How to deal with Sines and Cosines

Excerpts

How to deal with Sines and Cosines
What we have to investigate is the value of $\dfrac{d(\sin \theta)}{d \theta}$; or, in other words, if the angle $\theta$ varies, we have to find the relation between the increment of the sine and the increment of the angle, both increments being indefinitely small in themselves.
How to deal with Sines and Cosines
But if we regard $d \theta$ as indefinitely small, then in the limit we may neglect $\frac{1}{2} d \theta$ by comparison with $\theta$, and may also take $\sin\frac{1}{2} d \theta$ as being the same as $\frac{1}{2} d \theta$.
How to deal with Sines and Cosines
Now $\cos \theta=\sin\left(\dfrac{\pi}{2}-\theta\right)$.
How to deal with Sines and Cosines
Passing now from the inverse function to the original one, we get
How to deal with Sines and Cosines
*Sines and cosines are the only functions of which the second differential coefficient is equal *(and of opposite sign to)* the original function.*
How to deal with Sines and Cosines
So we have this curious result that we have found a function such that if we differentiate it twice over, we get the same thing from which we started, but with the sign changed from $+$ to $-$.
How to deal with Sines and Cosines
If the *frequency*, or number of periods per second, be denoted by $n$, then $n = \dfrac{1}{T}$, and we may then write:

Equations

How to deal with Sines and Cosines
y= \sin \theta
The height y is defined as the sine of the angle theta.
How to deal with Sines and Cosines
dy = \sin(\theta + d \theta)- \sin \theta
The increment dy of the sine equals the sine of the increased angle minus the sine of the original angle.
How to deal with Sines and Cosines
\sin M - \sin N = 2 \cos\frac{M+N}{2}·\sin\frac{M-N}{2}
The difference of two sines equals twice the cosine of half the sum times the sine of half the difference of the angles.
How to deal with Sines and Cosines
dy = \cos \theta · d \theta
In the limit of an indefinitely small angle, the increment of the sine is cosine theta times the increment of the angle.
How to deal with Sines and Cosines
\cos \theta=\sin\left(\dfrac{\pi}{2}-\theta\right)
Cosine of an angle equals sine of its complement, pi/2 minus the angle.
How to deal with Sines and Cosines
\frac{dy}{d\theta} = -\sin \theta
The derivative of cosine theta with respect to theta is minus sine theta.
How to deal with Sines and Cosines
\frac{dy}{d\theta} = \sec^2 \theta
The derivative of tangent theta with respect to theta is secant squared theta.
How to deal with Sines and Cosines
-(1+\cot^2 \theta) = -\cosec^2 \theta
Minus one plus cotangent squared equals minus cosecant squared, the step by which the derivative of cotangent is reduced.
How to deal with Sines and Cosines
\frac{1}{\sin\theta} × \cos\theta = \cot\theta
Cosine divided by sine equals cotangent, which gives the derivative of log of sine in the example.
How to deal with Sines and Cosines
\theta = 2\pi\frac{t}{T}
The angle moved through in time t, for a motion with period T, is two pi times t over T in radians.
How to deal with Sines and Cosines
\theta = 360\frac{t}{T}
The angle moved through in time t, for a motion with period T, is 360 times t over T in degrees.
How to deal with Sines and Cosines
n = \dfrac{1}{T}
The frequency n, the number of periods per second, is the reciprocal of the period T.
How to deal with Sines and Cosines
\theta=2\pi nt.
The angle equals two pi times the frequency times the time.
How to deal with Sines and Cosines
y = \sin 2\pi nt.
A sine varying with time as sine of two pi n t, a simple periodic motion.
How to deal with Sines and Cosines
\frac{dy}{dt} = \frac{dy}{d\theta} · \frac{d\theta}{dt}
The rate of change of y with time is the rate with respect to the angle times the rate of the angle with time.
How to deal with Sines and Cosines
\frac{d(\cos 2\pi nt)}{dt} = -2\pi n · \sin 2\pi nt
The derivative of cosine of two pi n t with respect to time is minus two pi n times sine of two pi n t.
How to deal with Sines and Cosines
\frac{d^2(\DPtypo{\cos \theta}{\sin \theta})}{d\theta^2} = -\sin \theta
The second derivative of sine theta with respect to theta is minus sine theta, so sine is a function whose second derivative is its own negative. The source's DPtypo markup shows cos theta struck for sin theta; the corrected reading (sin theta) is what the text's argument uses, and this is a typesetting note to flag rather than a silent fix.
How to deal with Sines and Cosines
\frac{d^2(\cos\theta)}{d\theta^2} = -\cos\theta
The second derivative of cosine theta with respect to theta is minus cosine theta.
How to deal with Sines and Cosines
\frac{dy}{dx}=\cos(x+a)
The derivative of sine of x plus a with respect to x is cosine of x plus a.
How to deal with Sines and Cosines
\frac{dy}{d\theta}=3 \sec^2 3\theta
The derivative of tangent of three theta with respect to theta is three times secant squared of three theta.

Problems

Exercise XIV

Exercise XIV, problem 101, p. 174
Differentiate $y=\epsilon^x \sin^2 x$.
Printed answer:
- $\epsilon^x \left(\sin^2 x + \sin2x\right)$; $\epsilon^x \left(\sin^2 x + 2\sin2x + 2\cos2x\right)$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes exp(x)*(sin(x)**2 + sin(2*x))
Exercise XIV, problem 102, p. 174
Differentiate $y=\epsilon^x \sin^2 x$.
Printed answer:
- $\epsilon^x \left(\sin^2 x + \sin2x\right)$; $\epsilon^x \left(\sin^2 x + 2\sin2x + 2\cos2x\right)$.
verified: the printed answer passed a computed check
How it was checked
- differentiate2: passes exp(x)*(sin(x)**2 + 2*sin(2*x) + 2*cos(2*x))
Exercise XIV, problem 111, p. 174
Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.
Printed answer:
- $\left(i\right) \dfrac{dy}{dx} = \dfrac{ab}{\left(x + b\right)^2}$; (ii) $\dfrac{a}{b} \epsilon^{-\efrac{x}{b}}$; (iii) $\dfrac{1}{90}° × \dfrac{ab}{\left(b^2 + x^2\right)}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check a*b/(x + b)**2
Exercise XIV, problem 112, p. 174
Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.
Printed answer:
- $\left(i\right) \dfrac{dy}{dx} = \dfrac{ab}{\left(x + b\right)^2}$; (ii) $\dfrac{a}{b} \epsilon^{-\efrac{x}{b}}$; (iii) $\dfrac{1}{90}° × \dfrac{ab}{\left(b^2 + x^2\right)}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check a/b*exp(-x/b)
Exercise XIV, problem 113, p. 174
Differentiate the three equations of Exercises XIII. (XIII:4), No. 4, and compare their differential coefficients, as to whether they are equal, or nearly equal, for very small values of $x$, or for very large values of $x$, or for values of $x$ in the neighbourhood of $x=30$.
Printed answer:
- $\left(i\right) \dfrac{dy}{dx} = \dfrac{ab}{\left(x + b\right)^2}$; (ii) $\dfrac{a}{b} \epsilon^{-\efrac{x}{b}}$; (iii) $\dfrac{1}{90}° × \dfrac{ab}{\left(b^2 + x^2\right)}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check a*b/(90*(b**2 + x**2))
Exercise XIV, problem 12i, p. 174
Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*
Printed answer:
- (i) $\dfrac{dy}{dx} = \sec x \tan x$; (ii) $\dfrac{dy}{dx} = - \dfrac{1}{\sqrt{ 1 - x^2}}$; (iii) $\dfrac{dy}{dx} = \dfrac{1}{ 1 + x^2}$; (iv) $\dfrac{dy}{dx} = \dfrac{1}{x \sqrt{ x^2 - 1}}$; (v) $\dfrac{dy}{dx} = \dfrac{\sqrt{ 3\sec x} \left(3\sec^2 x - 1\right)}{2}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes tan(x)/cos(x)
Exercise XIV, problem 12ii, p. 174
Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*
Printed answer:
- (i) $\dfrac{dy}{dx} = \sec x \tan x$; (ii) $\dfrac{dy}{dx} = - \dfrac{1}{\sqrt{ 1 - x^2}}$; (iii) $\dfrac{dy}{dx} = \dfrac{1}{ 1 + x^2}$; (iv) $\dfrac{dy}{dx} = \dfrac{1}{x \sqrt{ x^2 - 1}}$; (v) $\dfrac{dy}{dx} = \dfrac{\sqrt{ 3\sec x} \left(3\sec^2 x - 1\right)}{2}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes -1/sqrt(1 - x**2)
Exercise XIV, problem 12iii, p. 174
Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*
Printed answer:
- (i) $\dfrac{dy}{dx} = \sec x \tan x$; (ii) $\dfrac{dy}{dx} = - \dfrac{1}{\sqrt{ 1 - x^2}}$; (iii) $\dfrac{dy}{dx} = \dfrac{1}{ 1 + x^2}$; (iv) $\dfrac{dy}{dx} = \dfrac{1}{x \sqrt{ x^2 - 1}}$; (v) $\dfrac{dy}{dx} = \dfrac{\sqrt{ 3\sec x} \left(3\sec^2 x - 1\right)}{2}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 1/(1 + x**2)
Exercise XIV, problem 12iv, p. 174
Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*
Printed answer:
- (i) $\dfrac{dy}{dx} = \sec x \tan x$; (ii) $\dfrac{dy}{dx} = - \dfrac{1}{\sqrt{ 1 - x^2}}$; (iii) $\dfrac{dy}{dx} = \dfrac{1}{ 1 + x^2}$; (iv) $\dfrac{dy}{dx} = \dfrac{1}{x \sqrt{ x^2 - 1}}$; (v) $\dfrac{dy}{dx} = \dfrac{\sqrt{ 3\sec x} \left(3\sec^2 x - 1\right)}{2}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 1/(x*sqrt(x**2 - 1))
Exercise XIV, problem 12v, p. 174
Differentiate the following: align*%[** TN: Reformatted in two columns] (i) y &= x. & (ii) y &= x. (iii) y &= x. & (iv) y &= x. (v) y &= x × 3 x. && align*
Printed answer:
- (i) $\dfrac{dy}{dx} = \sec x \tan x$; (ii) $\dfrac{dy}{dx} = - \dfrac{1}{\sqrt{ 1 - x^2}}$; (iii) $\dfrac{dy}{dx} = \dfrac{1}{ 1 + x^2}$; (iv) $\dfrac{dy}{dx} = \dfrac{1}{x \sqrt{ x^2 - 1}}$; (v) $\dfrac{dy}{dx} = \dfrac{\sqrt{ 3\sec x} \left(3\sec^2 x - 1\right)}{2}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes sqrt(3/cos(x))*(3/cos(x)**2 - 1)/2
Exercise XIV, problem 13, p. 174
Differentiate $y=\sin(2\theta +3)^{2.3}$.
Printed answer:
- $\dfrac{dy}{d\theta} = 4.6\left(2\theta + 3\right)^{1.3} \cos\left(2\theta + 3\right)^{2.3}$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes Rational(23,5)*(2*theta + 3)**Rational(13,10)*cos((2*theta + 3)**Rational(23,10))
Exercise XIV, problem 14, p. 174
Differentiate $y=\theta^3+3 \sin(\theta+3)-3^{\sin \theta} - 3^\theta$.
Printed answer:
- $\dfrac{dy}{d\theta} = 3\theta^2 + 3\cos \left( \theta + 3 \right) - \log_\epsilon 3 \left( \cos\theta × 3^{\sin\theta} + 3\theta \right)$.
unverified: no computed check settled this one (yet)
How it was checked
- differentiate: the printed answer does not match the problem 3*theta**2 + 3*cos(theta + 3) - log(3)*(cos(theta)*3**sin(theta) + 3*theta)
Exercise XIV, problem 15a, p. 174
Find the maximum or minimum of $y=\theta \cos \theta$.
Printed answer:
- $\theta = \cot\theta; \theta = ±0.86$; is max. for $+\theta$, min. for $-\theta$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem 0.8603335890193798
- evaluate: the printed answer does not match the problem 0.8603335890193798
Exercise XIV, problem 15b, p. 174
Find the maximum or minimum of $y=\theta \cos \theta$.
Printed answer:
- $\theta = \cot\theta; \theta = ±0.86$; is max. for $+\theta$, min. for $-\theta$.
unverified: no computed check settled this one (yet)
How it was checked
- extremum: the printed answer does not match the problem -0.8603335890193798
- evaluate: the printed answer does not match the problem -0.8603335890193798
Exercise XIV, problem 1a, p. 173
Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*
Printed answer:
- (i) $\dfrac{dy}{d\theta} = A \cos \left( \theta - \dfrac{\pi}{2} \right)$; (ii) $\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$ and $\dfrac{dy}{d\theta} = 2\cos2\theta$; (iii) $\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$ and $\dfrac{dy}{d\theta} = 3\cos3\theta$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes A*cos(theta - pi/2)
Exercise XIV, problem 1b, p. 173
Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*
Printed answer:
- (i) $\dfrac{dy}{d\theta} = A \cos \left( \theta - \dfrac{\pi}{2} \right)$; (ii) $\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$ and $\dfrac{dy}{d\theta} = 2\cos2\theta$; (iii) $\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$ and $\dfrac{dy}{d\theta} = 3\cos3\theta$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*sin(theta)*cos(theta)
Exercise XIV, problem 1c, p. 173
Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*
Printed answer:
- (i) $\dfrac{dy}{d\theta} = A \cos \left( \theta - \dfrac{\pi}{2} \right)$; (ii) $\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$ and $\dfrac{dy}{d\theta} = 2\cos2\theta$; (iii) $\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$ and $\dfrac{dy}{d\theta} = 3\cos3\theta$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*cos(2*theta)
Exercise XIV, problem 1d, p. 173
Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*
Printed answer:
- (i) $\dfrac{dy}{d\theta} = A \cos \left( \theta - \dfrac{\pi}{2} \right)$; (ii) $\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$ and $\dfrac{dy}{d\theta} = 2\cos2\theta$; (iii) $\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$ and $\dfrac{dy}{d\theta} = 3\cos3\theta$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 3*sin(theta)**2*cos(theta)
Exercise XIV, problem 1e, p. 173
Differentiate the following: align* (i) y &= A (- 2). (ii) y &= ^2 ; and y = 2. (iii) y &= ^3 ; and y = 3. align*
Printed answer:
- (i) $\dfrac{dy}{d\theta} = A \cos \left( \theta - \dfrac{\pi}{2} \right)$; (ii) $\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$ and $\dfrac{dy}{d\theta} = 2\cos2\theta$; (iii) $\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$ and $\dfrac{dy}{d\theta} = 3\cos3\theta$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 3*cos(3*theta)
Exercise XIV, problem 2, p. 173
Find the value of $\theta$ for which $\sin\theta × \cos\theta$ is a maximum.
Printed answer:
- $\theta = 45°$ or $\dfrac{\pi}{4}$ radians.
verified: the printed answer passed a computed check
How it was checked
- extremum: passes pi/4
Exercise XIV, problem 3, p. 173
Differentiate $y=\dfrac{1}{2\pi} \cos 2\pi nt$.
Printed answer:
- $\dfrac{dy}{dt} = -n \sin 2\pi nt$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes -n*sin(2*pi*n*t)
Exercise XIV, problem 4, p. 174
If $y = \sin a^x$, find $\dfrac{dy}{dx}$.
Printed answer:
- $a^x \log_\epsilon a \cos a^x$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes a**x*log(a)*cos(a**x)
Exercise XIV, problem 5, p. 174
Differentiate $y=\log_\epsilon \cos x$.
Printed answer:
- $\dfrac{\cos x}{\sin x} = \cotan x$
unverified: no computed check settled this one (yet)
How it was checked
- differentiate: the printed answer does not match the problem cos(x)/sin(x)
Exercise XIV, problem 6, p. 174
Differentiate $y=18.2 \sin(x+26°)$.
Printed answer:
- $18.2 \cos \left(x + 26° \right)$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes, with the problem read into an equation 18.2*cos(x + 26*pi/180)
Exercise XIV, problem 7a, p. 174
Plot the curve $y=100 \sin(\theta-15°)$; and show that the slope of the curve at $\theta = 75°$ is half the maximum slope.
Printed answer:
- 0.5em plus 0.5em minus 0.25emThe slope is $\dfrac{dy}{d\theta} = 100\cos\left(\theta - 15° \right)$, which is a maximum when $(\theta -15°) = 0$, or $\theta = 15°$; the value of the slope being then ${}= 100$. When $\theta = 75°$ the slope is $100\cos(75° - 15°) = 100\cos 60° = 100 × \frac{1}{2} = 50$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes, with the problem read into an equation 100*cos(theta - 15*pi/180)
- evaluate: passes 100*cos(theta - 15*pi/180)
- evaluate: passes 100*cos(theta - 15*pi/180)
Exercise XIV, problem 7b, p. 174
Plot the curve $y=100 \sin(\theta-15°)$; and show that the slope of the curve at $\theta = 75°$ is half the maximum slope.
Printed answer:
- 0.5em plus 0.5em minus 0.25emThe slope is $\dfrac{dy}{d\theta} = 100\cos\left(\theta - 15° \right)$, which is a maximum when $(\theta -15°) = 0$, or $\theta = 15°$; the value of the slope being then ${}= 100$. When $\theta = 75°$ the slope is $100\cos(75° - 15°) = 100\cos 60° = 100 × \frac{1}{2} = 50$.
unverified: no computed check settled this one (yet)
How it was checked
- other: no printed answer to check
Exercise XIV, problem 8, p. 174
If $y=\sin \theta·\sin 2\theta$, find $\dfrac{dy}{d\theta}$.
Printed answer:
- $\begin{aligned}[t] \cos\theta \sin2\theta + 2\cos2\theta \sin\theta &= 2\sin\theta\left(\cos^2 \theta + \cos2\theta\right) \\ &= 2\sin\theta\left(3\cos^2 \theta - 1\right). \end{aligned}$
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes 2*sin(theta)*(3*cos(theta)**2 - 1)
Exercise XIV, problem 9, p. 174
If $y=a·\tan^m(\theta^n)$, find the differential coefficient of $y$ with respect to $\theta$.
Printed answer:
- $amn\theta^{n-1} \tan^{m-1}\left(\theta^n\right)\sec^2 \theta^n$.
verified: the printed answer passed a computed check
How it was checked
- differentiate: passes a*m*n*theta**(n-1)*tan(theta**n)**(m-1)/cos(theta**n)**2