Trigonometric Identities Notes

Standard

Intro to Trigonometry:

There are 6 trigonometric functions, which take angles as their input and output to a real number. An angle is said to be in standard position: x-axis, with it vertex at the origin.  If the angle opens clockwise then the angle is considered to be negative angle. If the angle opens counter-clockwise it is considered to be a positive angle.

Two angles are called coterminal if they share a terminal side in standard position. For example, the angles 150 degrees, 510 degrees, and -210 degrees are all coterminal.

Radians are another unit of measure for angles. There are 360 degrees in a complete angle

so 360 degrees is equal to 2 pi radians; 180 degrees is equal to pi radians; 180 degrees divided by pi is equal to 1 or equal to pi divided by 180 degrees.

The unit circle is the circle centered at the origin with a radius the length of one unit.

The definition of the trig functions in the context of a point on the plane is suppose $\theta$ is in standard position, and p= (x,y) is a point on the terminal side of theta, and r is the distance between P and the origin. Then the six trig functions are defined as follows:

sin $\theta$ = $\frac {y}{r}$

cos $\theta$ = $\frac {x}{r}$

tan $\theta$ = $\frac {y}{x}$

csc $\theta$ = $\frac {r}{y}$

sec $\theta$ = $\frac {r}{x}$

cot $\theta$ = $\frac {x}{y}$

In the context if a point is on the unit circle. Suppose $\theta$ is in standard position , and P= (x,y) is a point on the unit circle. Then the 6 trig functions are defined as follows:

sin $\theta$ =  y

cos $\theta$ = x

tan $\theta$ = $\frac {y}{x}$

csc $\theta$ = $\frac {1}{y}$

sec $\theta$ = $\frac {1}{x}$

cot $\theta$ = $\frac {x}{y}$

Definition of the trig functions in the context of a right triangle. Suppose $\theta$ is an acute angle of a right triangle. Trig functions are defined as follows:

sin $\theta$ = $\frac {opposite}{hypotenuse}$

cos $\theta$ = $\frac {adjacent}{hypotenuse}$

tan $\theta$ = $\frac {opposite}{adjacent}$

csc $\theta$ = $\frac {hypotenuse}{opposite}$

sec $\theta$ = $\frac {hypotenuse}{adjacent}$

cot $\theta$ = $\frac {adjacent}{opposite}$

Basic Identities:

$sin ^2$ x + $cos^2$ x = 1

tanx= $\frac {sinx}{cosx}$

sin(x + y) = sin x cos y + cos x sin y

sin (x – y) = sin x cos y – cos x sin y

cos (x + y) = cos x cos y – sin x sin y

cos (x – y) = cos x cos y + sin x sin y

The derivative of y = sin x also depends on the value of

$\overset {lim}{ h \rightarrow 0}$ $\frac {sin(x+h)-sinx}{h}$

To esitmate the limit, find the qoutient $\frac {sinx}{x}$ for various calues of x close to 0. ( make sure to set your calculator to radian mode not in degree mode)

Derivatives of Trig Functions:

Dx(sin x) = cos x

Dx(cos x) = -sin x

Dx (tan x) = $sec^2$ x

Dx (cot x) = $-csc^2$ x

Dx (sec x) = sec x tan x

Dx (csc x) = -csc x cot x