Chapter 4 Review

 Chapter 4

Chapter 4 focuses on trigonometric functions.

4.1 - Radian and Degree Measure

Angle: two rays with a common endpoint

Angles are measured in degrees or radians.

• To convert to degrees, multiply radians by    

• To convert to radians, multiply degrees by     

To obtain the length of an arc length, you can use the equation  , where s = arc length, r = radians, and  = the angle measure in radians.

• When two angles are supplementary, their measures add up to 180 degrees
• When two angles are complementary, their measures add up to 90 degrees
• When angles are congruent, they have the same measure
• When angles are coterminal, they have the same initial and ending sides.

4.2 - The Unit Circle

The unit circle has a radius of one, and can be used to solve trigonometric functions of angles.

The sin value corresponds to the y-coordinate on the unit circle, and the cos value corresponds to the x-coordinate.

For example, the angle  has a sin value of  , and a cos value of 

Cosine and secant functions are even, which means that

 

4.3 - Right Triangle Trigonometry

Using the various trigonometric functions and the rules of SOHCAHTOA, it's possible to find side lengths or angles of right triangles.

• When you are looking for a trigonometric function that is different from the ones you have, the trigonometric identities that relate them can be used.

• Using these identities, you can transform one side of an equation to the other by only manipulating one side.

   

1 = 1

4.4 - Trigonometric Functions of Any Angle

Any angle  on the unit circle has x and y values for each trigonometric function

On the unit circle, reference angles can be found by evaluating the angle between the x - axis and the terminal side of an angle. They're always acute, and differ for each quadrant

For example, for an angle  in Quadrant II, you would find the reference angle with 

4.5 - Graphs of Sine and Cosine Functions

When the graph makes one full cycle, it is called the period.

The equations for the sin and cos graphs are

y = a sin (bx - c) + d

y = a cos (bx - c) + d

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a = the amplitude of the function (vertical stretching). It basically tells you how high and low the graph should reach

b = the value that helps to find the period. For sin and cos, the period is 

c = phase shift (horizontal shifting). When subtracting c, the graphs shifts right. When adding it, it shifts left.

d = vertical shift

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4.6 - Graphs of Other Trigonometric Functions

The graphs of tan, cot, sec, and csc share the same equation as the sin and cos graphs

* In the tan and cot graphs, the period is determined by   

  

  The csc and sec graphs run tangent to the sin and cos graphs, so it may be useful to graph those functions as well.

4.7 - Inverse Trigonometric Functions

When we restrict the domain of  y = sin x to      , we can use the inverse sin function, denoted by 

This can be done with all six functions

THE OUTPUT OF AN INVERSE TRIG FUNCTION SHOULD ALWAYS BE AN ANGLE MEASURE

Inverse functions may also be denoted with y = arcsin x,  y = arccos x, etc.

In a triangle such as this, we can use inverse functions to find 

4.8 - Applications and Models

There are many pointless ways to apply trigonometric functions in real life.

Bearings:

Since bearings are usually given in degrees, it's a natural fit for trigonometry.

Bearings are typically written like this:  N  43° E  

This means 43° north of east

Angles of Elevation/depression:

Many story problems call for an angle to something higher or lower than another object

Using this angle is typically required for such problems, as it can be used to find other values.

For example, the angle of elevation (48°), can be used along with the tangent function to find x.

x = 20 ft.

4.8 - Applications and Models (Story Problems)

Sections 4.8 explains how to use trigonometric functions to model and solve "real-life" problems

Some useful things to know in this section are angle of elevation and angle of depression.

• Angle of Elevation  is the term that denotes the angle from the horizontal, upward to an object.

Picture Source

• Angle of Depression is the term that denotes the angle from the horizontal, downward to an object.

Picture Source

Here is an example problem from the textbook of when to use angle of elevation

• A safety regulation states that the maximum angle of elevation for rescue ladder is 72°. If a fire department's longest ladder is 110 feet, what is the maximum safe rescue height?

So from this picture and the problem we are solving for "a". From the equation sin A = a/c, we can conclude that a = c sinA. So a = 110 sin 72° ≈ 104.6. So, the maximum safe rescue height is about 104.6 feet above the height of the truck.

Trigonometry and Bearings

In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle a path or line of sight makes with a fixed north-worth line.

For instance, the bearing of S 35° E means 35 degrees east of south

When dealing with bearings often time you'll be given a measurement like this: 39° 45' l

In this case the angle measurement would be (39 + 45/60)°

4.7 Inverse Trigonmetric Functions

Section 4.7 explains inverse trigonometric functions as well as compositions of functions.  It is convenient for trigonometric functions to have an inverse because it makes solving for angles of a right triangle much easier, but trig functions are not one-to-one.  By restricting the domain of the trig functions, each function has an inverse.

Inverse Sine

For the y = sin x graph above, the shaded area represents a section of the graph where sin x is one-to-one and the entire range is represented.  The domain [ -π/2, π/2] is where sin x has an inverse function y = arcsin x, or y = sin^-1 x.

The above graph shows y = arcsin x.  Note that the domain and range have switched from the sin x to the arcsin x graphs.  Like all inverse functions, the input- x, and the output- y, switch.  

Inverse Cosine and Inverse Tangent

Inverse Cosine

Similar to y = sin x, y = cos x and y = tan x are not one-to-one unless the domain of these functions are restricted.

For the y = cos x graph the restricted domain is [ 0, π ].  When the x and y values are switched the y =  arccos x graph looks like:

Inverse Tangent

The domain of the y = tan x graph is (-π/2, π/2) to be one-to-one.  Note that unlike the restricted domain of y = sin x with brackets, the restricted domain of y = tan x has parenthesis because y = -π/2 and y = π/2 are vertical asymptotes.

Graph of y = tan x

Graph of y = arctan x

Compositions of Functions

When dealing with an inverse trig function composed of a trig function, or a trig function composed of an inverse trig function, the most important thing to remember is that an inverse trigonometric function equals an angle measurement.  

For example, to solve-

The first step is to recognize that

 

Next, cos x = adjacent/hypotenuse, so 3 is the measure of the adjacent side and 5 is the hypotenuse.  Using the Pythagorean Theorem or special right triangles, the opposite side is found to be 4.  Finally, sin x = opposite/hypotenuse, so 4/5 is the answer.

To sum it up,

Extra

For help with the range of inverse trigonometric functions (the restricted domain of trigonometric functions), here's a table-

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