Question

compare and contrast sine and cosine functions in standard form. apply: period, shape, minimum point, maximum point, domain, range x-intercept y intercept phase shift etc

Answer 1

The sine and cosine functions are both periodic functions that represent oscillatory motion. They have similarities in terms of period, shape, minimum and maximum points, domain and range, x-intercepts and y-intercepts, and phase shift.

Step-by-step explanation:

The sine and cosine functions are both periodic functions that represent oscillatory motion. They have several similarities and differences:

Period:

The period of the sine function is 2π, while the period of the cosine function is also 2π. This means that both functions repeat their values every 2π units of input.

Shape:

The sine function has a wave-like shape, oscillating between -1 and 1. The cosine function has a similar shape, but it is shifted to the left by 90 degrees or π/2 radians compared to the sine function.

Minimum and Maximum Points:

The minimum value of the sine function is -1, and the maximum value is 1. The minimum value of the cosine function is also -1, and the maximum value is also 1. The difference lies in the location of these points in the period.

Domain and Range:

The domain of both functions is the set of all real numbers. The range of the sine function is [-1, 1], while the range of the cosine function is also [-1, 1].

X-intercepts and Y-intercepts:

The sine function has x-intercepts at every integer multiple of π, while the cosine function has x-intercepts at every half-integer multiple of π. Both functions have a y-intercept at the origin (0, 0).

Phase Shift:

The phase shift of a function determines its horizontal shift. If the phase shift is zero, the sine and cosine functions will have the same initial conditions, such as the initial position, velocity, and acceleration. However, if there is a phase shift, the initial conditions will differ between the two functions.

Answer 2

The standard form of sine and cosine functions are given by these equations:

`Sine \ function: \\ y=d+asin(bx-c) \\ \\ Cosine \ function: \\ y=d+acos(bx-c)`

So let's compare each topic as follows:

1. Period

The period
`T` of these two functions is the same. So, let
`b` be a positive real number. The period of
`y=d+asin(bx-c)` and
`y=d+acos(bx-c)` is given by:

`Period=(2\pi)/(b)`

2. Shape

If you want to graph the sine function, you need to mark the angle along the horizontal
`x axis`, and for each angle, you put the sine of that angle on the vertical
`y-axis`. As a result, a smooth curve that varies from +1 to -1 is formed as indicated in the blue curve below. We call this type of curves sinusoidal after the name of the sine function. This shape is also called a sine wave.

On the other hand, if you want to graph the cosine function, you need to mark the angle along the horizontal
`x axis`, and for each angle, you put the cosine of that angle on the vertical
`y-axis`. As a result, a smooth curve that varies from +1 to -1 is formed as indicated in the red curve below. It is the same shape as the sine function but displaced to the left 90°. This is also called sinusoidal.

3. Maximum point

For the sine function the maximum point occurs when:

`bx-c=(\pi)/(2) \therefore x=(2c+\pi)/(2b)`

Therefore:

`Maximum \ point: ((2c+\pi)/(2b),d+a)`

Since this is a periodic function each maximum point occurs at:

`Maximum \ point: ((2c+\pi)/(2b)+kT,d+a) \\ \\ k=...-3,-2,-1,0,1,2,3... \\ T:Period`

On the other hand, for the cosine function we have:

`bx-c=0 \therefore x=(c)/(b)`

Therefore:

`Maximum \ point: ((c)/(b),d+a)`

Since this is a periodic function each maximum point occurs at:

`Maximum \ point: ((c)/(b)+kT,d+a) \\ \\ k=...-3,-2,-1,0,1,2,3... \\ T:Period`

4. Minimum Point

For the sine function the minimum point occurs when:

`bx-c=(3\pi)/(2) \therefore x=(2c+3\pi)/(2b)`

Therefore:

`Minimum \ point: ((2c+3\pi)/(2b),d-a)`

Since this is a periodic function each minimum point occurs at:

`Minimum \ point: ((2c+3\pi)/(2b)+kT,d-a) \\ \\ k=...-3,-2,-1,0,1,2,3... \\ T:Period`

On the other hand, for the cosine function we have:

`bx-c=\pi \therefore x=(c+\pi)/(b)`

Therefore:

`Minimum \ point: ((c+pi)/(b),d-a)`

Since this is a periodic function each maximum point occurs at:

`Minimum \ point: ((c+\pi)/(b)+kT,d-a) \\ \\ k=...-3,-2,-1,0,1,2,3... \\ T:Period`

5. Domain

The domain of the sine and cosine functions is the set of all real numbers, that is:

`Df=\mathbb{R}`

6. Range

The range of the sine and cosine function in its standard form is:

`d-a \leq y \leq d+a`

7. The x-intercept

For cosine and sine functions in its standard forms there are two possibilities:

a. The graph intersects the x-axis at infinitely many points.

b. The graph does not intersects the x-axis.

8. The y-intercept

For cosine function the y-intercept occurs at:

`when \ x=0 \\ y=d+acos(-c)`

On the other hand, for sine function the y-intercept occurs at:

`when \ x=0 \\ y=d+asin(-c)`

9. Phase shift

The constant
`c` in the equations

`y=asin(bx-c) \ and \ y=acos(bx-c)`

Creates a horizontal translation (shift) of the basic sine and cosine curves. So the graphs are shifted an amount
`c/b`, so this number is called the phase shift.

10. Amplitude

The amplitude of sine and cosine functions represents half the distance between the maximum and minimum values of the function and is given by:

`Amplitude=\left | a \right |`