Reference Sheet: Programme F.11 — Trigonometric and Exponential Functions

Rules, procedures and drill, in the order of the book's programme. Every rule links to the unit in the lesson that derives it — if a rule here looks arbitrary, the link is where it stops being arbitrary.

Notation, as in the book: \sin^{-1}x is the inverse sine, not 1/\sin x (some texts write \arcsin); \operatorname{cosec} is 1/\sin; \equiv marks an identity; limits are written \mathop{Lim}_{x\to x_0}; and families of solutions use n=0,\pm1,\pm2,\dots Set the calculator to degrees or radians to match the angle.

1. Sine and cosine for any angle

From Unit 01.

A unit line OA turns anticlockwise about O (B is the foot of the perpendicular from A to the horizontal axis). Then, for every angle, positive or negative:

• \sin\theta = the signed height AB of A above the axis (negative below it);
• \cos\theta = the signed distance OB (negative when B is left of O);
• a negative angle is a clockwise turn.

Both lie between -1 and 1, and \sin(\theta+2\pi)=\sin\theta, \cos(\theta+2\pi)=\cos\theta. The sine crosses the axis at every multiple of \pi, the cosine at every odd multiple of \frac\pi2. For acute angles this is the right-triangle ratio of Programme F.9.

Examples. \sin153^\circ=\sin27^\circ=0.4540 (height of the 27^\circ triangle, positive); \cos153^\circ=-\cos27^\circ=-0.8910 (B is left of O).

2. Calculator values

From Unit 01.

Check the mode first: \sin30=0.5 in degrees but -0.9880 in radians. To evaluate by hand, find the acute angle the line makes with the horizontal axis and attach the sign of the quadrant.

Examples. \sin153^\circ=0.4540; \sin\big(-\frac\pi4\big)=-0.7071; \cos(-272^\circ)=0.0349 (this is the line at 88^\circ, just short of straight up); \cos\frac{2\pi}3=-0.5; \tan333^\circ=-0.5095; \tan\big(-\frac{6\pi}5\big)=-0.7265.

3. The tangent

From Unit 01.

the slope of the turning line. It is undefined where \cos\theta=0, that is at \theta=\frac\pi2+n\pi, where the graph has vertical asymptotes. Each branch climbs from -\infty to +\infty and crosses the axis at every multiple of \pi. Its period is \pi: \tan(\theta+n\pi)=\tan\theta. Reciprocals: \sec\theta=\frac1{\cos\theta}, \operatorname{cosec}\theta=\frac1{\sin\theta}, \cot\theta=\frac1{\tan\theta}.

Examples. \tan45^\circ=\tan225^\circ=\tan405^\circ=1; \tan\theta is undefined at 90^\circ and 270^\circ.

4. Period

From Unit 02.

The period is the interval of the input after which the function repeats. Basic periods: 2\pi for \sin and \cos, \pi for \tan. For a scaled angle, add the basic period inside the argument and factor out the coefficient of \theta:

A constant added inside the argument does not change the period.

Examples. \cos3\theta: period \frac{2\pi}3. \tan5\theta=\tan5\big(\theta+\frac\pi5\big): period \frac\pi5. \sin\frac\theta3=\sin\frac13(\theta+6\pi): period 6\pi, not 2\pi. \cos\big(\frac\theta2+\frac\pi3\big): period 4\pi.

5. Amplitude

From Unit 02.

The amplitude is the maximum value minus the average value over one period. For A\sin(k\theta+\varphi) or A\cos(k\theta+\varphi) it is \lvert A\rvert, because the average is 0. The definition also fits waves that are not sinusoidal.

Examples. 4\cos(2\theta-3) ranges between 4 and -4 with average 0: amplitude 4. -7\cos5\theta: amplitude 7 (the sign flips the wave but the amplitude is positive).

6. Periodic functions given by a prescription

From Unit 02.

A periodic function of period P is given by one stretch and the repeat rule:

To evaluate far from the stretch, subtract whole periods until the input falls in it.

Examples. The sawtooth f(x)=x, 0\le x<1, f(x+n)=f(x): f(2.5)=f(0.5)=0.5; maximum 1, average \frac12, amplitude \frac12. For f(x)=4-x, 0\le x<3, f(x+3)=f(x): period 3; f(7)=f(1)=3; maximum 4, average \frac{4+1}2=2.5, amplitude 1.5.

7. Phase difference

From Unit 02.

The phase difference of a wave is the interval of the input by which it leads or lags a reference function. Positive means it leads: it reaches each value earlier, which is a shift to the left. To read it, factor out the coefficient of the angle:

The same curve read as a shift to the right: \sin(\theta-\varphi) is the reference shifted right by \varphi, so it lags.

Examples. \sin\big(x-\frac\pi6\big) relative to \sin x: -\frac\pi6 (lags). \cos x=\sin\big(x+\frac\pi2\big): leads \sin x by \frac\pi2. \sin\big(3x+\frac\pi8\big)=\sin3\big(x+\frac\pi{24}\big): leads \sin3x by \frac\pi{24}. \cos\big(\frac\theta2+\frac\pi3\big)=\cos\frac12\big(\theta+\frac{2\pi}3\big): leads by \frac{2\pi}3 (not \frac\pi3). 4\cos(2\theta-3)=4\cos2\big(\theta-\frac32\big): phase difference -\frac32.

8. Inverse trigonometric functions

From Unit 03.

The inverse of \sin is the reflection of the whole curve in y=x: many values for one x, not a function. Cutting the curve to a one-to-one piece and reflecting that gives the inverse function \sin^{-1} (calculator key):

\sin^{-1}x does not mean \frac1{\sin x}.

Examples. \sin^{-1}0.5=\frac\pi6=30^\circ. \tan^{-1}(-3.5)=-1.2925 rad =-74.05^\circ. A ramp rising 1 m in 12 m: \tan^{-1}\frac1{12}=4.764^\circ.

9. The reciprocal inverses

From Unit 03.

Examples. \sec^{-1}10=\cos^{-1}0.1=84.26^\circ=1.4706 rad. \operatorname{cosec}^{-1}(-4)=\sin^{-1}(-0.25)=-0.2527 rad.

10. Simple trigonometric equations

From Unit 04.

A simple equation has one trigonometric expression. Read the principal value from the inverse key, then add the partner and the repeats:

• \sin x=s: x=x_0+2n\pi and x=\pi-x_0+2n\pi, where x_0=\sin^{-1}s.
• \cos x=s: x=\pm x_0+2n\pi, where x_0=\cos^{-1}s.
• \tan x=s: x=x_0+n\pi, where x_0=\tan^{-1}s (one value per period).

Examples. \sin x=\frac12: x=\frac\pi6+2n\pi or \frac{5\pi}6+2n\pi. \cos x=\frac12: x=\pm\frac\pi3+2n\pi. \sin3x=0: 3x=n\pi, so x=\frac{n\pi}3. \cos2x=1: 2x=2n\pi, so x=n\pi.

11. General solutions with a scaled angle

From Unit 04.

Solve for the whole angle kx first, then divide every term, including the 2n\pi, by k.

Examples. 2\sin3x=1: \sin3x=\frac12, so 3x=\frac\pi6+2n\pi or \frac{5\pi}6+2n\pi, and x=\frac\pi{18}+\frac{2n\pi}3 or x=\frac{5\pi}{18}+\frac{2n\pi}3. \tan4x=1: 4x=\frac\pi4+n\pi, so x=\frac\pi{16}+\frac{n\pi}4. A quadratic in a trig function factorises first: 18\cos^2x+3\cos x-1=(6\cos x-1)(3\cos x+1)=0 gives \cos x=\frac16 or -\frac13, so x=\pm1.4033+2n\pi or \pm1.9106+2n\pi.

12. Equations of the form a\cos x+b\sin x=c

From Unit 04.

Write a\cos x+b\sin x=R\sin(x+\theta) with

choosing \theta in the quadrant that matches the signs of a and b. Solve \sin(x+\theta)=\frac cR (needs \lvert c\rvert\le R): principal value and partner, then subtract \theta from both families. The same wave is R\cos(x-\alpha) with R\cos\alpha=a, R\sin\alpha=b, \alpha=\frac\pi2-\theta.

Examples. 3\cos x+4\sin x=5: R=5, \theta=0.6435, \sin(x+\theta)=1, so x=\frac\pi2-0.6435+2n\pi=0.9273+2n\pi. 3\cos x+4\sin x=2.5: \cos(x-0.9273)=\frac12, so x=0.9273\pm\frac\pi3+2n\pi, that is 1.9745+2n\pi or -0.1199+2n\pi. \sin x-\sqrt2\cos x=1: R=\sqrt3, \theta=-0.9553, \sin(x+\theta)=\frac1{\sqrt3} with principal value 0.6155 and partner 2.5261, so x=1.5708+2n\pi and x=3.4814+2n\pi (the book lists only the first).

13. Exponential functions

From Unit 05.

y=e^x (or \exp x), with e=2.7182818\ldots, grows at a rate proportional to its own size; its graph lies wholly above the x-axis. A calculator builds it from

For a general base (a>0, a\ne1):

so a^x grows more slowly than e^x when a<e (\ln a<1) and faster when a>e.

Examples. e^1 from eleven terms of the series: 2.7182818. 12^{0.7}=e^{0.7\ln12}=5.69.

14. Logarithmic functions

From Unit 05.

y=\log_ax (with base e, y=\ln x) is the inverse of y=a^x, so the two are mutual inverses: \log_a(a^x)=x and a^{\log_ax}=x. The graph of \ln x is the graph of e^x reflected in y=x. Laws: \log AB=\log A+\log B, \log\frac AB=\log A-\log B, \log A^n=n\log A. There is no rule for \log(A+B).

Examples. e^{\ln5}=5; \ln e^{-2}=-2; \log_28=3 because 2^3=8.

15. Indicial equations

From Unit 05.

An equation with the unknown in an index. Take logs of both sides (the book uses base 10; any base gives the same x) and bring the index down with \log A^n=n\log A. Carry full precision and round once, at the end.

Examples. 12^{2x}=35.4: 2x\log12=\log35.4, so x=\frac{1.5490}{2\times1.0792}=0.7177. 4^{3x-2}=26^{x+1}: (3x-2)\log4=(x+1)\log26, so x=\frac{2\log4+\log26}{3\log4-\log26}=6.695 (rounding the logs to four places first gives 6.694 instead). Capacitor: 12e^{-t/2}=5 gives t=2\ln\frac{12}5=1.751 s.

16. Indicial equations quadratic in a^x

From Unit 05.

When a^x and a^{2x}=(a^x)^2 both appear, put y=a^x, solve the quadratic in y, reject any root y\le0 (a^x is always positive), then solve a^x=y for each remaining root separately.

Examples. 2^{2x}-6\cdot2^x+8=0: (y-2)(y-4)=0, so 2^x=2 or 4, giving x=1 or x=2. 2\cdot3^{2x}-6\cdot3^x+4=0: 2y^2-6y+4=2(y-1)(y-2)=0, so 3^x=1 or 2, giving x=0 or x=\frac{\ln2}{\ln3}=0.631. 7^{2x}-9\cdot7^x+14=0: (y-2)(y-7)=0, so x=\frac{\ln2}{\ln7}=0.356 or x=1. (The book's printed version of this last equation has the signs misprinted.)

17. Logarithmic equations

From Unit 05.

Combine logs into one, convert to the index form \log_ab=c\Leftrightarrow a^c=b, solve, and check every answer in the original equation: a log needs a positive argument and a base must be positive and not 1.

Examples. \log_x49=2: x^2=49, and the base must be positive, so x=7. \log_2x+\log_2(x-2)=3: \log_2x(x-2)=3, so x^2-2x-8=0 gives x=4 or -2; -2 is rejected, leaving x=4 (check: 2+1=3).

18. Odd and even functions

From Unit 06.

f is even if f(-x)=f(x) (graph symmetric in the vertical axis) and odd if f(-x)=-f(x) (graph unchanged by a half turn about the origin). x^2 and \cos x are even; x^3, \sin x and \tan x are odd. The reflection that sends \theta\to-\theta flips the height of the turning line but not its foot, which is why \sin(-\theta)=-\sin\theta and \cos(-\theta)=\cos\theta.

Examples. x\sin x is even (odd times odd). \frac x{x^2-1} is odd.

19. Odd and even parts

From Unit 06.

When f(-x) is defined,

Examples. 3x^2-2x+1: f_e=3x^2+1, f_o=-2x. \frac1{x-1}: f_e=\frac1{(x-1)(x+1)}, f_o=\frac x{(x-1)(x+1)}. See-saw loads 30 kg at x=-1 and 10 kg at x=1: f_e=20 on each side and f_o=\pm10.

20. Polynomials: even powers and odd powers

From Unit 06.

For a polynomial, the even part is the terms with even powers (including the constant) and the odd part is the terms with odd powers. A polynomial is even if it has only even powers and odd if it has only odd powers.

Examples. x^3-2x^2-3x+4: f_e=-2x^2+4, f_o=x^3-3x. x^3(x^2-3x+5)=x^5-3x^4+5x^3: f_e=-3x^4, f_o=x^5+5x^3.

21. Hyperbolic functions

From Unit 06.

The even and odd parts of e^x are the hyperbolic cosine and hyperbolic sine:

Beyond the book: \cosh^2x-\sinh^2x=1.

Examples. \cosh1=1.5431, \sinh1=1.1752, sum 2.7183=e. x^2e^x is neither even nor odd; f_e=x^2\cosh x and f_o=x^2\sinh x.

22. Functions with no even or odd part

From Unit 06.

The formulas need f(-x). If the domain is not symmetric about 0, there are no parts. \ln x is defined only for x>0, so \ln(-x) does not exist and \ln x is neither even nor odd; the same goes for x^4\ln x.

Examples. \log_ax and x^4\ln x have no even or odd part; \frac1{x-1} does, because f(-x)=-\frac1{x+1} exists for x\ne\pm1.

23. Limits

From Unit 07.

\mathop{Lim}_{x\to x_0}f(x)=A means f(x) can be made as close to A as you like by taking x close enough to x_0, without f having to be defined at x_0. In band-and-window form: for every band of width \varepsilon about A there is a window of width \delta about x_0, excluding x_0, inside which the graph stays in the band.

Examples. \mathop{Lim}_{x\to1}\frac{x^2-1}{x-1}=2: the graph is the line y=x+1 with a hole at (1,2), and \delta=\varepsilon works. A speedometer: s=t^2, average speed \frac{t^2-1}{t-1} over [1,t] tends to 2 m/s as t\to1.

24. Rules of limits: sum, difference, product, quotient

From Unit 07.

If \mathop{Lim}f=A and \mathop{Lim}g=B as x\to x_0:

The sum rule follows from adding two \varepsilon-bands to get 2\varepsilon; the product rule is an argument from fg=AB+Ae_2+Be_1+e_1e_2 with small e_1,e_2. If B=0 the quotient rule says nothing.

Examples. \mathop{Lim}_{x\to\pi}(x^2-\sin x)=\pi^2; \mathop{Lim}_{x\to\pi}x^2\sin x=0; \mathop{Lim}_{x\to\pi/4}\frac{\tan x}{\sin x}=\frac1{1/\sqrt2}=\sqrt2.

25. The composition rule and continuity

From Unit 07.

Without continuity the limit cannot be moved inside: f(y)=0 for y\ne0, f(0)=1 and g(x)=x^2 give \mathop{Lim}_{x\to0}f(g(x))=0 but f(0)=1.

Examples. \mathop{Lim}_{x\to1}\cos(x^2-1)=\cos0=1. \mathop{Lim}_{x\to\pi/2}2x^2\cos\big(3x-\frac\pi2\big)=\frac{\pi^2}2\cos\pi=-\frac{\pi^2}2.

26. Limits where numerator and denominator both vanish

From Unit 07.

When both go to 0 at x_0 the quotient rule does not apply. Factorise and cancel the common factor for x\ne x_0, then take the limit of what is left. The cancellation is legitimate near x_0 and never at it.

Examples. \mathop{Lim}_{x\to-3}\frac{x^2-9}{x+3}=\mathop{Lim}(x-3)=-6. \mathop{Lim}_{x\to-1}\frac{x^2+2x+1}{x^2+3x+2}=\mathop{Lim}\frac{x+1}{x+2}=0. \mathop{Lim}_{x\to2}\frac{x^2-4}{x^2-x-2}=\mathop{Lim}\frac{x+2}{x+1}=\frac43.

Traps

• \sin^{-1} is not 1/\sin. \sin^{-1}0.5=0.5236, but \frac1{\sin0.5}=2.0858. The reciprocal is \operatorname{cosec}.
• The period of \sin k\theta is not 2\pi. It is \frac{2\pi}{\lvert k\rvert}: \sin\frac\theta3 has period 6\pi and \tan5\theta has period \frac\pi5.
• Read a phase only after factoring out k. \cos\big(\frac\theta2+\frac\pi3\big) is shifted \frac{2\pi}3, not \frac\pi3.
• Do not stop at the calculator's one solution. Every equation \sin x=s has a partner \pi-x_0 and repeats 2n\pi. Dropping the partner loses half the answers: \sin x-\sqrt2\cos x=1 has two families.
• Divide the 2n\pi by k as well. For \sin3x=\frac12 the families step by \frac{2\pi}3, not by 2\pi.
• Check the calculator mode. \sin30 is 0.5 in degrees and -0.9880 in radians.
• Choose \theta in the right quadrant. In a\cos x+b\sin x the signs of a and b fix the quadrant of \theta; \tan^{-1}\frac ab alone can be off by \pi.
• There is no log of a sum. \log(A+B)\ne\log A+\log B. Take the log of a product or power, and reject any root of an indicial quadratic that is \le0.
• Cancelling at the point itself. \frac{x^2-1}{x-1}=x+1 only for x\ne1; the limit is 2 because the function approaches it, not because f(1)=2. And the quotient rule needs B\ne0.

Self-check

1. Evaluate \sin(-150^\circ), \cos\frac{7\pi}6 and \tan(-225^\circ) without a calculator, then check.
2. Find the amplitude, period and phase difference (from \sin4\theta) of 3\sin(4\theta-1).
3. Find the period of \cos\frac\theta4, of \tan2\theta, and the period and phase difference of \sin\big(\frac\theta2+\frac\pi3\big) from \sin\frac\theta2.
4. A periodic function has f(x)=4-x for 0\le x<3 and f(x+3)=f(x). Find f(7) and its amplitude.
5. Evaluate \sin^{-1}(-0.5), \cos^{-1}(-0.5), \tan^{-1}1, \sec^{-1}2 and \operatorname{cosec}^{-1}(-4).
6. Solve \tan4x=1.
7. Solve 18\cos^2x+3\cos x-1=0.
8. Solve 4\cos x+3\sin x=2.
9. Solve 3^{2x+1}=7^x.
10. Solve (a) 9^x-4\cdot3^x+3=0 and (b) \log_2x+\log_2(x-2)=3.
11. Find the even and odd parts of \frac1{1+x} and of e^{2x}.
12. Find (a) \mathop{Lim}_{x\to2}\frac{x^2-4}{x^2-x-2} and (b) \mathop{Lim}_{x\to\pi/3}\frac{\sin x}{1+\cos x}.

Fully worked solutions

1. -150^\circ is a clockwise turn of 150^\circ: the line points down and to the left, 30^\circ below the negative horizontal axis, so the height is negative: \sin(-150^\circ)=-\sin30^\circ=-0.5. \frac{7\pi}6=210^\circ points down and left, 30^\circ from the axis, with B left of O: \cos\frac{7\pi}6=-\cos30^\circ=-\frac{\sqrt3}2=-0.8660. -225^\circ is the same line as 135^\circ, up and to the left, 45^\circ from the axis, with rise \sin135^\circ=0.7071 and run -0.7071: \tan(-225^\circ)=-1.

2. Amplitude 3. Period \frac{2\pi}4=\frac\pi2. Factor: \sin(4\theta-1)=\sin4\big(\theta-\frac14\big), so the phase difference from \sin4\theta is -\frac14 (it lags by 0.25).

3. \cos\frac\theta4=\cos\frac14(\theta+8\pi): period 8\pi. \tan2\theta=\tan2\big(\theta+\frac\pi2\big): period \frac\pi2. \sin\big(\frac\theta2+\frac\pi3\big)=\sin\frac12\big(\theta+\frac{2\pi}3\big): period \frac{2\pi}{1/2}=4\pi; phase difference +\frac{2\pi}3 (leads).

4. Subtract whole periods: 7=1+2\times3, so f(7)=f(1)=4-1=3. On one period the function falls from 4 (at x=0) towards 1; its maximum is 4 and its average is \frac{4+1}2=2.5, so the amplitude is 4-2.5=1.5.

5. \sin^{-1}(-0.5)=-\frac\pi6 (-30^\circ). \cos^{-1}(-0.5)=\frac{2\pi}3 (120^\circ). \tan^{-1}1=\frac\pi4. \sec^{-1}2=\cos^{-1}\frac12=\frac\pi3. \operatorname{cosec}^{-1}(-4)=\sin^{-1}(-0.25)=-0.2527 rad.

6. \tan4x=1 gives 4x=\frac\pi4+n\pi (the tangent has one value per period \pi), so x=\frac\pi{16}+\frac{n\pi}4, n=0,\pm1,\pm2,\dots Check n=1: 4x=\frac\pi4+\pi and \tan\frac{5\pi}4=1. ✓

7. 18\cos^2x+3\cos x-1=(6\cos x-1)(3\cos x+1)=0. So \cos x=\frac16 or \cos x=-\frac13. \cos^{-1}\frac16=1.4033, giving x=\pm1.4033+2n\pi; \cos^{-1}\big(-\frac13\big)=1.9106, giving x=\pm1.9106+2n\pi. Check: 18\cdot\frac1{36}+3\cdot\frac16-1=0.5+0.5-1=0. ✓

8. a=4 (cosine), b=3 (sine): R=5, R\sin\theta=4 and R\cos\theta=3, so \theta=\tan^{-1}\frac43=0.9273. Then \sin(x+\theta)=\frac25=0.4, with principal value 0.4115 and partner \pi-0.4115=2.7301. Subtract \theta from both: x=0.4115-0.9273=-0.5158+2n\pi or x=2.7301-0.9273=1.8028+2n\pi. Check with the unrounded angles: both x=-0.515778 and x=1.802781 give 4\cos x+3\sin x=2.0000. ✓

9. Take natural logs: (2x+1)\ln3=x\ln7. So 2x\ln3+\ln3=x\ln7, then x(\ln7-2\ln3)=\ln3 and x=\frac{\ln3}{\ln7-2\ln3}=\frac{1.0986}{1.9459-2.1972}=\frac{1.0986}{-0.2513}=-4.371. Round once, at the end: carrying the logs to four places gives -4.37; full precision gives -4.3715.

10. (a) With y=3^x: y^2-4y+3=(y-1)(y-3)=0, so 3^x=1 or 3, giving x=0 or x=1. Check: 1-4+3=0 and 9-12+3=0. ✓ (b) \log_2x(x-2)=3 means x(x-2)=8, so x^2-2x-8=(x-4)(x+2)=0. x=-2 is rejected, since \log_2(-2) does not exist, so x=4. Check: \log_24+\log_22=2+1=3. ✓

11. f(x)=\frac1{1+x}, so f(-x)=\frac1{1-x}. f_e=\frac12\Big(\frac1{1+x}+\frac1{1-x}\Big)=\frac12\cdot\frac2{1-x^2}=\frac1{1-x^2} and f_o=\frac12\Big(\frac1{1+x}-\frac1{1-x}\Big)=\frac12\cdot\frac{-2x}{1-x^2}=-\frac x{1-x^2}. Check at x=0.5: f=0.6667, f_e=1.3333, f_o=-0.6667, and 1.3333-0.6667=0.6667. ✓ For e^{2x}: f(-x)=e^{-2x}, so f_e=\frac{e^{2x}+e^{-2x}}2=\cosh2x and f_o=\frac{e^{2x}-e^{-2x}}2=\sinh2x. Check at x=1: \cosh2+\sinh2=3.7622+3.6269=7.3891=e^2. ✓

12. (a) Both numerator and denominator vanish at x=2, so factorise: \frac{(x-2)(x+2)}{(x-2)(x+1)}=\frac{x+2}{x+1} for x\ne2, whose limit is \frac43. Check at x=2.001: 1.3332. ✓ (b) The denominator tends to 1+\cos\frac\pi3=\frac32\ne0, so the quotient rule applies: \frac{\sin\frac\pi3}{\frac32}=\frac{\sqrt3/2}{3/2}=\frac{\sqrt3}3=0.5774.