6.2 Identities

In this section, we learned all about identities.
Let us begin with the simplest- the reciprocal identities.

So, an example of using this would be:

There are also the quotient identities. 

And last but not least are the Pythagorean Identities!

We proved the first one in class, so here is the proof of that along with the other identities: 

And where the trigonometric functions are negative or positive: 

Alright, that's it! Sorry this took a while, I didn't get the e-mail for the blog!

MATHS IS VITAL!!!!!!!!!!

-Maggie Ridenour

6.3 Trigonometric Functions of Real Numbers

Definition of the Trigonometric Functions of Real Numbers: the value of a trigonometric function at a real number t is its value at an angle of t radians

because of this, "sin 2" could be the sine of the real number 2, OR the sine of an angle of 2 radians

You can interpret trigonometric functions of real numbers geometrically by using a unit circle: 





 The radius of the circle is 1, with the center at the origin of a rectangular coordinate plane. 

=  theta

We will use the variable t as the radian measure of the angle theta or as the lenght of the black-highlighted circular arc on the circle above.

Definition of the Trigonometric Functions in Terms of a Unit Circle:

sin t = y               cos t = x             tan t = y/x

csc t = 1/y           sec t = 1/x         cot t = x/y

Finding values of trigonometric functions:

If point P is at the coordinates (-3/5, -4/5), find the values of the trigonometric functions at t (theta).

Use the definition of trigonometric functions in terms of a unit circle to get your answers.....

sin t = y = -4/5               cos t = x = -3/5             tan t = y/x = (-4/5)/(-3/5) = 4/3

csc t = 1/y = 1/(-4/5)           sec t = 1/x = 1/(-3/5)        cot t = x/y = (-3/5)/(-4/5) = 3/4

Theorem on Repeated Function Values for sine and cosine:  If n is any integer, then

                                        sin (t + 2n) = sin t        and          cos (t = 2n) = cos t

Graphing Functions

Graph of y = sin x:

y = cos x:

The part of the graph of the sine or cosine function where x is greater or equal to 0 and less than or equal to 2 is called one cycle. These cycles are sometimes refered to as a sine wave  or a cosine wave.

y = tan x:

y = csc x:

y = sec x:

y = cot x:

Formulas for Negatives:  

                            

                           

Theorem on Even and Odd Trigonometric Functions: 

1. The cosine and secant functions are even.

2. The sine, tangent, cosecant, and cotangent functions are odd.

That's All Folks!

-Henry Burg 





Chapter 6.1 ANGLES

FISRT!
Angles have two sides: the Initial Side and the Terminal Side. 

The rotation of the rays always starts on the intial side, and always ends on the terminal side.
Many angles can have the same initial and terminal sides. These are called Coterminal Angles.

When it is rotated counter-clockwise to the terminal side, the angle is positive. When it is rotated clockwise, the angle is negative.


Standard Position:
The vertex is on the origin of the graph, and the initial side is on the positive x-axis.

Finding Cotermianl Angles

is standard position, find two positive angles and two negative angles that are coterminal with

 to find positive coterminal angles, you can add any positive integer multiple of 360° to 

 to find negative coterminal angles, you can add any negative integer multiple  of 360° to

Radians


Definition: one radian is the measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle.
The length of the radius is eqaul to the arc length.

Realtionships Between Degrees and Radians

1. 
2. 
3. 

Changing Angular Measures

Degrees to Radians: 

Radians to Degrees: 

Formula for the Length of a Circular Arc

If an arc of length s on a circle of radius r subtends a central angle or a radian measure θ, then s=rθ.

Formula for the Length of a Circular Sector

If θ is the radian measure of a central angle of a circle of radius r and if A is the area of the circular sector determined by θ, then A=1/2 r²θ.

-Austin A.K.A. the Hiphopopotamus