Show:
7.5-- Unit Circle
7.8-- Phase Shift and Sinusoidal Curve Fitting
8.5-- Sum and Difference Formulas
8.7-- Product-to-Sum and Sum-to-Product Formulas
Personal Reflection
100
What is the Unit Circle, and what is it used for?
The Unit Circle is a circle whose origin is at the center of a coordinate plane (0,0), whose radius is one unit, and whose circumference is 2πr. It can be used to find the exact values of trigonometric functions.
100
Define the following terms: Amplitude, period, vertical shift, phase shift, and reflection.
Amplitude: The vertical stretch or shrink of a graph. Period: The horizontal stretch or shrink of a graph; how many units it takes for the graph to complete one period (cycle). Vertical shift: How far up or down (on the Y-axis) the parent function is moved. Phase shift: How far to the left or right (on the X-axis) the parent function is moved. Reflection: When a figure, in this case a graph, is reflected across an axis, creating a mirror image.
100
List the sum and difference formulas for the cosine and sine functions.
cos(α + β) = [cos(α) cos(β)] - [sin(α) sin(β)] cos(α - β) = [cos(α) cos(β)] + [sin(α) sin(β)] cos[(π/2) - θ] = sin(θ) sin(α + β) = [sin(α) cos(β)] + [cos(α) sin(β)] sin(α - β) = [sin(α) cos(β)] - [cos(α) sin(β)] sin[(π/2) - θ] = cos(θ)
100
What are the product-to-sum formulas?
sin(α) sin(β) = 1/2 [cos(α - β) - cos (α + β)] cos(α) cos(β) = 1/2 [cos(α - β) + cos (α + β)] sin(α) cos(β) = 1/2 [sin(α + β) - sin(α - β)]
100
Starting assignments as soon as I get them, and generally procrastinating less.
What is something that I have learned I need to do to be the most successful I can be?
200
What are the even functions and what are the odd functions? What does it mean for a function to be even or odd and why is that important?
Even functions: cosine and secant. They are even because they have the same graph regardless of whether θ is positive or negative. This comes in handy to know when simplifying trigonometric expressions. Therefore, cos(-θ)=cos(θ) and sec(-θ)=sec(θ). Odd functions: Sine, cosecant, tangent and cotangent. They are odd because their graphs change depending on whether θ is positive or negative. This is also handy when simplifying trigonometric expressions. Therefore, sin(-θ)=-sin(θ), csc(-θ)=-csc(θ), tan(-θ)=-tan(θ), and cot(-θ)=-cot(θ).
200
How would one find the amplitude, period, vertical shift, phase shift, and reflection of a basic sinusoidal function? Given function is f(x) = A sin (ωθ - ϕ) + B
Amplitude = |A| Period = 2π/ω Vertical shift = B Phase shift = ϕ/ω Reflection = none
200
List the sum and difference formulas for the tangent function.
tan(α + β) = [tan(α) + tan (β)] / [1- tan(α) tan(β)] tan(α - β) = [tan(α) - tan (β)] / [1 + tan(α) tan(β)]
200
What are the sum-to-product equations?
sin(α) + sin(β) = 2 sin [(α + β) / 2] cos [(α - β) / 2] sin(α) - sin(β) = 2 sin [(α - β) / 2] cos [(α + β) / 2] cos(α) + cos(β) = 2 cos[(α +β) / 2] cos [(α - β) / 2] cos(α) - cos(β) = -2 cos[(α +β) / 2] cos [(α - β) / 2]
200
Something I have to not be angry at myself for having to do, especially when it comes to math.
What is asking for extra help?
300
Using the Unit Circle, find the exact value of sin(405°).
√2/2
300
Find the amplitude, period, vertical shift, and horizontal shift of the following function, and determine whether or not it will be reflected: f(θ) = 2 sin [θ - (π/4)] + 1
Amplitude = 2 Period = 2π Vertical shift = 1 Horizontal shift = π/4 Reflection = none
300
Using one of the sum or difference formulas, find the exact answer of cos(105°).
(-√6 - √2) / 4
300
Find the simplest version of sin(-6θ) cos (2θ).
1/2 [-sin(4θ) + sin(8θ)]
300
Something I learned about others this year.
What is realizing that no one expects me to be perfect the first time I do something?
400
Using the Unit Circle, find the exact value of cos(33π/4).
√2/2
400
Find the amplitude, period, vertical shift, and horizontal shift of the following function, determine whether it will be reflected, and then graph it: f(θ) = (1/2) sin (2θ - π)
Amplitude = 1/2 Period = π Vertical shift = none Phase shift = π/2 Reflection = none (See annotated solution page for graph.)
400
Using one of the sum or difference formulas, find the exact value of sin(5π/12).
(√2 + √6) / 4
400
Find the simplest version of sin(4θ) + sin(6θ).
2 sin (5θ) cos (θ)
400
Something that I absolutely hated when I learned it in 9th grade, but now actually find enjoyable because I see it as a puzzle.
What is using trigonometric equations to find the measurements of triangles?
500
DOUBLE GEOPARDY: Using the Unit Circle, find the exact value of sin(-π) + cos(5π).
-1
500
Find the amplitude, period, vertical shift, and horizontal shift of the following function, determine whether it will be reflected, and then graph it: f(θ) = -3 cos (2θ + π) - 3
Amplitude = 3 Period = π Vertical shift = -3 Phase shift = -π/2 Reflection = yes (See annotated solution page for graph.)
500
Using sum and difference formulas, find the exact solution of tan(75°) + cos(165°).
(8 + 4√3 - √6 - √2) / 4
500
Find the exact value of sin(195°) cos(75°).
(-2 - √3) / 4
500
Something that I used to think I was hopelessly bad at, but now realize that if I put the work into understanding it, I'm actually pretty good at it.
What is trigonometry?