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

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