Unit 2: Unit Circle Trigonometry

Looking Back

Austin Zhuang and Elijah Lee

Introduction

Welcome back! Below, you can find information to remind you of important concepts, formulas and diagrams that might be able to help you recall the topics covered in this unit.

The Unit Circle

Characteristics

It has a radius of 1 unit.
The center of the circle is at the origin.

Angles

To find a positive angle, start from the point (1, 0) and rotate counter-clockwise around the circle.
For negative angles, go clockwise around the circle instead.

Trigonometric Values

Each angle or θ value in the unit circle corresponds with an ordered pair.
The x-value of the pair is cos(θ).
The y-value of the pair is sin(θ).

Radians vs. Degrees

Radians

There are 2π radians in a circle.
1 radian is the measure of the angle formed from the endpoints of an arc on the circle with the length of the radius.
Advanced math uses radians over degrees.

Degrees

There are 360 degrees in a circle.
The number 360 was chosen only because it was close to the days in a calendar year (365.25) and it is divisible by many numbers.

Connections to the Unit Circle

To prove there are 2π radians in a circle, you can use the unit circle.

First, the circumference of the unit circle is 2πr, so 2π.

Since 1 radian in the unit circle would have an arc of 1 unit, 2π radians would have an arc of 2π units.

Thus, a full rotation would be 2π radians as the arc should be 2π units.

Converting Between Radians and Degrees

360 degrees are equivalent to 2π radians.

Thus, 1 degree is 2π/360 radians

Also, 1 radian is 360/2π degrees.

Common angles used in problems are showed in the image in the above section, like π/3 radians, 30 degrees, and π/2

Trigonometric Functions

Basic Trig Functions

Sine - To find sin(θ), find the ratio between the side opposite θ and the hypotenuse, or opposite/hypotenuse.

In the unit circle, the hypotenuse of a right triangle is the radius, so sin(θ) = opposite/1. The opposite side is the rise, or y-value, of the coordinate point, so sin(θ) = y-value.

Cosine - To find cos(θ), find the ratio between the side adjacent θ and the hypotenuse, or adjacent/hypotenuse.

Hypotenuse has length 1, so cos(θ) = adjacent/1. The adjacent side is the run, or x-value, of the coordinate point, so cos(θ) = x-value.

Tangent - To find tan(θ), find the ratio between the side opposite θ and the side adjacent θ, or opposite/adjacent.*

In the unit circle, because the opposite side is the y-value, the adjacent side is the x-value, and the hypotenuse originates at the origin, tan(θ) is the slope of the hypotenuse (rise/run).

*Another method if you know the sine and cosine of θ is to use the ratio sin(θ)/cos(θ). You can do this because when you divide the ratios (opposite/hypotenuse and adjacent/hypotenuse), the hypotenuse value cancels out, resulting in opposite/adjacent.

Reciprocal Trig Ratios

Cosecant - To find csc(θ), take the reciprocal of sin(θ), or hypotenuse/opposite and 1/sin(θ).

Also equals 1/y-value in the unit circle as it is the reciprocal function of sine.

Secant - To find sec(θ), take the reciprocal of cos(θ), or hypotenuse/adjacent and 1/cos(θ).

In the unit circle, it is also 1/x-value as it is the reciprocal function of cosine.

Cotangent - To find cot(θ), take the reciprocal of tan(θ), or adjacent/opposite and 1/tan(θ).

Also x-value/y-value in the unit circle as it is the reciprocal of tangent.

*You can also use the ratio cos(θ)/sin(θ) as it is the reciprocal of tan(θ).

Inverse Trig Functions

Given the value of the ratio, x, inverse trig functions find the respective angle, θ.

Domain: [-1, 1]

Range: [-π/2, π/2]

Domain: [-1, 1]

Range: [0, π]

Domain: (-∞, ∞)

Range: [-π/2, π/2]

Domain: (−∞,−1]∪[1,∞)

Range: [0,π/2)∪(π/2,π]

Domain: (−∞,−1]∪[1,∞)

Range:[−π/2,0)∪(0,π/2]

Domain: (-∞, ∞)

Range: (0,π)

Inverse vs. Arc

Since none of the 6 trig functions are one-to-one, their inverses have restricted domains to remain functions. Using arc lets us find all values of θ in the range (-∞, ∞) whose sin, cos, or tan is equal to x. Because trig functions repeat, we can represent the additional solutions outside of the range of the inverses by adding 2πn (sin, cos, sec, and csc) or πn (tan and cot) to the solutions, where n is a real whole number.

Example:

Sinusoidal Functions

Amplitude

Controlled by the parameter a
Amplitude is the distance from the sinusoidal axis (the middle line) and the maximum or minimum of the graph.
When |a| = n, the amplitude is also n.*

*Absolute valued needed as 3 and -3 both have an amplitude of 3.

Angular Frequency

Controlled by the parameter b
A cycle is one repetition of the pattern of the graph
Angular frequency is the number of cycles in 2π units.
When b = n, there are n cycles in 2π units.

Horizontal Shift

Controlled by the parameter h
The shift moves the graph either to the left or the right
When h = n, the graph shifts -n units horizontally.

Vertical Shift

Controlled by the parameter k
The shift moves the graph either up or down.
When k = n, the graph shifts n units vertically.

Other Features

When a is negative, the graph is reflected across the sinusoidal axis.
The parent function y = sin(θ) always starts at the origin because, when θ = 0, sin(θ) = 0
After the origin, the graph of sine increases first because, on the unit circle, rotating counter-clockwise and increasing the angle initially increase the y-value. (See image)
The parent function y = cos(θ) always starts at (1, 0) because, when θ = 0, cos(θ) = 0
After (1, 0), the graph of cosine decreases first because increasing the angle initially decreases the x-value on the unit circle. (See image)
The functions are all periodic, meaning the graph repeats after a given length, or a period.
- Period = length of one cycle.

Trig Identities

Pythagorean Identities

Half Angle Formula

Angle Sum Formulas

Angle Difference Formulas

Double Angle Formula

Rotary Motion

Linear Velocity

a is arc length
t is time
Speed in a straight line (dist/time)
ft/sec, m/s, etc.
Pulleys, belts, and gears share linear velocity

Angular Velocity

θ is the angle measure
t is time
Change in angle over time
RPMs, rad/min
Axles share angular velocity

Other Equations

Useful when you need to convert units in a problem.

Useful when finding linear velocity as you need the arc length.

Also helpful when finding linear velocity, and it shows the correlation between the two velocities.

The key to solving rotary motion problems is keeping track of units as you convert between different units and between linear and angular velocities.

Concepts of rotary motion are very applicable to everyday situations. It allows for a mathematical understanding of rotating objects and how they interact with each other. Any problem where rotating objects and their speeds are involved is a rotary motion problem, using these ideas. For example, the blades of a fan, a bicycle, or a robotics system with gears and belts can all be modeled with the concepts of rotary motion.

Sinusoidal Functions as Mathematical Models

Applicable Situations

The functions of sine and cosine are most effective at modeling situations with a repeating pattern. For example, mathematicians can use these models to predict when high tide and low tide will occur. Also, temperature fluctuations during the day follow a sinusoidal shape. Because the graphs are periodic, every repetition of the real-life event can be replicated as one cycle in the sinusoid. Normally, the models are functions of time.

Finding the Equation

Determine the maximum and minimum values
Take the average of the maximum and minimum to find the sinusoidal axis
Find the time it takes to complete one repetition of the pattern (one cycle)
Determine how many cycles are in 2π units
Find the value of the function at time 0.

After these steps, you have the information to find the parameters a, b, h, and k.

Parameter a, or the amplitude, is the maximum y-value minus the y-value of the sinusoidal axis.
Parameter b, or the angular frequency, is the number of cycles in 2π.
- Also, if you know the period, b = 2π/period because period is just the length of one cycle
Parameter k, or the vertical shift, is the y-value of the sinusoidal axis.

For parameter h:

If you are using cosine as the parent function, h will equal the x-value of the closest minimum or maximum to the y-axis.
- If taking the closest minimum, parameter a must be negative as a positive parameter a, like the parent function, will have a maximum (the point (1,0)) closest to the y-axis, so a minimum would mean a reflection.
If you are using sine as the parent function, h will equal the x-value of the closest point on the sinusoidal axis to the y-axis.
- If the graph decreases after the closest point to the axis, parameter a must be negative. In the parent function, after the closest point (the origin), the graph increases, so a decrease would mean a reflection.

Example

Cosine or sine?

For this model, use cosine as the pattern starts out decreasing, which is a characteristic of the parent function y=cos(θ).

Parameter a

Maximum is 10 units

Minimum is 0 units

Sinusoidal axis is 5 units

10-5 = 5, so the parameter a is 5

Parameter b

The p is period, which is 2π/3 minus π/6 = π/2.

b = 2π/p

b = 2π/π/2 = 4

Parameter h

Closest minimum/maximum is (-π/12, 10).

x-value is -π/12, so h = -π/12*

*parameter a is still positive because a maximum is closest to the y-axis

Parameter k

The y-value of the sinusoidal axis is 5 units.

Thus, parameter k = 5

Plugging in the values

Sample Problems

Trig Values

Solution

https://i.imgur.com/lUBfMRb.png

Solve the following equation on the given interval.

Solution

https://i.imgur.com/QjYKnwN.png

Rotary Motion

A bicycle wheel has a diameter of 78 cm. If the wheel revolves at a rate of 120 revolutions per minute, what is the linear velocity of the bike, in kilometers per hour?

A wheel has a diameter of 100 centimeters. If the wheel is supporting a cart moving at 45 kilometers per hour, then what is the rpm of the wheel?

Solution

https://i.imgur.com/F2AKbST.png

Solution

https://i.imgur.com/uCB3KxN.png

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