Sixth Term Examination Paper (STEP) Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers Syllabus

Give the exact values of $\sin\theta$, $\cos\theta$, $\tan\theta$ for $\theta = 30^\circ$ ($\tfrac{\pi}{6}$).

$\sin 30^\circ = \tfrac{1}{2}$, $\cos 30^\circ = \tfrac{\sqrt{3}}{2}$, $\tan 30^\circ = \tfrac{1}{\sqrt{3}} = \tfrac{\sqrt{3}}{3}$.

Give the exact values of $\sin\theta$, $\cos\theta$, $\tan\theta$ for $\theta = 45^\circ$ ($\tfrac{\pi}{4}$) and $60^\circ$ ($\tfrac{\pi}{3}$).

$45^\circ$: $\sin = \cos = \tfrac{1}{\sqrt{2}}$, $\tan = 1$. $60^\circ$: $\sin = \tfrac{\sqrt{3}}{2}$, $\cos = \tfrac{1}{2}$, $\tan = \sqrt{3}$.

What are the periods of $\sin x$, $\cos x$ and $\tan x$?

$\sin x$ and $\cos x$ have period $2\pi$; $\tan x$ has period $\pi$. $\tan x$ has vertical asymptotes at $x = \tfrac{\pi}{2} + n\pi$.

Define the reciprocal trigonometric functions $\sec$, $\csc$ (cosec) and $\cot$.

$$\sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}, \quad \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$$

State the domain and range of $\arcsin x$ (inverse sine).

Domain $-1 \le x \le 1$; range $-\tfrac{\pi}{2} \le \arcsin x \le \tfrac{\pi}{2}$. It is the inverse of $\sin$ restricted to $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$.

State the principal-value ranges of $\arccos x$ and $\arctan x$.

$\arccos x \in [0, \pi]$ for domain $[-1,1]$; $\arctan x \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ for all real $x$.

State the three Pythagorean identities.

$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$

State the addition formulas for $\sin(A \pm B)$ and $\cos(A \pm B)$.

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$; $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

State the addition formula for $\tan(A \pm B)$.

$$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$

State the double-angle formulas for $\sin 2\theta$ and $\tan 2\theta$.

$\sin 2\theta = 2\sin\theta\cos\theta$; $\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$.

Give the three equivalent forms of $\cos 2\theta$.

$$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

State the factor formulae for $\sin P + \sin Q$ and $\cos P + \cos Q$.

$\sin P + \sin Q = 2\sin\!\left(\tfrac{P+Q}{2}\right)\cos\!\left(\tfrac{P-Q}{2}\right)$; $\cos P + \cos Q = 2\cos\!\left(\tfrac{P+Q}{2}\right)\cos\!\left(\tfrac{P-Q}{2}\right)$.

State the product-to-sum identity for $2\sin A \cos B$ and for $2\cos A\cos B$.

$2\sin A\cos B = \sin(A+B) + \sin(A-B)$; $2\cos A\cos B = \cos(A-B) + \cos(A+B)$. Also $2\sin A\sin B = \cos(A-B)-\cos(A+B)$.

Express $a\sin x + b\cos x$ in the harmonic form $R\sin(x + \alpha)$.

$R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \dfrac{b}{a}$ (with $R > 0$). Then $a\sin x + b\cos x = R\sin(x + \alpha)$.

For $f(x) = a\sin x + b\cos x = R\sin(x+\alpha)$, what are the maximum and minimum values and where do they occur?

Maximum $R = \sqrt{a^2+b^2}$ when $\sin(x+\alpha)=1$; minimum $-R$ when $\sin(x+\alpha)=-1$. Useful for solving and for extrema problems.

Describe the general process for solving a trigonometric equation over a given interval.

Find the principal solution, use the function's symmetry/periodicity to get all solutions in one period (e.g. $\sin$: $x$ and $\pi - x$; $\cos$: $x$ and $-x$; $\tan$: add $\pi$), then add multiples of the period and keep only those inside the interval.

How is a complex number written in Cartesian (algebraic) form, and how do you add and multiply $z_1=a+bi$, $z_2=c+di$?

$z = a + bi$ with $i^2 = -1$. Sum: $(a+c) + (b+d)i$. Product: $(ac - bd) + (ad + bc)i$.

How do you divide complex numbers, e.g. compute $\dfrac{a+bi}{c+di}$?

Multiply numerator and denominator by the conjugate $c - di$: $$\frac{(a+bi)(c-di)}{c^2 + d^2}$$ giving a real denominator.

What does the Argand diagram represent, and how is $z = a + bi$ plotted?

It is the complex plane: the horizontal axis is the real part and the vertical axis the imaginary part. $z = a+bi$ is the point $(a, b)$ (or the vector from the origin to it).

Define the modulus and argument of $z = a + bi$.

Modulus $|z| = \sqrt{a^2 + b^2}$ (distance from origin). Argument $\arg z = \theta$ is the angle from the positive real axis, with $\tan\theta = \tfrac{b}{a}$, usually taken in $(-\pi, \pi]$.

Write the modulus-argument (polar) form of a complex number.

$$z = r(\cos\theta + i\sin\theta), \quad r = |z|,\ \theta = \arg z$$

When multiplying two complex numbers in modulus-argument form, what happens to their moduli and arguments?

Moduli multiply and arguments add: $|z_1 z_2| = |z_1||z_2|$ and $\arg(z_1 z_2) = \arg z_1 + \arg z_2$. Geometrically multiplication is a rotation (by $\arg z_2$) and scaling (by $|z_2|$).

When dividing complex numbers, what happens to moduli and arguments?

$\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|}$ and $\arg\!\left(\dfrac{z_1}{z_2}\right) = \arg z_1 - \arg z_2$ (moduli divide, arguments subtract).

State De Moivre's theorem.

$$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$$ for all integer $n$ (and extends to rational $n$ for roots).

How do you find the $n$ distinct $n$th roots of a complex number $z = r(\cos\theta + i\sin\theta)$?

$$z^{1/n} = r^{1/n}\left(\cos\frac{\theta + 2k\pi}{n} + i\sin\frac{\theta + 2k\pi}{n}\right), \quad k = 0, 1, \dots, n-1.$$ The roots lie equally spaced on a circle of radius $r^{1/n}$.

State Euler's relation and the exponential form of a complex number.

Euler: $e^{i\theta} = \cos\theta + i\sin\theta$. Exponential form: $z = r e^{i\theta}$ where $r = |z|$, $\theta = \arg z$. Special case: $e^{i\pi} + 1 = 0$.

What locus in the complex plane is described by $|z - a| = r$, and by $|z - a| = |z - b|$?

$|z - a| = r$ is a circle of radius $r$ centred at the point $a$. $|z - a| = |z - b|$ is the perpendicular bisector of the segment joining $a$ and $b$.

What region/locus does $\arg(z - a) = \theta$ describe, and what does $|z - a| \le r$ describe?

$\arg(z - a) = \theta$ is a half-line (ray) from $a$ at angle $\theta$ to the positive real axis. $|z - a| \le r$ is the closed disc (interior and boundary) of the circle centre $a$, radius $r$.

For a polynomial with real coefficients, what is the conjugate root theorem?

Complex roots occur in conjugate pairs: if $z = a + bi$ is a root, so is $\bar{z} = a - bi$. Hence a real polynomial of odd degree has at least one real root, and complex roots come paired.