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A guide for those starting the Pre-Calculus curriculum
By Damian Ferraro and Drew Davenport
Table of Contents
Table of Contents
Introduction to Unit Circle Trigonometry
Introduction to Unit Circle Trigonometry
Concepts of the Unit:
Concepts of the Unit:
- Unit Circle
- Unit Circle
- Trig Functions
- Trig Functions
- Sinusoids
- Sinusoids
- Trig Identities
- Trig Identities
- Rotary Motion
- Rotary Motion
Real Life Applications
Real Life Applications
Sample Problems
Sample Problems
Introduction
Introduction
For all of you, this unit is rich with familiar terms that you have heard of before. Remember 30 - 60 - 90 triangles and sine and cosine? Those will all be in this unit. What the bulk of this area of Pre-Calc talks about is the unit circle itself, and how it reflects a true diagram of sin, cos, tan, and the inverse functions of csc, sec, and cot. In this unit, you will discover news ways to incorporate these trig functions into your wide range of knowledge in mathematics.
Some unfamiliar terms that you will learn are sinusoidal graphs, rotary motion, and the double and half angle formulas. All of these incorporate the six trig functions, and especially sin and cos. You will learn how the graphs of sine and cosine play a role in the way that they look, and learn ways to use these equations to solve for variables.
Now, we know that was a lot of different topics and aspects of Pre-Calc that you guys do not understand yet, but we hope that through this website, we can give you an overview on Unit Circle Trigometry, and how it plays a major role in understand trig functions and problems that come along with them.
Concepts of the Unit
Concepts of the Unit
Let's start with the Unit Circle!
Let's start with the Unit Circle!
As you can see, there are a lot of numbers, but its a lot less complicated when you think about how we got these numbers:
- The radius is always 1 and the center is always at (0,0)
- Sine and cosine act as x and y, almost as if they are creating axises for themselves.
- Radians (or the values inside the circle) are an alternate way of thinking about degrees.
- There are so many other values that can be calculated, just knowing the knowledge of 45-45-90 triangles and 30-60-90 triangles, and applying them to the unit circle
- The values on the outside of the circle are an ordered pair (x, y), but in unit circle trigonometry, the ordering pair is (cos, sin)
This unit circle lays out a completely new structure to understanding how concepts that you learned in Geometry and Algebra II have a significance in your Pre-Calc class.
Remember these?
Remember these?
Both of these triangle properties that you learned in Geometry have a role in finding the values on the Unit Circle. Since the hypotenuse is 1, the unknown values are then introduced through finding the cosine and sine values. Hand and hand with this is the set of trig functions that we also learn next.
Main Trig Functions
Main Trig Functions
Basic Trig Functions
Basic Trig Functions
Sine
Sine
- Triangle ratio (sinθ): opposite / hypotenuse
- Always acts as the y-value / vertical values
Cosine
Cosine
- Triangle ratio (cosθ): adjacent / hypotenuse
- Always acts as the x-value / horizontal values
- sinθ^2 + cosθ^2 = 1 (look familiar?)
Tangent
Tangent
- Triangle ratio (tanθ): opposite / adjacent
- Also known as sinθ / cosθ
- Almost acts as a reflected hypotenuse (see image)
Reciprocal Trig Functions
Reciprocal Trig Functions
Cosecant
Cosecant
- Triangle Ratio: hypotenuse / opposite
- cscθ = 1 / sinθ
Secant
Secant
- Triangle Ratio: hypotenuse / adjacent
- secθ = 1 / cosθ
Cotangent
Cotangent
- Triangle Ratio: hypotenuse / (opposite / adjacent)
- cotθ = 1 / tanθ (cosθ / sinθ)
Now that you've heard of the trig functions that we use, what happens when we graph them?
Now that you've heard of the trig functions that we use, what happens when we graph them?
Sinusoids
Sinusoids
y = a sin(b(x-h)) + k
y = a sin(b(x-h)) + k
y = a cos(b(x-h)) + k
y = a cos(b(x-h)) + k
Variables
Variables
- a = amplitude (vertical distance from either max or min to the sinusoidal axis
- b = frequency (number of cycles in 360° or 2π)
- h = horizontal shift (opposite since it is in the paratheneses)
- k = vertical shift
Other Things to Note
Other Things to Note
- Finding the period
- Frequency * Period = 360° or 2π
- Simplified: Frequency / 2π = Period
- Period: Distance from one max or min to the next
- Sinusoidal Axis: The point at which the middle of the sinusoid in located in the y - axis.
- Phase Displacement: The length from the top of the sinusoid to the bottom
The Difference between Arcs and Inverse Trig Functions
The Difference between Arcs and Inverse Trig Functions
Though on the calculator, these two functions seem to be the same thing, there is a very important distinction between the two. As you explore problems that ask for a value of x including these functions, you must be very careful to interpret which function is which.
Arcsin(x), Arccos(x), Arctan(x)
Arcsin(x), Arccos(x), Arctan(x)
These functions include all real numbers and be thought of almost as if there are infinite values and rotations around the unit circle. When determining values for this function, first find all of the answers within the range of [0, 360). After that, be sure to add your answers by 360*n, where n = all integers. This will help understand that not just one value can be determined, and you can infinitely calculate more values as you keep rotating around the unit circle
Sin^-1(x), Cos^-1(x), Tan^-1(x)
Sin^-1(x), Cos^-1(x), Tan^-1(x)
On the other hand, these functions are known as the inverse functions of sin, cos, and tan. Each of these have a specific domain. There are graphs for each of these inverse functions, which are all difference in terms of their shape and domain. What make these functions interesting is that sometimes you will come upon these, and you must remember to apply the specific domain, when finding the answer to a problem involving these inverse trig functions.
Sin^-1(x)
Sin^-1(x)
This function has a domain of [-1, 1] and a range of (-π/2, π/2). This function acts as a 90° rotation to the sin(x) = y function.
Cos^-1(x)
Cos^-1(x)
This function has a domain of [-1, 1] and a range of [0, π]. This function almost is rotated 90° from the cos(x) = y function.
Tan^-1(x)
Tan^-1(x)
This function, unlike sin and cos, have a domain of (-∞,∞). The range however, is limited to just (-π/2, π/2). This tangent function is similar to that of the tan(x) = y function.
Trig Identities
Trig Identities
As you have became familiar with these trig functions, you will also have to evalute them and learn the main trig identities that come with these functions. This will further your understand on the propertes of the six trig functions, and in turn give you a broadened view on mathematics as a whole.
Pythagorean Identities
Pythagorean Identities
Half Angle Formulas
Half Angle Formulas
Angle Sum Formulas
Angle Sum Formulas
Angle Difference Formulas
Angle Difference Formulas
Double Angle Formula
Double Angle Formula
Rotary Motion
Rotary Motion
Overview
Overview
This concept of Unit 2 is one of the most complex, but also the most relatable towards real world Applications. Rotary Motion takes into account objects when they are spun, while also implementing time and velocity into the problem. The first thing that you must understand about rotary motion is the difference between angular velocity and linear velocity.
Angular Velocity
Angular Velocity
Angular Velocity applies to certain objects that are moving along a circular path. Great examples of objects that have angular velocity are ferris wheels and racecars on a circular track. The formula of this form of velocity is w = θ / t, where w = angular velocity, θ = the position angle, and t = time. The angular velocity of an object is their displacement with respect to time. Axils are used in determining angular velocity.
- Units:
- rev / min
- rad / min
Linear Velocity
Linear Velocity
On the other hand, linear Velocity applies to certain objects that are moving on a straight line. Great examples of objects that have linear velocity are speed driving on the road, and the speed of airplanes flying off the runway. The formula of this form of velocity is v = a / t, where v = the linear velocity, a = distance travelled, and t = time. The linear velocity of an object is their rate of change with respect to time. Pulleys, belts, and gears are used in determining linear velocity.
- Units:
- cm / sec, m / sec, ft / sec, etc.
One main thing to consider in rotary motion are these two facts:
One main thing to consider in rotary motion are these two facts:
2π radians = 1 revolution
1 revolution = 1 circumference
In rotary motion, you will have to use dimensional analysis to go from certain units to the correct units of angular and linear velocity to determine the answers in certain problems.
Real Life Applications
Real Life Applications
Many of the concepts in this unit can play a large role in real life applications. From the unit circle to rotary motion, this unit is full of ideas that will help you all in your future careers. Below are the two main topics that are catagorized into the main real life applications that come wtih Unit 2 Unit Circle Trigometry.
Sinusoidal Functions
Sinusoidal Functions
This topic is full of real life world applications, since the sinusoid itself represents a variety of different uses in daily life. From sound waves to metrology, sinusoidal functions play a large role in understanding real life applications in the mathematical world. For the rest of your math career, a concept like this will help you understand certain topics to a full extent. These real world topics include health, sea levels, sound and music, and many more. The four main properties within a sinusoid will also prove to be important in terms of understanding variables that could change your final answers. Sinusoids will prove to be an important part of your understanding of the relation of math to the real world.
Rotary Motion
Rotary Motion
Rotary motion is entirely made up of real life applications. From the wheel, to propellers, to extremely complicated machinery, rotary motion is extremely important when it comes to technology and speed calculation. Knowing rotary motion gives you the ability to use measurements to calculate speed, which can be critical information. Knowing how hard you have to pedal to get your car to go a certain speed, or knowing how long it will take your car to stop depending on your speed are pieces of information that could potentially save your life. Rotary Motion is without a doubt one of the most important mathematical topics you will ever come across.
Sample Problems
Sample Problems
Unit Circle Trigonometry
Unit Circle Trigonometry
Fill in all of the blank spots based on the grid. You will probably get a bunch of these coming into this year!
Sinusoidal Functions
Sinusoidal Functions
Determine the equation of the sinusoid, along with the amplitude, period, phase displacement, and sinusoidal axis.
Solution
Solution
Equation: y = -3 + 5 cos 3(θ-10°)
Amplitude = 5
Period = 120°
Phase Displacement = 10
Sinusoidal Axis = -3
Rotary Motion
Rotary Motion
Solution
Solution
Question A: w = 900 in / s * 1rad / 6in = 150 rad / sec
w = 900 in / s * 1rev / 12(π)in * 60s / min = 4500 rev / (π)min = 1432.39 rev/min
Question B: 4500rev / (π)min * 19(π)in / rev * 1min / 60s = 1425 in/s
Conclusion
Conclusion
After reading this page, you all are probably a bit confused with the amount of topics that you have to understand, but it will all come with time and commitment. Just take your time and be sure to ask help when you need it. We hope that this page will prove to be helpful in both your learning and understanding of mathematics! Good luck this year in Pre-Calculus!
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