1. AP Calculus AB Summer Review 2012
Trigonometry
AP Calculus AB
Summer Review 2012

2. Trigonometry and Calculus
Calculus requires a thorough knowledge of the basic functions, which includes the transcendental functions
Thus, you are expected to understand polynomial, exponential, logarithmic, and trigonometric functions
Trigonometry occurs very often in calculus, and without knowing the algebra of trig functions several calculus concepts will be very difficult
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3. Algebra of Trigonometry
The most basic trigonometry is understanding the algebra trig functions
This means the methods of rearranging trig functions (as a whole) in addition/subtraction, multiplication/division, and exponents to solve equations
This also includes the use of “inverse trig functions” to pull information from a trigonometric equation
Several trig equations require a calculator to solve angles in either degrees or radians
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4. Degrees
Degree measurement is a standard measurement for angles in some applied sciences, such as mechanics or optics
It is based on the idea that a circle turns across 360° for one revolution
Degrees are a completely made up unit; they have no physical basis, but are still a useful reference unit
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5. Radians
The radian is based on the proportion of a circle length to its radius
Every circle has this relationship, and it helps to define a circle
This is not the definition of a circle!
By definition, there 2π radians in one revolution of any circle
You can also find the length of a sector, area of a sector, and area of a circle (as well as several other quantities) using angles in radians
Most math (and theoretical science) is done in radians
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6. Conversion of degrees to radians
Since we have the following formulae for 1 revolution:
Degrees:
Radians:
We can deduce that the following is true:
Notice that radians have no units when written down, but if you wish you may write “rad” as a unit (Ex: 2π rad)
Thus, if we ever need to convert, we can use either of the following factors
Note: you may have learned a less significant, but still correct, relationship of 180°=π; this proportion works, but it loses the definition of a revolution
It is better to think of angles and radians in terms of revolutions, not just a meaningless simplified proportion
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7. Exact Values
There are few things that need to be memorized in math, and exact values of some angles are a necessity
You should become familiar with the following degree measurements in terms of radians: 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°
Although these degree measurements appear to have no significant connections, they are very closely connected in radians
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8. Connections of the 30n° angles
In radian measurement, the 30n°angles take on reduced values of
Notice that 30° is one-sixth of a semi-circle (pi radians)
By this logic, all 60n° angles take on reduced values of
This makes 60° one-third of a semi-circle
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9. Connections of the 45n° angles
In radian measurement, the 45n°angles take on reduced values of
45° is one-fourth of a semicircle
By this logic, all 90n° angles take on reduced values of
90° is half of a semi-circle
By extension, all angles 180n° take on reduced values of
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10. Summary of Exact Values
Degree 30 45 60 90
120
135
150
180
Radian
π/6
π/4
π/3
π/2
2π/3
3π/4
5π/6 π
Degree
210
225
240
270
300
315
330
360
Radian
7π/6
5π/4
4π/3
3π/2
5π/3
7π/4
11π/6 2π
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11. Graphical representation of Exact Values
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Thirds
Fourths

12. What Should We Use: Degrees or Radians?
The only exception is if the problem explicitly uses degrees in its statement, or asks for an answer in degrees
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13. Triangles in the Coordinate Plane
We are given a right triangle of hypotenuse r superimposed onto an x,y-plane; there is an angle θ between the hypotenuse and the x side
When you draw this triangle, draw an angle arrow on the x-axis, draw a ray starting at the origin in the direction of that angle with length r, and drop a perpendicular to the x-axis
Note: r is always positive, but x and y can be positive or negative, depending on the quadrant
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14. Ex: Draw a triangle in the x,y-plane with hypotenuse 2 at an angle at 50°.
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15. Ex: Draw a triangle in the x,y-plane with hypotenuse 3 at an angle at 290°.
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16. Trigonometric Functions
For a triangle in the x,y-coordinate plane, the three trigonometric functions are defined as:
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17. The Reciprocal Functions
Each trig function has an associated reciprocal function:
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18. Right Triangle Geometry
We may use any (x,y) coordinate to solve for the hypotenuse by the Pythagorean Theorem
Ex: Given point (-10,24), find all six trig functions.
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-10

19. Quadrants and Signs
To analyze what sign to expect, just consider the following:
Cosine is related to the x-coordinate; thus, any triangle to the right (of the y-axis) has a positive cosine and anything to the left is negative
Cuz positive is to the right, and negative is to the left for x
Sine is related to the y-coordinate; thus, any triangle pointing up (above the x-axis) has a positive sine and anything down is a negative
Cuz up is positive and down is negative for y
Tangent is a fraction of the y over x; fractions are positive if top and bottom are both positive or negative
So triangles in the (+,+) and (-,-) regions have a positive tangent; this is Quadrants I and III (respectively)
Since the others are mixed, that is QII(-,+) and QIV(+,-), then tan is negative in those quadrants
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20. Summary of quadrant signsII
III IV
Sin + -
Cos
Tan
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21. Domain and Range Function Domain Range Sin Cos Tan Csc Sec Cot
The trig functions all have an infinite domain (except some undefined points); their ranges significantly differ, but you can likely see the relationships in this table:
Function
Domain
Range
Sin
Cos
Tan
Csc
Sec
Cot
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22. There are two common special right triangles: Special Right Triangles
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23. Ex: Use the special right triangles to evaluate each trig function
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24. Wait, so what’s the point of exact values if we just need to know trig functions?
Exact values are great to work with since each class of exact values gives the same answer for a trig function
The only difference is the sign!
Notice that the angles of the exact values apply to special right triangles
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25. Ex: Evaluate sin(30°), sin(150°), sin(210°), sin(330°).
These seem to be a random collection of angles in degrees
If we switch to radians we see they fall into the pattern of
We already know sin π/6 = ½; so each of these will have the value ½, and the sign depends on the quadrant
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26. Ex: Evaluate tan(45°), tan(135°), tan(225°), and tan(315°)
Switching to radians, we see there all are fractions of fourths, and thus must have the same value
We know tan π/4=1 , so all the values are 1 and the signs are dependent on the quadrant
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27. Negative angles
A negative angle means the direction of sweep from 0° is clockwise
These follow the same rules as exact values, in that the values of your function is the same but the sign has to do with the quadrant rules
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28. Ex: Evaluate sin(120°), sin(-120°), cos(120°), and cos(-120°).
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29. “Triangles” of only one line
A triangle by definition has three lines and three angles, with angles that add up to 180°
The situation arises of what happens if two sides have the same length
Then it’s not technically a triangle since you would have two angles of 90°
This is a purely theoretical case, but let’s just look at it in terms of x and y
If you are given a point that does not make a proper triangle, then look at the coordinates to help you evaluate the trig functions
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30. Ex: 90° angle Suppose we want to find the sin 90°, cos90°, and tan90°
If we draw a 90° angle and an arbitrary hypotenuse r in that direction, we get a line pointing up (not a triangle)
We still have enough information to evaluate, since we know a few things:
1. In this case, there is no x-coordinate
2. In this case, if did have a “triangle”, x=0 and the “hypotenuse” r is the same as our y side
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31. Ex: 90° angle Now we can plug in this information:
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32. Ex: sin, cos, and tan of 180°
This is the same problem, but instead our angle gives the following information:
1. There is no y-coordinate, so
2. The “hypotenuse” r has the same length as x, but x is clearly a negative number since the ray points backwards
Now we can formulate:
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33. Do not memorize these! Just remember how to get them if you need them!
Trig Identities
Since trig functions are a result of triangles, it is worth relating them to the Pythagorean Theorem; we write in terms of x, y, and r
If we rewrite this by dividing by x, y, or r, we get our three trig identities:
Do not memorize these! Just remember how to get them if you need them!
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34. Proofs Involving Trig Functions
Since trig functions are related to one another by x, y, and r, it is an instructive exercise to rearrange a given expression of trig functions into another equivalent expression
To do this, use your basic definitions and your exact values
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35. Ex: Prove the following statement by rearranging the left side only.
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36. Trigonometric Equations and Inverse Trigonometry
A trigonometric equation is an equation containing one or more trig functions
Simple trig equations can be solved analytically
Equations containing several different trig functions are often only possible to solve with a computer
Once the equation is in the form that the trig function has a value, you may use the “inverse trig function” to get the answer
The inverse trig function simply means the operation (not function) used to convert your output value to your input angle
Algebraically, it is more correct to use an inverse trig function to solve an equation; logically, you may just infer your answer from the information
These are sometimes referred to as “arc” (like arcsine)
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37. Definition of an Inverse Trig Function
By definition, an inverse trig function is defined at the points its parent function has a value at; its value is the angle input of the parent function
Thus, the function and inverse switch their domain (x) and range (y) coordinates
Because inverse trig functions “undo” the function, we get the property:
This works both ways!
Ex: Cosine vs Arccosine
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38. Domain of Inverse Trig Functions
Since these functions must have well defined values, it its essential to restrict the domains
Why? Because otherwise we wouldn’t know which value to choose!
Ex:
Which value do we pick? Is the answer 0, 2π, 4π, or something else
Actual answer is
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39. Domain and Range of Inverse Trig Functions
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40. Simple Example of an Inverse Trig Function
For what value of θ does sin θ = 1?
We know that sin π/2 = 1, so logically θ= π/2
Algebraically, the process goes:
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41. Is it Necessary to Use Inverse Trig Functions
No, not if you can logically deduce the answer yourself
Because of this tedious process, and the fact that inverse functions are difficult to conceptualize, most people prefer to use the logical approach
You do need to use inverse trig functions if you are solving for angles not included in the exact values set
In that case, you would need to use your calculator’s inverse trig button
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42. Not So Simple Inverse Trig Problem
This is tricky! Arcsine is defined only in Quadrants I and IV.
Since the arcsine is negative, we assume it’s a fourth quadrant angle.
The question asks: what is the cosine of the angle defined by the inverse sine of (-2/5).
So the answer is just the cosine of this triangle!
We need to solve for x, but other than that it’s just writing the cosine.
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43. Ex: 11 sin x = 2 Let’s get this into a useable form:
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So our answer is radians, or 10.5°

44. Ex:
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45. Ex: First step is to isolate the sine:
This doesn’t make sense. No matter what, sine is between -1 and 1. Since 5/4 is out of our range for sine, this doesn’t have a solution.
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46. Ex:
Anytime we mix trig functions, the equation is more difficult. Always check which functions they are, because some are related by the trig identities. In this case, we can see that tan and sec are related (1+tan2x=sec2x). You should replace the squared term and see where that takes you.
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47. Example Continued
This is a difference of two squares now, so separate into factors. You’ll see it simplifies quickly:
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48. Ex: As stated this is difficult to see
Let’s make a simple variable substitution
Our new equation is easy; just factor and solve
This solves “a” but we need to solve for our original variable θ
Let’s replace with our substituted variable
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49. Ex: Consider each trig function to be its own variable!
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50. Solve the triangle
This means to solve all parts, typically given only two parts
In this class, we will focus on right triangles
No Law of Sines or Cosines!
Given two sides
Solve third side using Pythagorean Theorem
Solve angles using inverse functions
Given a side and angle
Given right triangles, the last angle is obvious
Use trig functions to solve another side
Solve for last side with:
Pythagorean Theorem (long way)
Any other trig function (much faster)
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51. Solve the triangle The third angle completes the triangle:
180 ° – 90 ° – 53 ° = 37°
To solve the sides, choose a trig function that includes your given side
- here, you are given a y-coordinate
We can use Pythagorean Theorem for the last side, or another trig function
- let’s use cosine
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52. Solve the triangle
54.2°
54.6
31.9
35.8°
44.3
We can use the Pythagorean Theorem for the last side:
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For the angles, choose any trig function and take its inverse
The last angle is simple: 180° - 90° ° = 54.2 °

53. Trigonometry Assignment
The quiz opens July 29 and closes August 5
The next will also open July 29, but is not due until August 12
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54. End
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