Definition: The unit circle is a circle that is centered at the origin (0,0) & has a radius of one unit, and can be used to directly measure sine, cosine, & tangent.
Equation: x2 + y2 = 1 (x squared + y squared = 1)
Arc length: s =θ --> the radian measure of angle θ
Circumfrence: C = 2πr = 2π (1) = 2π --> circumference of 2π is also the radian measure of the angle corresponding to 360°:
2π radians = 360 degrees
π radians = 180 degrees
Arc length: s =θ --> the radian measure of angle θ
Circumfrence: C = 2πr = 2π (1) = 2π --> circumference of 2π is also the radian measure of the angle corresponding to 360°:
2π radians = 360 degrees
π radians = 180 degrees
How to use:
1) Start from point 0, which is located on the x-axis between Quadrant One & Quadrant 4
2) Begin counting the given units (either degrees, or radians) in a counter-clockwise manner
3) Once the reference point is determined, draw a line to the nearest x-axis to get the reference triangle
4) With the reference triangle, determine what you are trying to solve by applying you knowledge of special right triangles to determine values
2) Begin counting the given units (either degrees, or radians) in a counter-clockwise manner
3) Once the reference point is determined, draw a line to the nearest x-axis to get the reference triangle
4) With the reference triangle, determine what you are trying to solve by applying you knowledge of special right triangles to determine values
Understanding the Coordinates:
For any ordered pair (x,y) on the unit circle: cos θ= x and sin θ= y and θ is any central angle with: 1) initial side = positive x axis 2) terminal side = radius through the determine point
Trick!
Use the saying "All Students Take Calculus" to remeber that appropriate signs used for reference angles within the quadrant.
Quadrant 1: All = A = All are positive :
sin--> positive
cos--> positive
tan--> positive
Quadrant 2: Students = S = Sine is positive
sin--> positive
cos--> negative
tan--> negative
Quadrant 3: Take = T = Tangent is positive
sin--> negative
cos--> negative
tan--> positive
Quadrant 4: Calculus = C = Cosine is positive
sin--> negative
cos--> positive
tan--> negative
Quadrant 1: All = A = All are positive :
sin--> positive
cos--> positive
tan--> positive
Quadrant 2: Students = S = Sine is positive
sin--> positive
cos--> negative
tan--> negative
Quadrant 3: Take = T = Tangent is positive
sin--> negative
cos--> negative
tan--> positive
Quadrant 4: Calculus = C = Cosine is positive
sin--> negative
cos--> positive
tan--> negative
Example
Using the Unit circle evaluate the value of:
**remember cos θ= x and sin θ= y**
cos(5π/3)=
sin(5π/3)=
Answers:
cos(5π/3)= 1/2
sin(5π/3)= -√3/2
**remember cos θ= x and sin θ= y**
cos(5π/3)=
sin(5π/3)=
Answers:
cos(5π/3)= 1/2
sin(5π/3)= -√3/2
Breaking Down The Unit Circle
Counting by π/6 :
When counting by π/6, each quadrant is divided into 3 equivalent sections, resulting in the entire unit circle being split up into 12 sections each measuring 30°. Note the x and y axis are included when counting by π/6.
Counting by π/3 :
When counting by π/3, the unit circle is divided into 6 equivalent sections each measuring 60°. Note ONLY the x-axis is included when counting by π/3. Also note that counting by π/3 isn't necessary if measurements are simplified when counting by π/6. For example, 2π/6 is the same as π/3.
Counting by π/4 :
When counting by π/4, each quadrant is divided into 2 equivalent sections, resulting in the entire unit circle being split into 8 sections each measuring 45°. Note the x and y axis are included when counting by π/4.
Final product:
*note the black lines representing the axis are simplified measurements that are sometimes included when counting by certain measurement, as previously stated*
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