~ 1$.
(Hint: Use Exercise 9 and $C=180^\circ - (A+B)$.)

3.2.20 For any positive angles $A$, $B$, and $C$ such that $A+B+C=90^\circ$, show that

$$\tan\;A~\tan\;B ~+~ \tan\;B~\tan\;C ~+~ \tan\;C~\tan\;A ~=~ 1 ~. \nonumber$$

3.2.21 Prove the identity $\;\sin\;(A+B)~\cos\;B ~-~ \cos\;(A+B)~\sin\;B ~=~ \sin\;A$. Note that the right side depends only on $A$, while the left side depends on both $A$ and $B$.

3.2.22 A line segment of length $r > 0$ from the origin to the point $(x,y)$ makes an angle $\alpha$ with the positive $x$-axis, so that $(x,y) = (r\;\cos\;\alpha,r\;\sin\;\alpha)$, as in the figure below. What are the endpoint's new coordinates $(x',y')$ after a counterclockwise rotation by an angle $\beta\;$? Your answer should be in terms of $r$, $\alpha$, and $\beta$.

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3.3 Exercises

For Exercises 1-8, prove the given identity.

3.3.1 $\cos\;3\theta ~=~ 4\;\cos^3 \;\theta ~-~ 3\;\cos\;\theta$

3.3.2 $\tan\;\tfrac{1}{2}\theta ~=~ \csc\;\theta ~-~ \cot\;\theta$

3.3.3 $\dfrac{\sin\;2\theta}{\sin\;\theta} ~-~ \dfrac{\cos\;2\theta}{\cos\;\theta} ~=~ \sec\;\theta$

3.3.4 $\dfrac{\sin\;3\theta}{\sin\;\theta} ~-~ \dfrac{\cos\;3\theta}{\cos\;\theta} ~=~ 2$

3.3.5 $\tan\;2\theta ~=~ \dfrac{2}{\cot\;\theta \;-\; \tan\;\theta}$

3.3.6 $\tan\;3\theta ~=~ \dfrac{3\;\tan\;\theta \;-\; \tan^3 \;\theta}{1 \;-\; 3\;\tan^2 \;\theta}$

3.3.7 $\tan^2 \;\tfrac{1}{2}\theta ~=~ \dfrac{\tan\;\theta \;-\; \sin\;\theta}{\tan\;\theta \;+\; \sin\;\theta}$

3.3.8 $\dfrac{\cos^2 \;\psi}{\cos^2 \;\theta} ~=~ \dfrac{1 \;+\; \cos\;2\psi}{1 \;+\; \cos\;2\theta}$

3.3.9 Some trigonometry textbooks used to claim incorrectly that $\;\sin\;\theta ~+~ \cos\;\theta ~=~ \sqrt{1 \;+\; \sin\;2\theta}$ was an identity. Give an example of a specific angle $\theta$ that would make that equation false. Is $\;\sin\;\theta ~+~ \cos\;\theta ~=~ \pm\;\sqrt{1 \;+\; \sin\;2\theta}$ an identity? Justify your answer.

3.3.10 Fill out the rest of the table below for the angles $0^\circ < \theta < 720^\circ$ in increments of $90^\circ$, showing $\theta$, $\tfrac{1}{2}\theta$, and the signs ($+$ or $-$) of $\sin\;\theta$ and $\tan\;\tfrac{1}{2}\theta$.

3.3.11 In general, what is the largest value that $\;\sin\;\theta~\cos\;\theta\;$ can take? Justify your answer.

For Exercises 12-17, prove the given identity for any right triangle $\triangle\,ABC$ with $C=90^\circ$.

3.3.12 $\sin\;(A-B) ~=~ \cos\;2B$

3.3.13 $\cos\;(A-B) ~=~ \sin\;2A$

3.3.14 $\sin\;2A ~=~ \dfrac{2\;ab}{c^2}$

3.3.15 $\cos\;2A ~=~ \dfrac{b^2 - a^2}{c^2}$

3.3.16 $\tan\;2A ~=~ \dfrac{2\;ab}{b^2 - a^2}$

3.3.17 $\tan\;\tfrac{1}{2}A ~=~ \dfrac{c - b}{a} ~=~ \dfrac{a}{c + b}$

3.3.18 Continuing Exercise 20 from Section 3.1, it can be shown that

$$\begin{align*} r\;(1 \;-\; \cos\;\theta) ~&=~ a\;(1 \;+\; \epsilon)\,(1 \;-\; \cos\;\psi) ~,~\text{and}\\ r\;(1 \;+\; \cos\;\theta) ~&=~ a\;(1 \;-\; \epsilon)\,(1 \;+\; \cos\;\psi) ~, \end{align*} \nonumber$$

where $\theta$ and $\psi$ are always in the same quadrant. Show that $\;\tan\;\tfrac{1}{2}\theta ~=~ \sqrt{\frac{1 \;+\; \epsilon}{1 \;-\; \epsilon}}~ \tan\;\tfrac{1}{2}\psi\;$.

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3.4 Exercises

3.4.1 Prove formula 3.38.

3.4.2 Prove formula 3.39.

3.4.3 Prove formula 3.40.

3.4.4 Prove formula 3.41.

3.4.5 Prove formula 3.42.

3.4.6 Prove formula 3.44.

3.4.7 Prove Mollweide's second equation: For any triangle $\triangle\,ABC$, $~\dfrac{a+b}{c} ~=~ \dfrac{\cos\;\tfrac{1}{2}(A-B)}{\sin\;\tfrac{1}{2}C}$.

3.4.8 Continuing Example 3.21, use Snell's law to show that the p-polarization reflection Fresnel coefficient

$$\tag{3.46} r_{1\;2\;p} ~=~ \frac{n_2 ~\cos\;\theta_1 ~-~ n_1 ~\cos\;\theta_2}{n_2 ~\cos\;\theta_1 ~+~ n_1 ~\cos\;\theta_2} \nonumber$$

can be written as:

$$r_{1\;2\;p} ~=~ \frac{\tan\;(\theta_1 - \theta_2)}{\tan\;(\theta_1 + \theta_2)} \nonumber$$

3.4.9 There is a more general form for the instantaneous power $p(t) = v(t)\;i(t)$ in an electrical circuit than the one in Example 3.22. The voltage $v(t)$ and current $i(t)$ can be given by

$$\begin{align*} v(t) ~&=~ V_m \;\cos\;(\omega t + \theta)~,\\ i(t) ~&=~ I_m \;\cos\;(\omega t + \phi)~, \end{align*} \nonumber$$

where $\theta$ is called the phase angle. Show that $p(t)$ can be written as

$$p(t) ~=~ \tfrac{1}{2}\,V_m \; I_m \;\cos\;(\theta - \phi) ~+~ \tfrac{1}{2}\,V_m \; I_m \;\cos\;(2\omega t + \theta + \phi) ~. \nonumber$$

For Exercises 10-15, prove the given identity or inequality for any triangle $\triangle\,ABC$.

3.4.10 $\sin\;A \;+\; \sin\;B \;+\; \sin\;C ~=~ 4\;\cos\;\tfrac{1}{2}A~\cos\;\tfrac{1}{2}B~\cos\;\tfrac{1}{2}C$ (Hint: Mimic Example 3.18 using $(\sin\;A \;+\; \sin\;B) \;+\; (\sin\;C \;-\; \sin\;(A+B+C))$.)

3.4.11 $\cos\;A \;+\; \cos\;(B-C) ~=~ 2\;\sin\;B~\sin\;C$

3.4.12 $\sin\;2A \;+\; \sin\;2B \;+\; \sin\;2C ~=~ 4\;\sin\;A~\sin\;B~\sin\;C$ (Hints: Group $\sin\;2B$ and $\sin\;2C$ together, use the double-angle formula for $\sin\;2A$, use Exercise 11.)

3.4.13 $\dfrac{a-b}{a+b} ~=~ \dfrac{\sin\;A \;-\; \sin\;B}{\sin\;A \;+\; \sin\;B}$

3.4.14 $\cos\;\tfrac{1}{2}A ~=~ \sqrt{\dfrac{s\;(s-a)}{bc}}~~$ and $~~\sin\;\tfrac{1}{2}A ~=~ \sqrt{\dfrac{(s-b)\;(s-c)}{bc}}\;$, \;where $s=\tfrac{1}{2}(a+b+c)$$ (Hint: Use the Law of Cosines to show that $2bc\;(1 + \cos\;A) ~=~ 4s\;(s-a)$.)

3.4.15 $\tfrac{1}{2}\;(\sin\;A \;+\; \sin\;B) ~\le~ \sin\;\tfrac{1}{2}(A+B)$ (Hint: Show that $\sin\;\tfrac{1}{2}(A+B) \;-\; \tfrac{1}{2}\;(\sin\;A \;+\; \sin\;B) \;\ge\; 0$.)

3.4.16 In Example 3.20, which angles $A$, $B$, $C$ give the maximum value of $\cos\;A \;+\; \cos\;B \;+\; \cos\;C\;$?