
# 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*}
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. ## 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
$\nonumber -\sqrt{2} ~\le~ \;\sin\;\theta ~+~ \cos\;\theta ~\le~ \sqrt{2}$
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
$\tag{3.22} t_{1\;2\;s} ~=~ \frac{2\;n_1 ~\cos\;\theta_1}{n_1 ~\cos\;\theta_1 ~+~ n_2 ~\cos\;\theta_2}$
can be written as:
$\nonumber t_{1\;2\;s} ~=~ \frac{2\;\cos\;\theta_1~\sin\;\theta_2}{\sin\;(\theta_2 + \theta_1)}$

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

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

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 $$\beta$$.

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

## 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
$\tag{3.46} r_{1\;2\;p} ~=~ \frac{n_2 ~\cos\;\theta_1 ~-~ n_1 ~\cos\;\theta_2}{n_2 ~\cos\;\theta_1 ~+~ n_1 ~\cos\;\theta_2}$
can be written as:
$r_{1\;2\;p} ~=~ \frac{\tan\;(\theta_1 - \theta_2)}{\tan\;(\theta_1 + \theta_2)}$
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*}
where $$\theta$$ is called the phase angle. Show that $$p(t)$$ can be written as
$p(t) ~=~ \tfrac{1}{2}\,V_m \; I_m \;\cos\;(\theta - \phi) ~+~ \tfrac{1}{2}\,V_m \; I_m \;\cos\;(2\omega t + \theta + \phi) ~.$

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 $$\cos\;A \;+\; \cos\;B \;+\; \cos\;C\;$$?