Embedded Files
by Teddy Malone and Bryan Li, class of 2022
The Unit Circle
The Unit Circle
- The center of the unit circle is at the origin
- The unit circle has a radius of 1
- Positive angle measurements start from 0 degrees as shown in the diagram and rotates counter-clockwise (negative angle measurements go the other way)
- Angles can be measured in degrees or radians
- a radian is the angle whose arc is equal to the radius of the circle, in this case, 1
- The main angles and their calculable coordinate values are shown in this diagram
Main Trig Ratios
Main Trig Ratios
Sin
Sin
- Sin θ = opposite leg/hypotenuse
- Sin refers to the vertical and y-values on the unit circle
- This is because the hypotenuse is the radius, which is equal to 1
Cos
Cos
- Cos θ = adjacent leg/hypotenuse
- Cos refers to the horizontal and x-values on the unit circle
- This is because the hypotenuse is the radius, which is equal to 1
Tan
Tan
- Tan θ = opposite leg/adjacent leg
- Tan θ = sin θ/cos θ = y/x
- This is because sin θ = opposite = y, cos θ = adjacent = x
Reciprocal Trig Ratios
Reciprocal Trig Ratios
Csc
Csc
- Csc θ = hypotense/opposite leg
- Csc θ = 1/sin θ
Sec
Sec
Sec θ = hypotenuse / adjacent leg
Sec θ = 1/cos θ
Cot
Cot
Cot θ = cos θ/sin θ
Cot θ = 1/tan θ
Sinusoidal Functions
Sinusoidal Functions
y = a sin (b(x-h)) +k
y = a sin (b(x-h)) +k
- IaI = amplitude
- b = frequency (number of cycles in 360 degrees or 2π)
- h = horizontal shift
- k = vertical shift
y = a cos (b(x-h)) +k
y = a cos (b(x-h)) +k
- fequency * period = 360 degrees or 2π
Inverse Trig and Arcs
Inverse Trig and Arcs
Inverse Trig Functions
Inverse Trig Functions
θ = sin^-1 (x)
θ = sin^-1 (x)
- find the value of θ whose sine is equal to x
- the range of θ is between -90 and 90 degrees
- the domain of x is between -1 and 1
θ = cos^-1 (x)
θ = cos^-1 (x)
- find the value of θ whose cosine is equal to x
- the range of θ is between 0 and 180 degrees
- the domain of x is between -1 and 1
θ = tan^-1 (x)
θ = tan^-1 (x)
- find the value of θ whose tangent is equal to x
- the range of θ is between -90 and 90 degrees
- the domain of x is between -∞ and ∞
Arcs
Arcs
θ = arcsin (x), θ = arccos (x), θ = arctan (x) (not as important)
θ = arcsin (x), θ = arccos (x), θ = arctan (x) (not as important)
- find all values of θ within the range of (-∞, ∞) whose sine, cosine, or tangent is equal to the value givin as x
- this is different from the inverses because the inverses have a smaller range for θ
- the answer should be written in the form: θ + 360*n, where n = all integers
- it iis important that one calculates all values of θ within the range of [0, 360) before adding "360*n" as there may be one or two values that satisfy the equation
Trig Identities
Trig Identities
Pythagorean Identities
Pythagorean Identities
sin^2(θ) + cos^2(θ) = 1
tan^2(θ) + 1 = sec^2(θ)
cot^2(θ) + 1 = csc^2(θ)
Angle Sum Formulas
Angle Sum Formulas
sin(a+b) = sin(a) cos(b) + cos(a) sin(b)
cos(a+b) = cos(a) cos(b) - sin(a) sin(b)
Angle Difference Formulas
Angle Difference Formulas
sin(a-b) = sin(a) cos(b) - cos(a) sin(b)
cos(a-b) = cos(a) cos(b) + sin(a) sin(b)
Double Angle Formulas
Double Angle Formulas
sin(2a) = 2 sin(a) cos(a)
cos(2a) = cos^2(a) - sin^2(a) or
2 cos^2(a) -1 or
1 - 2 sin^2(a)
Half Angle Formulas
Half Angle Formulas
Rotary Motion
Rotary Motion
Rotary Motion: What is it?
Rotary Motion: What is it?
Rotary motion refers to the motion that an object undergoes when spun or rotated. Whether you know it or not, rotary motion is everywhere in our daily lives and it has numerous applications
Rotary Motion: Different Types of Velocity
Rotary Motion: Different Types of Velocity
Angular Velocity
Angular Velocity
- The number of degrees/radians per unit of time
- The angular velocity of a point on a rotating object is the number of degrees through which the points turns per unit of time.
- When there are two rotating objects connected by an axle, the two objects will have the same angular velocity
Linear Velocity
Linear Velocity
- The distance per unit of time
- The linear velocity of a point on a rotating object is the distance the point travels along its circular path per unit of time.
- When there are two rotating objects connected by their rims, then the two objects will have the same linear velocity.
- This can also be true for pulley systems, belts, chains, and gears.
Rotary Motion Velocity Equations
Rotary Motion Velocity Equations
Angular Velocity
Angular Velocity
Applications (Sinusoidal Functions as Mathematical Models)
Applications (Sinusoidal Functions as Mathematical Models)
Sample Problems
Sample Problems
Google Sites
Report abuse