Comprehensive Trigonometry Notes

Sec. 9.3: De Moivre’s Theorem; Complex Roots

De Moivre’s Theorem:
- Formula for raising a complex number $z$ to the power of $n$ , where $n ≥ 1$ is a positive integer.
Examples:
- Write $3 \cos 40^\circ + i \sin 40^\circ$ in standard form $a + bi$ .
- Write $1 - \sqrt{3}i$ in the standard form $a + bi$ .

Complex Roots

Examples:
- Find the complex cube roots of $-8$ .
- Find the complex fourth roots of $3-i$ . Leave answers in polar form, with the argument in degrees.

Sec. 9.3: Product and Quotient of Complex Numbers

Multiplying Complex Numbers in Polar Form

Proof
Example 1:
- Find the product $\left(4 \cos 200^\circ + i \sin 200^\circ\right) \cdot \left(6 \cos 185^\circ + i \sin 185^\circ\right)$ . Then express the product in rectangular form.
Example 2:
- Find the product $\left(7 \sqrt{2} \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \cdot \left(2 \sqrt{3} \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$ . Then express the product in rectangular form.

Dividing Complex Numbers in Polar Form

Proof
Example 3:
- Find the quotient $\frac{4 \left(\cos 60^\circ + i \sin 60^\circ\right)}{10 \left(\cos 90^\circ + i \sin 90^\circ\right)}$ . Then express the quotient in rectangular form.
Example 4:
- Find the quotient $\frac{12 \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)}{4 \left(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}\right)}$ . Then express the quotient in rectangular form.

Sec. 9.3: The Complex Plane and Polar Form

A complex number $z = x + yi$ can be interpreted geometrically as the point $(x, y)$ in the xy-plane.
Each point in the plane corresponds to a complex number, and conversely, each complex number corresponds to a point in the plane.
The collection of such points is referred to as the complex plane.
The x-axis is referred to as the real axis, because any point that lies on the real axis is of the form $z = x + 0i = x$ , a real number.
The y-axis is referred to as the imaginary axis, because any point that lies on it is of the form $z = 0 + yi = yi$ , a pure imaginary number.
Example 2:
- Graph each number in the complex plane and find its modulus.
  - a. $z = 4 - 3i$
  - b. $z = -3 + 4i$
Example 3:
- Express each complex number in polar form using exact values when necessary.
  - a. $1 + i$
  - b. $\sqrt{3} + i$
Example 4:
- Plot the point corresponding to $z = 11 \left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)$ in the complex plane, then express it in rectangular form using exact values.
Example 5:
- Graph $z = 3 \cos 60^\circ + i \sin 60^\circ$ in the complex plane and write an expression for $z$ in rectangular form.

Sec. 9.2: Polar Systems

Solving a system of equations with 2 unknowns involves finding the pair(s) of numbers that satisfy both equations.
Use the graphing calculator to look for potential solutions.
Use algebra and trig to find the exact solutions.
Solve equations for solutions between 0° and 360°.
Example 1:
- Solve
  - $r = \sin \theta$
  - $r = 1$
Example 2:
- Solve
  - $r = 8\cos \theta$
  - $r = 5$
Example 3:
- Solve
  - $r = 6 + 4 \cos \theta$
  - $r = 5 + 8 \cos \theta$
Example 4:
- Solve
  - $r = 5\sin \theta$
  - $r = 2\cos \theta$

Sec. 9.2: Graphs of Polar Equations

In Lesson 9.1, graphing polar equations that were simple in form like $r = k$ and $\theta = k$
In this lesson, you will learn how to graph more complicated types of polar equations.
The best way to graph a “new” type of equation is to use a table of values.
Example 1:
- Graph $r = \sin \theta$ . Hint: Let the polar axis scale be 0.25
- Just like the graph of $r = \sin \theta$ above, it illustrates a circle with a diameter of 1 that passes through the origin.
- Just like with any graph, you can alter its shape and position by multiplying the function by a number and/or adding to it.
- You can also alter the graph by multiplying or adding numbers to $\theta$ .
- However, you may be very surprised to find that the results are vastly different from the graphing transformations you have learned previously.
Example 2:
- Graph $r = -3 - 5\cos \theta$ Hint: Let the polar axis scale be 2

Use Graphing Calc to Sketch Classical Curves

Ex. Sketch $r = 2 + 2 \cos \theta$
Ex. Sketch $r = 4 \theta$
Ex. Sketch $r = 5\sin 2\theta$
Ex. Sketch $r^2 = 4\cos 2\theta$

Sec. 9.1: Converting from Polar to Rectangular and back…

Suppose a rectangular coordinate system is superimposed on a polar coordinate system so that the origins coincide and the x-axis aligns with the polar axis.
Let P be any point in the plane.
Polar coordinates: $(r, \theta)$
Rectangular coordinates: $(x, y)$
Trigonometric functions can be used to convert from polar coordinates to rectangular coordinates.
Example 1:
- Find the rectangular coordinates of each point. Express as exact values when possible, otherwise round coordinate values to the nearest hundredth.
  - a. $\left(5, \frac{\pi}{3}\right)$
  - b. $Q(-13, 70^\circ)$
If a point is named by the rectangular coordinates $(x, y)$ , you can find the corresponding polar coordinates by using the Pythagorean Theorem and the Arctangent function.
Since the Arctangent function only determines angles in the first and fourth quadrants, you must add $\pi$ radians to the value of $\theta$ for points with coordinates $(x, y)$ that lie in the second or third quadrants.
Example 2:
- Find the polar coordinates of $R(-8, -12)$ rounded to the nearest hundredth.
Example 3:
- The conversion equations can also be used to convert equations from one coordinate system to the other.
Example 4:
- Write the polar equation $r = 6\cos \theta$ in rectangular form.
Example 5:
- Write the rectangular equation $x^2 + (y + 3)^2 = 9$ in polar form.
Example 6:
- Transform the equation $y = \frac{x}{4}$ from rectangular coordinates to polar coordinates.

Sec. 9.1/9.2: Polar Coordinates

Recording the position of an object using the distance from a fixed point and an angle made with a fixed ray from that point uses a polar coordinate system.
When surveyors record the locations of objects using distances and angles, they are using polar coordinates.
The Polar Coordinate Plane
- In a polar coordinate system, a fixed point O is called the pole or origin.
- The polar axis is usually a horizontal ray directed toward the right from the pole.
- The location of a point P in the polar coordinate system can be identified by polar coordinates in the form $(r, \theta)$ .
- If a ray is drawn from the pole through P, the distance from the pole to point P is $r$ .
- The measure of the angle formed by $OP$ and the polar axis is $\theta$ .
- The angle can be measured in degrees or radians.
- It is important to consider both positive and negative values of r.
- Suppose r > 0. Then $\theta$ is the measure of any angle in standard position that has $OP$ as its terminal side.
- Suppose r < 0. Then $\theta$ is the measure of any angle that has the ray opposite $OP$ as its terminal side.
Example 1:
- Graph each point.
  - a. $P(3, 60^\circ)$
  - b. $Q\left(-1.5, \frac{7\pi}{6}\right)$
You may have noticed that the r-coordinate can be any real value.
The angle $\theta$ can also be negative.
If \theta > 0, then $\theta$ is measured counterclockwise from the polar axis.
If \theta < 0, then $\theta$ is measured clockwise from the polar axis.
Example 2:
- Graph $R(-2, -135^\circ)$
Example 3:
- Name four different ordered pairs of polar coordinates that represent point S on the graph with the restriction that $-360^\circ \le \theta \le 360^\circ$ .
An equation expressed in terms of polar coordinates is called a polar equation. For example, $r = 2\sin \theta$ is a polar equation.
A polar graph is the set of all points whose coordinates $(r, \theta)$ satisfy a given polar equation.
Example 4:
- Graph each polar equation.
  - a. $r = 3$
  - b. $\theta = \frac{4\pi}{3}$
Note: In Example 2, the location of point R suggests that it could be called $R(2, 45^\circ)$ . Polar points are not unique. They can be represented by infinitely many different ordered pairs.
Given two points $P1(r1, \theta1)$ and $P2(r2, \theta2)$ in the polar plane, draw $P1OP2$ . $\angle P1OP2$ has measure $|\theta1 - \theta2|$ . Apply the Law of Cosines to $\triangle P1OP2$ .

$P1P2^2 = OP1^2 + OP2^2 - 2(OP1)(OP2)\cos(\theta2 - \theta1)$
$P1P2^2 = r1^2 + r2^2 - 2r1r2 \cos(\theta2 - \theta1)$

Example 5:
- If two landmarks are 700 feet away and 40° to the left, and 350 feet away and 35° to the right, what is the distance between the landmarks?

Sec. 10.7: Parametric Equations and Projectile Motion

Objects that are launched are called projectiles. The path of a projectile is called its trajectory. The horizontal distance that a projectile travels is its range. Physicists describe the motion of a projectile in terms of its position, velocity, and acceleration. All these quantities can be expressed by vectors.
Parametric equations can represent the position of the ball relative to the starting point in terms of the parameter time.
In order to find parametric equations that represent the path of a projectile like a football, we must write the horizontal $vx$ and vertical $vy$ components of the initial vector.
Because the horizontal velocity is unaffected by gravity, it is the magnitude of the horizontal component of the initial velocity. Therefore, the horizontal position of a projectile, after $t$ seconds is given by the following equation.

x = v_x t

Since vertical velocity is affected by gravity, we must adjust the vertical component of the initial velocity. By subtracting the vertical displacement due to gravity from the vertical displacement caused by the initial velocity, we can determine the height of the projectile after t seconds. The height, in feet, or meters, of a free-falling object affected by gravity is given by the equation $h = \frac{1}{2}gt^2$ , where $g \approx 9.8 m/s^2$ or $32 ft/s^2$ and $t$ is the time in seconds.
Example 1:
- Jana hit a golf ball with an initial velocity of 102 ft/s at an angle of 67° to the ground.
  - a. Write the parametric equations.
  - b. After 2 seconds, how far has the ball traveled horizontally?
  - c. After 0.8 seconds, how high is the ball?
  - d. How long will the ball be in the air?
  - e. How far did the ball travel off the tee?
  - f. What was the maximum height of the ball?
Example 2:
- A pitcher on a professional baseball team throws a ball at an angle of 5.1° with the horizontal at a speed of 85 mph. The distance of the pitcher’s mound to home plate is 60.5 feet. The pitcher releases the ball 2.9 feet above the ground.
  - a. How far from the ground is the ball when it crossed home plate?
  - b. If it is a passed ball, how long will the ball be in the air?

Vectors and Parametric Equations

If a line passes through the points $P1$ and $P2$ and is parallel to the vector $\textbf{a} = \langle a1, a2 \rangle$ , the vector $\overrightarrow{P1P2}$ is also parallel to $\textbf{a}$ .
Thus $\overrightarrow{P1P}$ must be a scalar multiple of $\textbf{a}$ . Using the scalar $t$ , we can write the equation $\overrightarrow{PP1} = t\textbf{a}$ . This is called the vector equation of the line.
Since $\textbf{a}$ is parallel to the line, it is called the direction vector. The scalar $t$ is called the parameter.
Example 1:
- Write a vector equation describing a line passing through $(1, 4)$ and parallel to $\textbf{a} = \langle -3, 2 \rangle$ .
Example 2:
- Find the parametric equations for a line parallel to $\textbf{q} = \langle -6, 3 \rangle$ and passing through the point $(-2, -4)$ .
Example 3:
- Write parametric equations of $y = -x + 4$ .
Example 4:
- Write an equation in slope-intercept form of the line whose parametric equations are