Trigonometry: reciprocal functions and identities

Secant, cosecant and cotangent Paper 2

Paper 1 gave you $\sin$, $\cos$ and $\tan$ for angles of any size. Paper 2 adds their three reciprocals:
$$\sec \theta =\frac{1}{\cos \theta },\mathrm{cosec}\theta =\frac{1}{\sin \theta },\cot \theta =\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta }$$
A useful way to remember which is which: the third letter of each name is the first letter of the function it is the reciprocal of (sec with $\cos$, cosec with $\sin$, cot with $\tan$).

Each is not defined where its denominator is zero: $\sec \theta$ where $\cos \theta =0$ ($\theta ={90}^{°},{270}^{°},\dots$), and $\mathrm{cosec}\theta$ and $\cot \theta$ where $\sin \theta =0$ ($\theta =0^{°},{180}^{°},\dots$). Note that $\cot {90}^{°}=\frac{\cos {90}^{°}}{\sin {90}^{°}}=0$, even though $\tan {90}^{°}$ is not defined.

A reciprocal has the same sign as the function it comes from, so the quadrant rules you already know still decide the sign. For an angle of any size, find $\sin$, $\cos$ or $\tan$ first, then take the reciprocal:

• $\sec {120}^{°}=\frac{1}{\cos {120}^{°}}=\frac{1}{-\frac{1}{2}}=-2$;
• $\cot {30}^{°}=\frac{1}{\tan {30}^{°}}=\sqrt{3}$;
• $\mathrm{cosec}(-{150}^{°})=\frac{1}{\sin (-{150}^{°})}=\frac{1}{-\frac{1}{2}}=-2$.

Graphs of the six functions Paper 2

Each reciprocal graph can be drawn from its parent:

• where the parent is 1 or $-1$, so is the reciprocal;
• where the parent is small, the reciprocal is large, and where the parent is 0 the reciprocal has a vertical asymptote;
• the reciprocal keeps the parent's sign and its period.

So:

• $y=\sec x$ has period ${360}^{°}$, asymptotes at $x={90}^{°},{270}^{°},\dots$, and takes only values $y⩾1$ or $y⩽-1$. Like $\cos x$ it is symmetrical about the $y$-axis.
• $y=\mathrm{cosec}x$ has period ${360}^{°}$, asymptotes at $x=0^{°},{180}^{°},{360}^{°},\dots$, and also $\left|y\right|⩾1$.
• $y=\cot x$ has period ${180}^{°}$, asymptotes at $x=0^{°},{180}^{°},\dots$, crosses the axis where $\tan x$ has its asymptotes, and takes every real value. It decreases between asymptotes.

The ranges matter when solving: $\sec \theta =\frac{1}{2}$ and $\mathrm{cosec}\theta =-0.4$ have no solutions.

$\sec x$ meets $\cos x$ where $\cos x=\pm 1$ and has an asymptote wherever $\cos x=0$.

$\mathrm{cosec}x$ touches $\sin x$ at $x={90}^{°}$ and ${270}^{°}$. $\cot x$ repeats every ${180}^{°}$ and is zero where $\tan x$ is not defined.

The identities for sec and cosec Paper 2

Divide every term of ${\sin }^2\theta +{\cos }^2\theta \equiv 1$ by ${\cos }^2\theta$:
$${\tan }^2\theta +1\equiv {\sec }^2\theta$$
Divide it by ${\sin }^2\theta$ instead:
$$1+{\cot }^2\theta \equiv {\mathrm{cosec}}^2\theta$$
Both are in the formula list. Their main use is in equations that mix two functions: an equation containing $\tan \theta$ and ${\sec }^2\theta$ can be written entirely in $\tan \theta$, and one containing ${\cot }^2\theta$ and $\mathrm{cosec}\theta$ entirely in $\mathrm{cosec}\theta$. The result is usually a quadratic. Solve it for the function, then find every angle in the interval, rejecting any value outside the function's range.

Compound angle formulae Paper 2

$\sin (A+B)$ is not $\sin A+\sin B$: for example $\sin {90}^{°}=1$ but $\sin {30}^{°}+\sin {60}^{°}=1.37$. The correct expansions are in the formula list:
$$\sin (A\pm B)\equiv \sin A\cos B\pm \cos A\sin B$$
$$\cos (A\pm B)\equiv \cos A\cos B\mp \sin A\sin B$$
$$\tan (A\pm B)\equiv \frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$$
Watch the signs: in the cosine formula the sign in the middle is the opposite of the sign in the bracket.

Why they hold. Put the points $(\cos A,\sin A)$ and $(\cos B,\sin B)$ on a unit circle. The angle between their radii is $A-B$, so the square of the distance between them is $2-2\cos (A-B)$ by the cosine rule. Working the same distance out from the coordinates gives $2-2(\cos A\cos B+\sin A\sin B)$. Comparing gives $\cos (A-B)$, and the other formulae follow by replacing $B$ by $-B$, by using $\sin \theta =\cos ({90}^{°}-\theta )$, and by dividing $\sin$ by $\cos$.

Uses:

• exact values of angles such as ${15}^{°}={45}^{°}-{30}^{°}$ and ${105}^{°}={60}^{°}+{45}^{°}$;
• simplifying expressions such as $\sin (x+{30}^{°})+\cos (x+{60}^{°})$, by expanding and collecting the terms in $\sin x$ and $\cos x$;
• finding $\sin (A+B)$ from given values of $\sin A$ and $\cos B$, after finding $\cos A$ and $\sin B$ with ${\sin }^2+{\cos }^2=1$ and the quadrant.

Double angle formulae Paper 2

Put $B=A$ in the compound angle formulae:
$$\sin 2A\equiv 2\sin A\cos A$$
$$\cos 2A\equiv {\cos }^{2}A-{\sin }^{2}A\equiv 2{\cos }^{2}A-1\equiv 1-2{\sin }^{2}A$$
$$\tan 2A\equiv \frac{2\tan A}{1-{\tan }^{2}A}$$
The second and third forms of $\cos 2A$ come from replacing ${\sin }^{2}A$ or ${\cos }^{2}A$ using ${\sin }^{2}A+{\cos }^{2}A=1$.

Choosing the form of $\cos 2A$. In an equation, pick the form that matches the other terms: with $\sin \theta$ elsewhere use $1-2{\sin }^2\theta$, and with $\cos \theta$ use $2{\cos }^2\theta -1$. The equation becomes a quadratic in one function.

Rearranged, the same identities give ${\cos }^{2}A\equiv \frac{1}{2}(1+\cos 2A)$ and ${\sin }^{2}A\equiv \frac{1}{2}(1-\cos 2A)$, used for exact values and later in integration.

Do not divide by $\sin \theta$ or $\cos \theta$ when it could be zero. In $\sin 2\theta =3\sin \theta$, write $2\sin \theta \cos \theta -3\sin \theta =0$ and factorise: $\sin \theta (2\cos \theta -3)=0$.

Expressing a sin θ + b cos θ in a single form Paper 2

A sum such as $3\sin \theta +4\cos \theta$ is a single wave in disguise. Expanding $R\sin (\theta +\alpha )=R\cos \alpha \sin \theta +R\sin \alpha \cos \theta$ and comparing coefficients with $a\sin \theta +b\cos \theta$ (with $a$ and $b$ positive) gives
$$R\cos \alpha =a,R\sin \alpha =b$$
Squaring and adding: $R^2=a^2+b^2$, so $R=\sqrt{a^2+b^2}$. Dividing: $\tan \alpha =\frac{b}{a}$.

The four forms, each with $R>0$ and $\alpha$ acute when $a$ and $b$ are positive:

• $a\sin \theta +b\cos \theta =R\sin (\theta +\alpha )$;
• $a\sin \theta -b\cos \theta =R\sin (\theta -\alpha )$;
• $a\cos \theta +b\sin \theta =R\cos (\theta -\alpha )$;
• $a\cos \theta -b\sin \theta =R\cos (\theta +\alpha )$.

Choose the form whose expansion has the same signs as the expression; then $R\cos \alpha$ and $R\sin \alpha$ are both positive and $\alpha$ is acute.

Why it helps.

• Equations: $a\sin \theta +b\cos \theta =c$ becomes $\sin (\theta +\alpha )=\frac{c}{R}$, one function of one angle. Find every value of $\theta +\alpha$ in the shifted interval, then subtract $\alpha$.
• Greatest and least values: $R\sin (\theta +\alpha )$ lies between $-R$ and $R$. So $\frac{1}{a\sin \theta +b\cos \theta +k}$ is greatest when the denominator is least.

Give $R$ exactly, as a surd if necessary, and keep $\alpha$ unrounded in later working.

Choosing an identity Paper 2

In an equation or a proof, look at what is there and choose the identity that removes the mixture:

• ${\sec }^2$ with $\tan$, or ${\tan }^2$ with $\sec$: use ${\sec }^2\theta \equiv 1+{\tan }^2\theta$;
• ${\mathrm{cosec}}^2$ with $\cot$, or ${\cot }^2$ with $\mathrm{cosec}$: use ${\mathrm{cosec}}^2\theta \equiv 1+{\cot }^2\theta$;
• a $2\theta$ with a $\theta$: use a double angle formula, choosing the form of $\cos 2\theta$ to match;
• $(\theta +{30}^{°})$ and similar: expand with the compound angle formulae;
• $a\sin \theta +b\cos \theta =c$: use the $R$ form;
• $\tan \theta$ with $\cot \theta$: write $\cot \theta =\frac{1}{\tan \theta }$ and multiply through, or write both in $\sin$ and $\cos$.

The last case has a neat result: $\tan \theta +\cot \theta =\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }=\frac{1}{\sin \theta \cos \theta }=\frac{2}{\sin 2\theta }$, that is $2\mathrm{cosec}2\theta$.

The same ideas work in radians: an interval such as $0<x<2\pi$ means every answer is in radians, exact multiples of $\pi$ where the values are standard.