6.4 Graphs of The Sine and Cosine Functions

If ω>0, the amplitude and the period of y = A sin(ωx) and y= A cos (ωx) are given by ______________

Amplitude = |A|, and Period = T = 2π/ω

*The number |A| is called the __________ of y = A sin x or y = A cos x

amplitude

Steps for Graphing a Sinusoidal Function of the Form y=A sin(ωx) or y= A cos(ωx) Using Key Points: 1. Determine the _______ and _______ of the sinusoidal function 2. Divide the interval ________ into four subintervals of the same length 3. Use the endpoints of these subintervals to obtain ____ key points on the graph 4. Plot the five key points and draw a sinusoidal graph to obtain the graph of one _____, Extend the graph in one direction to make it complete.

amplitude, period, [0, 2π/ω], five

Properties of the Cosine Function: 1. The domain is the ________________ 2. The range consists of all real numbers _______________ 3. The cosine function is an ____________, as the symmetry of the graph with respect to the y-axis indicates 4. The cosine function is periodic, with period __ 5. The x-intercepts are _____________ the y-intercept is __ 6. The maximum value is 1, and occurs at x= _________________, the minimum value is -1, and occurs at x= ..., -π, π, 3π, 5π,......

set of all real numbers, 1 to 1 (incusive), even function, 2π, .., -3π/2, -π/2, π/2, 3π/2, 5π/2,...;, 1, ..., -2π, 0, 2π, 4π, 6π,......, ..., -π, π, 3π, 5π,......

cos(x-π/2) = ______

sin x

Since we want to graph the trigonometric functions in the xy-plane, we shall use the traditional symbols _ for the independent variable (or argument) and _ for the dependent variable (or value at x) for each function

x, y

Graphs of the Form ___________, can be graphed using transformations

y = A cos(ωx)

"Graphing Terms of Cosine Function": __________

y = f(x) = cos x

"Graphing Terms of Cotangent Function": ______________

y = f(x) = cot x

"Graphing Terms of Cosecant Function": ___________

y = f(x) = csc x

"Graphing Terms of Secant Function": _____________

y = f(x) = sec x

"Graphing Terms of Sine Function": _________

y = f(x) = sin x

"Graphing Terms of Tangent Function": ___________

y = f(x) = tan x

Properties of the Sine Function y = sin x 1. The domain is the __________________________ 2. The range consists of all real numbers from __________ 3. The sine function is an ___________________, as the symmetry of the graph with respect to the origin indicates 4. The sine function is ________, with period 2π 5. The x-intercepts are ______________________, the y-intercept (singular) is __ 6. The maximum value is 1 and occurs at x= __________________ The minimum value is at -1 and occurs at x= __________________

set of all real numbers, -1 to 1 (inclusive), odd function, ...,-2π, -π, 0, π, 2π, 3π,...., 0, ..., -3π/2, π/2, 5π/2, 9π/2,..., ...., -π/2, 3π/2, 7π/2, 11π/2,...