A unit circle is the circle with its center at the origin and a radius of 1 unit.
The figure here shows all the measurements of the unit circle:
Positive distance is measured in a counterclockwise direction; while negative distance is measured in a clockwise direction.
SINE, COSINE AND TANGENT
We can recall cosθ as the x-axis and sinθ as the y-axis
P(θ) is used to point the terminal point, where the terminal arm of angle θ intercepts the unit circle. P(θ) can be defined as P(x,y).
Booklet
sin θ = O/H => sin θ = y/1 = y
cos θ= A/H => cos θ = x/1 = x
tan θ = O/A => tan θ = y/x
In this case, P(x,y) = P(cosθ,sinθ)
Pythagoras Theorem says that In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The equation of a unit circle is x ² + y ² = r ², and since x=cos and y= sin we can say that
cos² θ+sin²θ = 1² ,Note* cos θ²+sin θ² = 1² is different from cos² θ+sin²θ = 1²
Quadrant I is located at the top right corner (A) where all are positive.
Quadrant II is located at the top left corner (S) where only sine is positive leaving Tangent and Cosine negative, this means if you were to have a positive Sine value, it will be allowed in Quadrant I and Quadrant II.
Quadrant III is located at the bottom left corner (T) where T is positive.
Quadrant IV is located at the bottom right (C) where C is positive.
If you were to have a x-coordinate as ⅔, to figure out which Quadrant it is located on. We must find the y value = sin, P(⅔,?)
sin θ = O/H => sin θ = y/1 = y
cos θ= A/H => cos θ = x/1 = x
tan θ = O/A => tan θ = y/x
In this case, P(x,y) = P(cosθ,sinθ)
Pythagoras Theorem says that In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The equation of a unit circle is x ² + y ² = r ², and since x=cos and y= sin we can say that
cos² θ+sin²θ = 1² ,Note* cos θ²+sin θ² = 1² is different from cos² θ+sin²θ = 1²
The Cast Rule
The cast rule is a really simple way of knowing which quadrant a trigonometric function will be located.
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| Cast Rule - Simple Writes. |
Quadrant I is located at the top right corner (A) where all are positive.
Quadrant II is located at the top left corner (S) where only sine is positive leaving Tangent and Cosine negative, this means if you were to have a positive Sine value, it will be allowed in Quadrant I and Quadrant II.
Quadrant III is located at the bottom left corner (T) where T is positive.
Quadrant IV is located at the bottom right (C) where C is positive.
If you were to have a x-coordinate as ⅔, to figure out which Quadrant it is located on. We must find the y value = sin, P(⅔,?)
Using the unit circle equation = x ² + y ² = r ²
as the Radius is 1 unit.
x ² + y ² = 1
(⅔)^2 + y ² = 1
4/9 + y² = 1
y ² = 1 - 4/9
y ² = 9/9 - 4/9
√y²= ± √5/√9 or ±√5/3
P(⅔,-√5/3) will lay on Quadrant 4 since its sine value is negative and cos being positive.
x ² + y ² = 1
(⅔)^2 + y ² = 1
4/9 + y² = 1
y ² = 1 - 4/9
y ² = 9/9 - 4/9
√y²= ± √5/√9 or ±√5/3
P(⅔,-√5/3) will lay on Quadrant 4 since its sine value is negative and cos being positive.
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| P(⅔,±√5/3) That's basically all I have to say. Although here is a method that I found on the internet. Click Here |
And a video.
Graph Cosine and Sine Functions
Graphing Parent Sine and Cosine Functions Precalculus Trig Functions
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