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.

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.

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.

The period (2π) represents one full revolution around 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).

A phase shift represents a horizontal translation, corresponding to a different starting point on the unit circle.

By observing the signs of the y-values (sine) and x-values (cosine), you can determine the quadrant based on the ASTC rule.

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.

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.

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.

1. Use the identity $\sin^2(\theta) + \cos^2(\theta) = 1$. 2. Solve for sin(θ). 3. Determine the sign of sin(θ) based on the given quadrant.

1. Find the principal angle θ in the range [-π/2, π/2]. 2. Find the other angle in [0, 2π] using symmetry. 3. Add 2πk to each to find all coterminal angles.

1. If sin(θ) > 0 and cos(θ) > 0, the angle is in Quadrant I. 2. If sin(θ) > 0 and cos(θ) < 0, the angle is in Quadrant II. 3. If sin(θ) < 0 and cos(θ) < 0, the angle is in Quadrant III. 4. If sin(θ) < 0 and cos(θ) > 0, the angle is in Quadrant IV.

1. If angle is in QI, reference angle = angle. 2. If angle is in QII, reference angle = 180 - angle (in degrees) or π - angle (in radians). 3. If angle is in QIII, reference angle = angle - 180 (in degrees) or angle - π (in radians). 4. If angle is in QIV, reference angle = 360 - angle (in degrees) or 2π - angle (in radians).

1. Recall tan(θ) = sin(θ)/cos(θ). 2. Find angles where sin(θ) and cos(θ) are equal. 3. Identify θ = π/4 and θ = 5π/4 as solutions in [0, 2π).

1. Recognize that sine is an odd function, so sin(-θ) = -sin(θ). 2. Find the value of sin(θ). 3. Change the sign to find sin(-θ).

1. Recognize that cosine is an even function, so cos(-θ) = cos(θ). 2. Find the value of cos(θ). 3. The value is the same.

1. Recognize sin(θ) = 0.5 corresponds to π/6. 2. Identify the second quadrant angle where sin(θ) is also 0.5, which is 5π/6.

1. Identify the angle on the unit circle. 2. Determine the sine, cosine, and tangent values for that angle. 3. Substitute these values into the expression and simplify.

1. Recall cos(θ) represents the x-coordinate on the unit circle. 2. Identify the angle where the x-coordinate is -1. 3. θ = π.

Sine: y-coordinate on the unit circle, odd function | Cosine: x-coordinate on the unit circle, even function

Positive angles: Measured counterclockwise | Negative angles: Measured clockwise

Radians: Based on the radius of the circle, dimensionless | Degrees: Arbitrary division of a circle into 360 parts

sin(0) = 0 | cos(0) = 1

sin(π/2) = 1 | cos(π/2) = 0

Sine: [-1, 1] | Cosine: [-1, 1]

y = sin(x): Starts at (0, 0), odd function | y = cos(x): Starts at (0, 1), even function

Derivative of sin(x): cos(x) | Derivative of cos(x): -sin(x)

Integral of sin(x): -cos(x) + C | Integral of cos(x): sin(x) + C

Sine: Opposite/Hypotenuse | Cosine: Adjacent/Hypotenuse