17b. Differentiation of Trig Functions — Practice

Drill sheet for 17. Differentiation of Trig Functions. Book-style solved ideas rewritten as guided proofs and regenerable worksheets — not a verbatim dump.

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1. Prove $\lim_{\theta\to 0}\dfrac{\sin\theta}{\theta}=1$

The quotient is even, so treat $\theta\to 0^{+}$. On the unit circle, sector $COD$ ≤ $\triangle COB$ ≤ sector $AOB$ yields

$$\cos\theta\le\frac{\sin\theta}{\theta}\le\frac{1}{\cos\theta}\xrightarrow{\theta\to 0^{+}}1.$$

Fig. 17-11 — sector COD ≤ △COB ≤ sector AOB

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2. Prove $D_x(\sin x)=\cos x$

$$\frac{\Delta y}{\Delta x} =\cos x\cdot\frac{\sin\Delta x}{\Delta x} +\sin x\cdot\frac{\cos\Delta x-1}{\Delta x} \to\cos x.$$

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3. Prove $\tan'={\sec}^{2}$ and $\sec'=\tan\sec$

(a) Quotient rule on $\sin/\cos$.

(b) Two routes: reciprocal rule on $1/\cos$, or differentiate $\tan^{2}x+1=\sec^{2}x$:

$$2\tan x\sec^{2}x=2\sec x\cdot D_x(\sec x)\implies D_x(\sec x)=\tan x\sec x.$$

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4–5. Period of $\tan$ and $\tan(u-v)$

$\sin(x+\pi)=-\sin x$, $\cos(x+\pi)=-\cos x$ ⇒ $\tan(x+\pi)=\tan x$.

Divide $\sin(u-v)/\cos(u-v)$ by $\cos u\cos v$:

$$\tan(u-v)=\frac{\tan u-\tan v}{1+\tan u\tan v}.$$

(The plus formula flips signs: $(\tan u+\tan v)/(1-\tan u\tan v)$.)

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6. Nested chain-rule derivatives

Book. $2\cos 7x$, $\sin^{3}(2x)$, $\tan(5x)$, $\sec(1/x)$.

Nested trig derivatives

Outer power / trig × inner chain

(1)Differentiate: (a) $2\cos 7x$ (b) $\sin^{3}(2x)$ (c) $\tan(5x)$ (d) $\sec(1/x)$.

(a) $-14\sin 7x$.

(b) $6\sin^{2}(2x)\cos(2x)$.

(c) $5\sec^{2}(5x)$.

(d) $-(1/x^{2})\tan(1/x)\sec(1/x)$.

See steps.

(2)Differentiate $y=\cot(1-2x^{2})$.

$y'=-\csc^{2}(1-2x^{2})\cdot(-4x)=4x\csc^{2}(1-2x^{2})$.

$y'=4x\csc^{2}(1-2x^{2})$.

(3)Differentiate $y=\cot(1-2x^{2})$.

$y'=-\csc^{2}(1-2x^{2})\cdot(-4x)=4x\csc^{2}(1-2x^{2})$.

$y'=4x\csc^{2}(1-2x^{2})$.

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7. Solve $\cos x=\tfrac12$ (and kin)

Unit-circle points with $x$-coordinate $1/2$: angles $\pi/3$ and $5\pi/3$ in $[0,2\pi]$; then add $2\pi n$.

Trig equations — general solution

Principal solutions + period

(1)Find all solutions of $\cos x=1/2$.

On $[0,2\pi]$: $x=\pi/3$ and $x=5\pi/3$.

General: $\pi/3+2\pi n$ and $5\pi/3+2\pi n$.

$x=\pm\pi/3+2\pi n$ (equiv. form OK).

(2)Find all solutions of $\cos x=0$.

Odd multiples of $\pi/2$.

$x=(2n+1)\pi/2$.

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8. Limits via $(\sin u)/u\to 1$

Book. $(\sin 5x)/(2x)\to 5/2$; $(\sin 3x)/(\sin 7x)\to 3/7$; $(\tan x)/x\to 1$.

Trig limits as x→0

Rewrite to sin u / u (radians)

(1)Evaluate: (a) $\lim_{x\to 0}\dfrac{\sin 5x}{2x}$ (b) $\lim_{x\to 0}\dfrac{\sin 3x}{\sin 7x}$ (c) $\lim_{x\to 0}\dfrac{\tan x}{x}$.

(a) $\tfrac{5}{2}\lim(\sin 5x)/(5x)=5/2$.

(b) $(\sin 3x)/(3x)\cdot(7x)/(\sin 7x)\cdot 3/7\to 3/7$.

(c) $(\sin x)/x\cdot 1/\cos x\to 1$.

(a) $5/2$; (b) $3/7$; (c) $1$.

(2)Evaluate $\lim_{x\to 0}\dfrac{\sin(4x)}{2x}$.

$\dfrac{a}{2}\lim\dfrac{\sin(4x)}{4x}=\dfrac{a}{2}$.

$\dfrac{4}{2}$.

(3)Evaluate $\lim_{x\to 0}\dfrac{\sin(5x)}{2x}$.

$\dfrac{a}{2}\lim\dfrac{\sin(5x)}{5x}=\dfrac{a}{2}$.

$\dfrac{5}{2}$.

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9–12. Higher / implicit derivatives

Book 9. $y=x\sin x$ ⇒ $y'''=-x\cos x-3\sin x$.

Book 10. $y=\tan^{2}(3x-2)$ ⇒ first $y'=6\tan(\cdots)\sec^{2}(\cdots)$; then product for $y''$.

Book 11. $y=\sin(x+y)$ ⇒ $y'=\dfrac{\cos(x+y)}{1-\cos(x+y)}$.

Book 12. $\sin y+\cos x=1$ ⇒ $y'=\sin x/\cos y$; differentiate again for $y''$.

Higher and implicit

Repeated product / solve for y′

(1)If $y=x\sin x$, find $y'''$.

$y'=x\cos x+\sin x$.

$y''=-x\sin x+2\cos x$.

$y'''=-x\cos x-3\sin x$.

$y'''=-x\cos x-3\sin x$.

(2)If $y=\tan^{2}(3x-2)$, find $y'$ (first derivative only).

$y'=2\tan(3x-2)\sec^{2}(3x-2)\cdot 3=6\tan(3x-2)\sec^{2}(3x-2)$.

$y'=6\tan(3x-2)\sec^{2}(3x-2)$.

(3)If $\sin y+\cos x=1$, find $y'$ (first derivative).

$\cos y\,y'-\sin x=0$ ⇒ $y'=\sin x/\cos y$.

$y'=\dfrac{\sin x}{\cos y}$.

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13. Related rates — pilot sighting

Altitude $2$ mi, ground distance $x=2\cot\theta$, $dx/dt=-240$ mi/h. At $\theta=30°$,

$$\frac{dx}{dt}=-2\csc^{2}\theta\frac{d\theta}{dt}\implies\frac{d\theta}{dt}=30\text{ rad/h}.$$

Fig. 17-12 — x = 2 cot θ

Related rates with cot / csc

Differentiate the geometric relation in t

(1)A $50$ ft pole; sun elevation $\theta=45^{\circ}$ with $d\theta/dt=-1/4$ rad/h. Shadow length $s=50\cot\theta$. How fast is the shadow lengthening?

$ds/dt=-50\csc^{2}\theta\,d\theta/dt$.

At $45^{\circ}$, $\csc=\sqrt{2}$, so $-50\cdot 2\cdot(-1/4)=25$ ft/h.

$25$ ft/h.

(2)A $50$ ft pole; sun elevation $\theta=45^{\circ}$ with $d\theta/dt=-1/4$ rad/h. Shadow length $s=50\cot\theta$. How fast is the shadow lengthening?

$ds/dt=-50\csc^{2}\theta\,d\theta/dt$.

At $45^{\circ}$, $\csc=\sqrt{2}$, so $-50\cdot 2\cdot(-1/4)=25$ ft/h.

$25$ ft/h.

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14. Sketch $f(x)=\sin x+\cos x$

Period $2\pi$. $f'=\cos-\sin=0$ ⇒ $\tan x=1$ ⇒ $x=\pi/4$ (max $\sqrt{2}$), $x=5\pi/4$ (min $-\sqrt{2}$). Inflection where $\tan x=-1$: $3\pi/4$, $7\pi/4$.

Fig. 17-13 — y = sin x + cos x

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15. Sketch $f(x)=\cos x-\cos^{2}x$

Even and $2\pi$-periodic; work on $[0,\pi]$. $f'=\sin x(2\cos x-1)=0$ ⇒ $x=0,\pi/3,\pi$. Classify: mins at $0$ and $\pi$; max at $(\pi/3,1/4)$. Inflections from $f''=0$ (quadratic in $\cos x$).

Fig. 17-14 — y = cos x − cos² x

Trig extrema / absolute extrema

Critical numbers + endpoint table

(1)Critical numbers of $f(x)=\sin x+\cos x$ on $[0,2\pi]$ and classification.

$\tan x=1$ ⇒ $\pi/4$ (max $\sqrt{2}$), $5\pi/4$ (min $-\sqrt{2}$).

Max $(\pi/4,\sqrt{2})$; min $(5\pi/4,-\sqrt{2})$.

(2)Critical numbers of $f(x)=\sin x+\cos x$ on $[0,2\pi]$ and classification.

$\tan x=1$ ⇒ $\pi/4$ (max $\sqrt{2}$), $5\pi/4$ (min $-\sqrt{2}$).

Max $(\pi/4,\sqrt{2})$; min $(5\pi/4,-\sqrt{2})$.

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16. Absolute extrema of $\sin x+x$ on $[0,2\pi]$

$f'=\cos x+1=0$ ⇒ $x=\pi$. Table: $f(0)=0$, $f(\pi)=\pi$, $f(2\pi)=2\pi$ ⇒ abs min $0$ at $0$; abs max $2\pi$ at $2\pi$.

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17–18. Angle between lines / curves

Book 17. $m_1=1$, $m_2=-3$:

$$\tan(\alpha_2-\alpha_1)=\frac{m_2-m_1}{1+m_1 m_2}=2\quad(\approx 63.4^\circ).$$

Book 18. At $(1,1)$ on $y=x^{2}$ and $y=\sqrt{x}$: slopes $2$ and $1/2$; $\tan\alpha=3/4$ ($\approx 36.9^\circ$).

Fig. 17-15 — intersection angle from slopes

tan(α₂−α₁) from slopes

Use the tan(u−v) formula

(1)Angle between $y=x+1$ ($m_1=1$) and $y=-3x+5$ ($m_2=-3$): find $\tan(\alpha_2-\alpha_1)$.

$\tan(\alpha_2-\alpha_1)=\dfrac{m_2-m_1}{1+m_1 m_2}=\dfrac{-4}{-2}=2$.

$\tan=2$ ($\approx 63.4^{\circ}$).

(2)Slopes $m_1=1$, $m_2=0$: acute intersection angle?

$\tan\phi=1$ ⇒ $\phi=\pi/4$.

$\pi/4$.

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Supplementary drills

Period / amplitude / equations

Book 19–22. Periodicity of $\cot,\sec,\csc$; $5\sin(x/3)$ has $p=6\pi$, $f=1/3$, $A=5$; $\cos x=0$ ⇒ $(2n+1)\pi/2$; $\tan x=1$ ⇒ $(4n+1)\pi/4$.

Amplitude / period / frequency

y = A sin(bx) readout

(1)For $y=1.5\sin 4x$: amplitude, period, frequency.

Amplitude $|A|=1.5$.

Period $2\pi/b=2\pi/4=\pi/2$.

Frequency $b=4$.

amp $1.5$; $p=\pi/2$; $f=4$.

(2)For $y=1.5cos(6x)$: give amplitude, period, and frequency.

Amplitude $|A|=1.5$.

Period $2\pi/6=\dfrac{1}{3}\pi$.

Frequency $b=6$.

amp $1.5$; $p=\dfrac{1}{3}\pi$; $f=6$.

Sketch $\dfrac{\sin x}{2-\cos x}$

Domain all $x$ ($2-\cos x\ge 1$). Period $2\pi$; use quotient rule for critical numbers.

Fig. 17-16 — y = sin x / (2 − cos x)

More differentiation (Book 25)

Mixed differentiation

Product, quotient, and chain

(1)Differentiate $y=\sin x\cos x$.

Product: $\cos x\cos x+\sin x(-\sin x)=\cos^{2}x-\sin^{2}x=\cos 2x$.

$y'=\cos 2x$ (or $\cos^{2}x-\sin^{2}x$).

(2)Differentiate $y=\sin x\cos x$.

Product: $\cos x\cos x+\sin x(-\sin x)=\cos^{2}x-\sin^{2}x=\cos 2x$.

$y'=\cos 2x$ (or $\cos^{2}x-\sin^{2}x$).

(3)Differentiate $y=\sin x\cos x$.

Product: $\cos x\cos x+\sin x(-\sin x)=\cos^{2}x-\sin^{2}x=\cos 2x$.

$y'=\cos 2x$ (or $\cos^{2}x-\sin^{2}x$).

Basic chain compositions

sin(kx), tan(kx), …

(1)Differentiate: (a) $y=\sin 3x$ (b) $y=\cos(x^{2})$ (c) $y=\tan 2x$.

$D_x(\sin 3x)=\cos 3x\cdot 3=3\cos 3x$.

$D_x(\cos(x^{2}))=-\sin(x^{2})\cdot 2x=-2x\sin(x^{2})$.

$D_x(\tan 2x)=\sec^{2}(2x)\cdot 2=2\sec^{2}(2x)$.

(a) $3\cos 3x$; (b) $-2x\sin(x^{2})$; (c) $2\sec^{2}(2x)$.

(2)Differentiate $y=cos(5x)$.

Chain rule: outer trig derivative times $5$.

$y'=-5\sin 5x$.

Limits (Book 26)

$\lim(\sin ax)/(\sin bx)=a/b$; $\lim\sin^{3}(2x)/(x\sin^{2}(3x))=8/9$.

More trig limits

Factor constants into sin u / u

(1)Evaluate $\lim_{x\to 0}\dfrac{\sin(6x)}{\sin(7x)}$.

$\dfrac{\sin(6x)}{6x}\cdot\dfrac{7x}{\sin(7x)}\cdot\dfrac{a}{b}\to\dfrac{a}{b}$.

$\dfrac{6}{7}$.

(2)Evaluate $\lim_{x\to 0}\dfrac{\tan(4x)}{x}$.

$\dfrac{\sin(4x)}{x}\cdot\dfrac{1}{\cos(4x)}=a\cdot\dfrac{\sin(4x)}{4x}\cdot\dfrac{1}{\cos(4x)}\to a$.

$4$.

DE checks / sketch gallery (Book 27–29)

If $x=A\sin kt+B\cos kt$, then $x''=-k^{2}x$. For $y=3\sin(2x+3)$, $y''+4y=0$. Sketch gallery on $[0,2\pi]$: $\tfrac12\sin 2x$, $\cos^{2}x-\cos x$, $x-2\sin x$, etc. — find crits with $y'=0$ and classify.

Extrema / mono on trig graphs

Sign charts for y′ and y″

(1)Critical numbers of $\sec x$ in $(-\pi,\pi)$ where defined.

$\tan x\sec x=0$ ⇒ $\tan x=0$ ⇒ $x=0$ (also $\pm\pi$ on a closed picture).

First-derivative test: relative min at $0$.

Crit at $x=0$ (relative min).

(2)Find inflection points of $y=\sin x$ in $[0,2\pi]$.

$y''=-\sin x=0$ ⇒ $x=0,\pi,2\pi$; sign of $y''$ changes at each.

$(0,0)$, $(\pi,0)$, $(2\pi,0)$.

Related rates / implicit (Book 30–31)

Shadow of a $50$ ft pole with $\theta'= -1/4$ rad/h at $45^\circ$ lengthens at $25$ ft/h. Implicit: $\tan y=x^{2}$, $\cos(xy)=2y$.

Implicit y′

Differentiate both sides; solve for y′

(1)If $y=\sin(x+y)$, solve for $y'$.

$y'=\cos(x+y)(1+y')$.

$y'(1-\cos(x+y))=\cos(x+y)$.

$y'=\dfrac{\cos(x+y)}{1-\cos(x+y)}$.

(2)If $y=\sin(x+y)$, solve for $y'$.

$y'=\cos(x+y)(1+y')$.

$y'(1-\cos(x+y))=\cos(x+y)$.

$y'=\dfrac{\cos(x+y)}{1-\cos(x+y)}$.

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Mixed review set

Supplementary mixed set

Problems 6–31 style

(1)For $y=3cos(5x)$: give amplitude, period, and frequency.

Amplitude $|A|=3$.

Period $2\pi/5=\dfrac{2}{5}\pi$.

Frequency $b=5$.

amp $3$; $p=\dfrac{2}{5}\pi$; $f=5$.

(2)Differentiate: (a) $y=\sin 3x$ (b) $y=\cos(x^{2})$ (c) $y=\tan 2x$.

$D_x(\sin 3x)=\cos 3x\cdot 3=3\cos 3x$.

$D_x(\cos(x^{2}))=-\sin(x^{2})\cdot 2x=-2x\sin(x^{2})$.

$D_x(\tan 2x)=\sec^{2}(2x)\cdot 2=2\sec^{2}(2x)$.

(a) $3\cos 3x$; (b) $-2x\sin(x^{2})$; (c) $2\sec^{2}(2x)$.

(3)Differentiate $y=\sqrt{\sin x}$ (where $\sin x>0$).

Chain: $\dfrac{1}{2}(\sin x)^{-1/2}\cos x=\dfrac{\cos x}{2\sqrt{\sin x}}$.

$y'=\dfrac{\cos x}{2\sqrt{\sin x}}$.

(4)Slopes $m_1=2$, $m_2=-1/2$: are the lines perpendicular?

$m_1 m_2=2\cdot(-1/2)=-1$ ⇒ yes, $1+m_1 m_2=0$.

Yes — perpendicular.

(5)Evaluate $\lim_{x\to 0}\dfrac{\tan(4x)}{x}$.

$\dfrac{\sin(4x)}{x}\cdot\dfrac{1}{\cos(4x)}=a\cdot\dfrac{\sin(4x)}{4x}\cdot\dfrac{1}{\cos(4x)}\to a$.

$4$.

(6)Critical numbers of $\sec x$ in $(-\pi,\pi)$ where defined.

$\tan x\sec x=0$ ⇒ $\tan x=0$ ⇒ $x=0$ (also $\pm\pi$ on a closed picture).

First-derivative test: relative min at $0$.

Crit at $x=0$ (relative min).

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Self-check

Quiz

Trig differentiation review

1 / 5

lim θ→0 (sin θ)/θ equals…

Pick an answer to see if you are right.

Further Reading

• 17. Differentiation of Trig Functions — theory and explorers
• 16b. Trig Review — Practice — prior trig drills
• 16. Review of Trigonometry — radians and identities
• 14b. Max/Min — Practice — absolute extrema workflow
• 18. Inverse Trig Functions — next: arcsin / arctan