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 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:

$Period=\frac{2\pi}{b}$

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:

$bx-c=\frac{\pi}{2} \therefore x=\frac{2c+\pi}{2b}$

Therefore:

$Maximum \ point: (\frac{2c+\pi}{2b},d+a)$

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

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

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

$bx-c=0 \therefore x=\frac{c}{b}$

Therefore:

$Maximum \ point: (\frac{c}{b},d+a)$

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

$Maximum \ point: (\frac{c}{b}+kT,d+a) \\ \\ k=...-3,-2,-1,0,1,2,3... \\ T:Period$

4. Minimum Point

For the sine function the minimum point occurs when:

$bx-c=\frac{3\pi}{2} \therefore x=\frac{2c+3\pi}{2b}$

Therefore:

$Minimum \ point: (\frac{2c+3\pi}{2b},d-a)$

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

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

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

$bx-c=\pi \therefore x=\frac{c+\pi}{b}$

Therefore:

$Minimum \ point: (\frac{c+pi}{b},d-a)$

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

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

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:

$Amplitude=\left | a \right |$