Chapter 4.5 and 4.6

Okay so for 4.5 we learned about the basic since and cosine curves.The cosine and since graphs look like this:

So the x-coordinates on this graph represents the angle measure in radians on a unit circle, and the y-coordinates represents the points at which the angle equals the x-coordinate (if that makes sense). The cosine and sine functions are pretty much identical, except for the fact that the intercepts are different because they have 0s at different points on the graph. So as you can see from the graph, the sines and the cosines all have the same max and min, the same period length. The only difference is that the the y value of sine = 0 when x = 0, while the y value of cosine = 1 when x=0. This is because the sine represents the y value of the point on a unit circle, and

the cosine represents the x value of the point.

A period in the a graph is 2 pis long for a sine and cosine.

The equation for these graphs are in this format:

y = d+a sin(bx-c) or y = d+a cos(bx-c)

d moves the graph up and down

a vertically stretches and compresses a graph, meaning that it will increase or decrease the maximum and minimum, but leave the x intercepts alone.

b horizontally stretches and compresses a graph, meaning that it will stretch out the x intercepts or compress them together

c moves the graph left and right, but unlike d, c moves right when it is subtracting from x, and moves left when adding to x.

The next section, 4.6, was about the tangent function, cotangent, cosecant, and secant functions

The tangent function is the sine function over the cosine function, which looks like this:

This means that whenever the x makes the value of the cosine = 0, the answer for the tangent function will be undefined, meaning there will be an asymptote wherever cosine = 0. This also means that wherever sine = 0, the tangent value will also be 0, so that's where the graph will intercept the x axis.

Unlike cosine and sine, the period is only 1 pi.

The cotangent function is the cosine over sine, so for this when the sine value = 0, there is a asymptote and when the cosine value = 0, there is an intercept. They look like this:

Cosecant functions are parabolas that open up at the max and open down at the min of a sine function. Their asymptotes are when x= n(pi) and they look like this:

The equation for these are: csc x = 1/sinx

Finally, the last one is the secant function. This one is like the c

osecant, because it looks just like it except that it opens up and opens down at the max and min of a cosine function. Their asymptotes are when x = (pi) / 2 + n (pi)

The graph for this looks like this:

Inverse Trigonometric Functions

For a function to have an inverse, it must pass the Horizontal Line Test. As shown below, y=sinx does not pass the test.




The graph is one-to-one only from peak to trough.

Because of this, in order to have an inverse function, we must restrict the domain to the interval:

.

The range would then be [-1,1]



So the restricted domain of , y=sinx has an inverse called the inverse sine function...

The graph would look something like this...



***The output of an inverse trigonometric function is an angle. In this case the output of the inverse sine would be in either the first or the fourth quadrant.***

The cosine function is decreasing on the interval as shown below




As with the sine function, we must also restrict the domain of the cosine function in order to have an inverse cosine function.

The restricted domain would be: [-1,1]
The range would be:

This graph would look like this....

***Again, remember the output of an inverse trigonometric function is an angle. So for this case the output of the inverse cosine function would be in the first or second quadrant.***





Similar to the inverse sine and cosine functions, you can define an inverse tangent function also by restricting the domain.

We restrict the domain of the inverse tangent function to all real numbers.
The range being .

The graph would like like this....




***Don't forget the output of an inverse trigonometric function is an angle. In this case the output of the inverse tangent lies in the first or fourth quadrant.***

Section 4.4 Trigonometric Functions of Any Angle

Blogpost 3rd Hour Pre Calc.

Today we learned about Reference Angles and Inverse Trigonometric Functions

REFERENCE ANGLES

• The values of the trigonometric functions of angles greater than 90° (or less than 0° ) can be determined from their values at corresponding acute angles called REFERENCE ANGLES.

• Definition: Letθ be an angle in standard position. Its REFERENCE ANGLE is the acute angleθ'formed by the TERMINAL SIDE ofθ and the HORIZONTAL AXIS.

• In short terms, Reference Angles are always Acute and every angle has one. The reference angle is the angle formed between the horizontal X-axis and the terminal side of the angle that intersects the origin.

In order to find reference

angles you need to determine which quadrant you want the reference angle to be in.

Different Reference Angles exist in all of the 4 quadrants. The

way to find out the reference angle in the designated quadrant is in the following

formulas.

In quadrant 1

:

1=

In quadrant 2:

1=-

In quadrant 3:

1=-

In quadrant 4:

1=2-

Today we also learned about identity inverses:

For example......

If you were assigned to find the angle measure algebraically when given the equation sin

= 1 you wouldn't be able to find solve and find the correct angle measure.

Today we learned that by taking the sine or cosine inverse of whatever the equation equals, we can find the 2 separate corresponding angle measures.

For example sin

= 1

In order to determine

you would plug in sin-1(1) and it would give you 90 degrees or /2 radians.

Almost always, with the exception of 1,-1, and 0, there is a corresponding value in the circle where the sin-1 value is the same. One example is when sin = 1/2, there will be two angle measures; one in the first quadrant and one in the second quadrant where both sin values are positive.