All Flashcards
What are the differences between sine and cosine functions?
Sine: y-coordinate on the unit circle, odd function | Cosine: x-coordinate on the unit circle, even function
What are the differences between positive and negative angles on the unit circle?
Positive angles: Measured counterclockwise | Negative angles: Measured clockwise
What are the differences between radians and degrees?
Radians: Based on the radius of the circle, dimensionless | Degrees: Arbitrary division of a circle into 360 parts
Compare sin(θ) and cos(θ) at θ = 0.
sin(0) = 0 | cos(0) = 1
Compare sin(θ) and cos(θ) at θ = π/2.
sin(π/2) = 1 | cos(π/2) = 0
Compare the range of sine and cosine functions.
Sine: [-1, 1] | Cosine: [-1, 1]
Compare the graphs of y = sin(x) and y = cos(x).
y = sin(x): Starts at (0, 0), odd function | y = cos(x): Starts at (0, 1), even function
Compare the derivatives of sin(x) and cos(x).
Derivative of sin(x): cos(x) | Derivative of cos(x): -sin(x)
Compare the integrals of sin(x) and cos(x).
Integral of sin(x): -cos(x) + C | Integral of cos(x): sin(x) + C
Compare the use of sine and cosine in right triangles.
Sine: Opposite/Hypotenuse | Cosine: Adjacent/Hypotenuse
What does the graph of y = sin(x) tell us about its derivative?
The derivative, cos(x), represents the slope of the sine function. Where sin(x) increases, cos(x) is positive; where sin(x) decreases, cos(x) is negative; where sin(x) has a max/min, cos(x) is zero.
What does the graph of y = cos(x) tell us about its derivative?
The derivative, -sin(x), represents the slope of the cosine function. Where cos(x) increases, -sin(x) is positive; where cos(x) decreases, -sin(x) is negative; where cos(x) has a max/min, -sin(x) is zero.
How does the unit circle relate to the graphs of sine and cosine?
The y-coordinates of points on the unit circle correspond to the y-values of the sine graph, and the x-coordinates correspond to the y-values of the cosine graph, as the angle increases.
What does the period of the sine and cosine graphs represent on the unit circle?
The period (2π) represents one full revolution around the unit circle.
How does the amplitude of a sine or cosine graph relate to the unit circle?
The amplitude represents the maximum distance from the x-axis, which corresponds to the radius of the unit circle (1 in the standard case).
What does a phase shift in a sine or cosine graph represent on the unit circle?
A phase shift represents a horizontal translation, corresponding to a different starting point on the unit circle.
How can you identify the quadrant of an angle from the sine and cosine graphs?
By observing the signs of the y-values (sine) and x-values (cosine), you can determine the quadrant based on the ASTC rule.
What does the integral of sin(x) represent graphically?
The integral of sin(x) is -cos(x) + C, which represents the area under the sin(x) curve. The constant C shifts the cosine graph vertically.
What does the integral of cos(x) represent graphically?
The integral of cos(x) is sin(x) + C, which represents the area under the cos(x) curve. The constant C shifts the sine graph vertically.
How does the steepness of the sine or cosine graph relate to the unit circle?
The steepness is related to how quickly the y or x coordinate changes as you move along the unit circle, which is greatest near 0 and π for cosine and π/2 and 3π/2 for sine.
What is the unit circle?
A circle with a radius of 1, centered at the origin (0,0).
What is an angle in standard position?
An angle that starts from the positive x-axis and goes counterclockwise.
Define 'radians'.
A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius.
Define 'terminal ray'.
A line from the origin at a given angle that intersects the unit circle.
Define sine (sin θ) in the context of the unit circle.
The y-coordinate of the point where the terminal ray intersects the unit circle.
Define cosine (cos θ) in the context of the unit circle.
The x-coordinate of the point where the terminal ray intersects the unit circle.
Define tangent (tan θ) in the context of the unit circle.
The ratio of the y-coordinate to the x-coordinate (sin θ / cos θ) of the point where the terminal ray intersects the unit circle.
What are reflex angles?
Angles greater than 180° but less than 360°.
What is a 'full rotation' in radians?
2π radians.
What is the ASTC rule?
A mnemonic to remember which trig functions are positive in each quadrant: All, Sine, Tangent, Cosine.