1.7.4 Practice set: trigonometry and complex numbers

Angles, sine and cosine

Angles, the unit circle, sine and cosine (lesson 1.7.1)

0 of 6 correct

1.
Convert 135° to radians
2.
Convert $\frac{4\pi}{3}$ radians to degrees
3.
A wheel of radius 0.3 m turns through 10 radians. How far, in metres, does a point on its rim travel?
4.
Give $\tan 135°$ exactly
5.
$\theta$ is in quadrant III and $\cos\theta = -\frac{5}{13}$ . What is $\sin\theta$ ? (a fraction)
6.
Give $\sin(-420°)$ exactly

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Complex numbers

Complex numbers (lesson 1.7.2)

0 of 7 correct

1.
Write $(3 + 2i) - (1 - 4i)$ in the form $a + bi$
2.
Write $(1 + 2i)(3 + i)$ in the form $a + bi$
3.
Multiplying by $i$ is a quarter turn. Write $i(4 - i)$ in the form $a + bi$
4.
What is $|-6 + 8i|$ ?
5.
Write $\dfrac{2 - i}{1 + i}$ in the form $a + bi$ (decimals are fine)
6.
$x^2 - 4x + 13 = 0$ has two complex roots. Write the one with a positive imaginary part
7.
What is $\sqrt{-4} \cdot \sqrt{-9}$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Euler's formula

Euler's formula (lesson 1.7.3)

0 of 7 correct

1.
Write $e^{3i\pi/2}$ using $i$
2.
What is the imaginary part of $4e^{i\pi/6}$ ?
3.
What is the angle of $-\sqrt{3} + i$ , in radians between $-\pi$ and $\pi$ ?
4.
What is $3e^{i\pi/3} \cdot 2e^{2i\pi/3}$ ?
5.
What is $(1 + i)^4$ ?
6.
What is $|5e^{3i}|$ ?
7.
$-2e^{i\pi/3}$ in polar form has $r = 2$ . What is its angle, between $-\pi$ and $\pi$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Polar form by hand

Write three functions without using cmath.polar, cmath.rect or cmath.phase.

to_polar(z) returns (r, theta): the distance $\sqrt{a^2 + b^2}$ and the angle math.atan2(b, a), where a = z.real and b = z.imag.
from_polar(r, theta) returns the complex number $r(\cos\theta + i\sin\theta)$ .
multiply_polar(p, q) takes two pairs (r1, t1) and (r2, t2) and returns the pair for the product: multiply the distances, add the angles (lesson 1.7.3).

math.atan2(b, a) gives the angle of the point $(a, b)$ , between $-\pi$ and $\pi$ . Unlike math.atan(b / a), it looks at the signs of both parts, so it puts $-1 - i$ at $-\frac{3\pi}{4}$ , not $\frac{\pi}{4}$ .

The tests compare your answers with cmath in every quadrant and on the negative real axis.

See the tests

import cmath
import math


def test_polar_one_plus_i():
    """1 + i has distance √2 and angle π/4"""
    expect_close(to_polar(1 + 1j), (math.sqrt(2), math.pi / 4), what="to_polar(1 + 1j)")


def test_polar_matches_cmath():
    """to_polar agrees with cmath.polar, in every quarter of the plane"""
    for z in (3 + 4j, -2 + 1j, -1 - 1j, 2 - 5j):
        expect_close(to_polar(z), cmath.polar(z), what=f"to_polar({z})")


def test_polar_negative_real():
    """−2 has distance 2 and angle π"""
    expect_close(to_polar(-2 + 0j), (2.0, math.pi), what="to_polar(-2 + 0j)")


def test_from_polar():
    """Distance 2 at angle π/2 is 2i"""
    z = from_polar(2, math.pi / 2)
    expect_close((z.real, z.imag), (0.0, 2.0), what="from_polar(2, π/2)", abs_tol=1e-12)


def test_round_trip():
    """from_polar undoes to_polar"""
    z = -3.5 + 1.25j
    back = from_polar(*to_polar(z))
    expect_close((back.real, back.imag), (z.real, z.imag), what="from_polar(*to_polar(z))")


def test_multiply_polar():
    """2 at π/6 times 3 at π/3 is 6 at π/2"""
    expect_close(multiply_polar((2, math.pi / 6), (3, math.pi / 3)), (6.0, math.pi / 2), what="multiply_polar((2, π/6), (3, π/3))")


def test_multiply_matches_product():
    """Multiplying in polar form gives the same number as z * w"""
    for z, w in ((1 + 1j, 1 + math.sqrt(3) * 1j), (-2 + 1j, 3 - 4j), (-1 - 1j, -1 - 1j)):
        back = from_polar(*multiply_polar(to_polar(z), to_polar(w)))
        expect_close((back.real, back.imag), ((z * w).real, (z * w).imag), what=f"the product of {z} and {w}", abs_tol=1e-12)

Angles, sine and cosine

Complex numbers

Euler's formula

Coding exercise

Free further reading