Sine and Cosine Functions

SAT Questions that focus on Sine and Cosine Functions require knowledge of the following topics.

Sine and Cosine in Right Triangles

Consider the triangle in the following figure:
C1 and C2 are called catheti (the sides adjacent to the right angle);
H is called hypotenuse (the side opposite to the right angle);
"a" and "b" are the two angles adjacent to the hypotenuse.

The sine of an angle is the ratio of the length of the opposite cathetus to the length of the hypotenuse:

$\sin{a}=\frac{C1}{H}$

$\sin{b}=\frac{C2}{H}$

The cosine of an angle is the ratio of the length of the adjacent cathetus to the length of the hypotenuse:

$\cos{a}=\frac{C2}{H}$

$\cos{b}=\frac{C1}{H}$

Sine and Cosine in the Unit Circle

The unit circle is a circle with radius one and centered at the origin (0, 0) of the Cartesian coordinate system. In this system, the x-axis is the cosine, and the y-axis is the sine (figure below):

Triangle AOC is a right triangle, with catheti  AC and OC, and hypotenuse OA. OS is the projection of cathetus AC on the y-axis (AC=OS).

Since the radius of the Unit Circle is 1, $OA=1$.

Therefore:

$\sin{a}=\frac{AC}{OA}=\frac{AC}{1}=AC=OS$

$\cos{a}=\frac{OC}{OA}=\frac{OC}{1}=OC$.

In the Unit Circle the cosine of angle $a$ is the coordinate of point A on the x-axis (the cosine axis); the sine of angle $a$ is the coordinate of point A on the y-axis (the sine axis).

Graphing Sine and Cosine Functions

Consider the Unit Circle (figure below):

When angle $a$ is 0º, OC is the radius of the circle (=1) and OS is zero. Thus: $\cos{0}=1$ and $\sin{0}=0$.

When angle $a$ increases from 0º to 90º, OC decreases from 1 to zero and OS increases from 0 to the radius of the circle (=1). Thus: $\cos{90º}=0$ and $\sin{90º}=1$.

As angle $a$ keeps increasing from 90º to 180º, OC decreases from 0 to minus one (-1) and OS decreases from 1 to zero. Thus: $\cos{180º}=-1$ and $\sin{180º}=0$.

As we keep this journey around the Unit Circle, recording the values of sine and cosine of angle $a$, we can draw the following graph of the sine function (angles are measured in degrees in the x-axis):

And the following graph of the cosine function (angles are measured in degrees in the x-axis):

Both the sine and the cosine graphs have the same shape, same amplitude, and same period (360º). The only difference between them is that the sine graph is shiftted 90º to the right from the cosine graph, that is, $cos{x}=sin{x+90º}$.

Graphing $f(x)=a+(b)sin(cx+d)$ and $f(x)=a+(b)cos(cx+d)$

Coefficient "a" makes the graph shift up (if positive) or down (if negative).

Consider, for example, functions $f(x)=cos{x}$ and $g(x)=2+cos{x}$. For $x=60º$, we have $f(60º)=cos{60º}=0.5$, and $g(60º)=2+cos{60º}=2.5$.

Coefficient "b", if positive, changes the amplitude of the graph. If it is negative, it not only changes the amplitude, but also reflects the graph vertically across the x-axis.

Consider, for example, functions $f(x)=cos{x}$ and $g(x)=2cos{x}$.

For $x=0º$, we have $f(0º)=cos{0º}=1$, and $g(0º)=2cos{0º}=2$. The amplitude of $2cos{x}$ is twice that of $cos{x}$.

Now consider functions $f(x)=cos{x}$ and $g(x)=-2cos{x}$.

For $x=0º$, we have $f(0º)=cos{0º}=1$, and $g(0º)=-2cos{0º}=-2$. The amplitude of $-2cos{x}$ is twice that of $cos{x}$, and when $-2cos{x}$ reaches its lowest value, $cos{x}$ reaches its highest value.

The following graph depicts functions $cos{x}$, $2+cos{x}$ and $2cos{x}$:

Coefficient "c" in functions $f(x)=a+(b)sin(cx+d)$ and $f(x)=a+(b)cos(cx+d)$ changes the period of the graph.

Consider, for example, functions $f(x)=cos{x}$ and $g(x)=cos{(2x)}$.

For $x=0º$ and $x=360º$, we have $f(0º)=cos{0º}=1$ and $f(360º)=cos{360º}=1$. The period of this function is 360º.

Now for $x=0º$ and $x=180º$, we have $g(0º)=cos{(2*0º)}=cos{0º}=1$ and $g(180º)=cos{(2*180º)}=cos{360º}=1$. The period of this function is 180º, that is, half the period of  $f(x)=cos{x}$.

Coefficient "d" shifts the graph to the right (if negative) or to the left (if positive).

The following graph depicts functions $cos{x}$, $cos{(2x)}$ and $cos{(x+1)}$:

Pythagorean Identity

Consider the Unit Circle in the figure below:

Triangle AOC is a right triangle with hypotenuse OA. According to Pythagoras' Theorem:

$AC^2+OC^2=OA^2$

$sen^2{a}+cos^2{a}=1^2$

$sen^2{a}+cos^2{a}=1$

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