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

24 sample cards from the Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $x$.

2. State the three Pythagorean identities.

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

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

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

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

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

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

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

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

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

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

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

8. 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 $2\sin A\sin B = \cos(A-B)-\cos(A+B)$.

9. 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 $a\sin x + b\cos x = R\sin(x + \alpha)$.

10. 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.

11. 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.

12. 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: $(ac - bd) + (ad + bc)i$.

13. 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.

14. 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).

15. 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 $(-\pi, \pi]$.

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

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

17. 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 $|z_2|$).

18. 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).

19. 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).

20. 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 $r^{1/n}$.

21. 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: $e^{i\pi} + 1 = 0$.

22. 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 $b$.

23. 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 $r$.

24. 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.

What this deck covers

Pure Mathematics: Coordinate Geometry, Trigonometry and Complex Numbers flashcards FAQ