Master AP Calculus AB & BC
Part II. AP CALCULUS AB & BC REVIEW
CHAPTER 2. Calculus Prerequisites
It is extremely important to make sure you are very proficient in trigonometry, as a great deal of calculus uses trigonometric functions and identites. We have whittled the content down to only the most important topics. Familiarize yourself with all these basics. Learn them. Love them. Take them out to dinner, and pick up the tab without grimacing.
The study of triangles and angles is fundamental to trigonometry. When angles are drawn in a coordinate plane, their vertices sit atop the origin, and the angles begin on the positive x-axis. A positive angle proceeds counterclockwise from the axis, whereas a negative angle winds clockwise. Both angles in the below figure measure π/3 radians (60°), although they terminate in different quadrants because of their signs.
On the other hand, some angles, although unequal, look the same. In the figure below, the angles A = 5π/4 and B = 13π/4 terminate in the exact same spot, but B has traveled an extra time around the origin, completing one full rotation before coming to rest. Angles such as these are called coterminal angles (since they terminate at the same ray).
TIP. In order to convert from radians to degrees, multiply the angle by 180/π. To convert from degrees to radians, multiply the angle by π/180.
TIP. Radians are used almost exclusively on the AP test (in lieu of degrees).
Another major topic from yonder days of precalculus is the unit circle—the dreaded circle with center at the origin and radius one—that defines the values of many common sines and cosines. It is essential to memorize the unit circle in its entirety.
TOP. One real-life example of coterminal angles in action is a revolving door. You may enter such a door and proceed right into the building or continue to loop around and around before entering. The number of times you circled in the revolving door does not affect your final destination, although it may alarm the building’s security staff (who are typically much larger and stronger than you).
The points next to each of the angles are the cosine and sine of that angle, respectively. The remaining trigonometric functions are directly based on sine and cosine:
Example 11: Evaluate all six trigonometric functions if θ = 11π/6.
Solution: From the unit circle, and From there, we use the definitions of the other functions:
(both are correct)
as cotangent is the reciprocal of tangent
as secant is the reciprocal of cosine
as cosecant is the reciprocal of sine
The reason that coterminal angles are so useful in trigonometry is that all trigonometric functions are periodic functions, because after some period of time, the graphs will repeat themselves. That interval of time is called (surprise!) the period. Sine, cosine, secant, and cosecant all have a period of 2π, whereas tangent and cotangent have a period of π. Thus, since there is an interval of between the inputs, and tangent has begun to repeat itself.
ALERT! Same students record unit circle values in their calculator’s memory instead of memorizing them. Remember that approximately 50 percent of the AP exam is taken without a calculator, and those precious notes will be inaccessible. Make sure to memorize the unit circle!
TIP. You are not required to rationalize fractions on the AP test (like when calculating secant and tangent in Example 11), and it’s generally a good idea not to waste time doing it. However, a nonrationalized fraction may not always be listed among multiple-choice options, whereas a rationalized answer could be. Make sure you can express your answer either way.
Perhaps the most important aspects of trigonometry you will use are the trigonometric identities. These are used to rewrite expressions and equations that are unsolvable in their current form. In other cases, through the substitution of an identity, an expression becomes much simpler. You’ll need to memorize these formulas, too. Below are listed the most important trigonometric identites:
cos2x + sin2x = 1 (Mamma Theorem)
1 + tan2x = sec2x (Pappa Theorem)
1 + cot2x = csc2x (Baby Theorem)
Even and odd identities:
sin(—x) = —sin x
csc(—x) = —esc x
tan(—x) = —tan x
cot(—x) = —cot x
cos(—x) = cos x
sec(—x) = sec x
Sum and difference formulas:
TIP. The nicknames Mamma, Pappa, and Baby are not conventional but are good for quick reference. Don’t use these names on the AP Test, or no one will have any idea what you are talking about!
Example 12: Simplify the expression
Solution: The first order of business is common denominators, so we can add the terms—multiply the tan x term by to do so. You then use a result of the Big Pappa Theorem to simplify.
NOTE. Only cosine and secant are even functions; the other trigonometric functions are odd.
Example 13: Rewrite the expression in terms of sine.
Solution: If you multiply the second power-reducing formula by 2, the result is 1 — cos 2x = 2sin2 x. Therefore, This is still true, as the cosine angle is still twice as large as the sine angle. Therefore, we can substitute for 1 — cos x:
NOTE. The cosine has three possible substitutions for a double angle. Choosing the correct one depends on the circumstance.
Another major important trigonometric topic is inverse functions. Inverse trigonometric functions can be written one of two ways. For example, the inverse of cosine can be written cos-1x or arccos x. Because the first format looks like (cos x)-1 (which equals sec x), the latter format (with the arc- prefix) is preferred by many. Both mean the same thing. The trickiest part of inverse trig functions is knowing what answer to give.
If asked to evaluate arccos 0, you might answer that cosine is equal to 0 when x = π/2 or 3π/2. It is true that However, this means that the function y = arccos x has two outputs when x = 0, so arccos x is not a function! This is remedied by restricting the ranges of the inverse trig functions, as shown below.
The Restricted Ranges (The Bubbles)
NOTE. Because the Pappa Theorem says 1 + tan2x = sec2x, subtract tan2x from both sidesto get 1 = sec2x - tan2x. This substitution is made in the second step.
Any answer for arcsin x, arctan x, or arccsc x will have to fall within the interval [-π/2,π/2], whereas outputs for arccos x, arccot x, and arcsec x must fall in the interval [0,π]. For our previous example of arccos 0, the correct answer is π/2, as 3π/2 does not fall within the correct interval (or “bubble”—see Tip at left).
TIP. It helps to read “arc” as “where is...”. For example, the function arcsin 1 is asking “Where is sine equal to 1?”. The answer is π/2.
Example 14: Evaluate the following expressions without the use of a calculator:
(a) arcsin √3/2
(b) arcsec —2
(c) arctan —√3/3
TIP. We will refer to the restricted trig ranges as “bubbles” for convenience and, well yes, fun. For example, arctan 1 ≠ 5π/4 because 5π/4 is not in the arctan bubble (i.e., the angle does not fall in the intervals given by The Restricted Ranges diagram).
Solution: (a) The sine function has the value √3/2 when x = π/3 and 5π/3; only π/3 falls in the arcsin bubble.
(b) If an angle has a secant value of —2, it must have a cosine value of —1/2, as the functions are reciprocals. Cosine takes this value when x = 2π/3 and 4π/3. Only the first falls in the arccos bubble, so the answer is 2π/3.
(c) Notice that —√3/3 is the same as 1/—√3 (the latter is just not rationalized). This is the same as . Tangent is negative in the second and fourth quadrants, but only the fourth quadrant is in the arctan bubble. Thus, only angle 11π/6 has the appropriate values (recall that tangent is equal to sine divided by cosine). However, to graph 11π/6, you have to pass outside the arcsin bubble, since 11π/6 > π/2, the largest value allowed for arctan, and passing outside the bubble is not allowed. Therefore, the answer is the coterminal angle —π/6, which ends in the same spot.
TIP. When giving solutions to inverse trigonometric functions, always remember the bubble. The bubble is a happy place. Outside the bubble, the people are not nice, and the dogs bite. Also, it’s hard to find stylish shoes that are still comfortable outside the bubble.
Finally, as promised earlier in the chapter, here are the final six functions you need to know by heart. They are—big shocker—the trigonometric functions:
The Trigonometric Functions
Directions: Solve each of the following problems. Decide which is the best of the choices given and indicate your responses in the book.
DO NOT USE A GRAPHING CALCULATOR FOR THESE PROBLEMS.
1. If sin β = —2/5 and π < β < 3π/2, find the values of the other five trigonometric functions for β.
2. Use coterminal angles to evaluate each of the following:
(a) sin (11 π)
(b) cos 21π/4
(c) tan 13π/3
3. Evaluate cos(arcsin(—2x)).
5. Verify that
6. Solve this equation algebraically: and give answers on the interval [0,2π).
ANSWERS AND EXPLANATIONS
1. The given interval for (B makes it clear that the angle is in the third quadrant (sine is also negative in the fourth quadrant, so this information is necessary). This allows you to draw a reference triangle for [3, knowing that sine is the ratio of the opposite side to the hypotenuse in a right triangle. Be careful to make the legs of the triangle negative, as both x and y are negative in the third quadrant. Using the Pythagorean Theorem, the remaining (adjacent) side measures √21. From the diagram, you can easily find the other 5 trigonometric ratios.
3. This problem is similar to Number 1, but the additional restriction we had there is replaced by the bubble. You know that sine is positive, but since this is the arcsin funcion, only the fourth quadrant is in the bubble. Thus, your reference triangle is drawn as if in the fourth quadrant.
From the diagram,
4. (a) The quadratic expression can be factored to get (tan2x + 1)(tan2x + 1), or (tan2x + 1)2. By the Pappa Theorem, this equals (sec2x)2 = sec4x.
(b) Substituting the values of sec x and esc x, you get which equals 1 by the Mamma Theorem.
6. Note that this question specifies the interval [0,2π). Thus, we do not ignore answers outside of the arcsin bubbles in the final step but give all the answers on the unit circle. Note also that we made the problem easier by substituting 1 for cos2x + sin2x in the second step:
7. The coefficient of the x affects the graph by stretching or shrinking the period. The number—here 2—explains how many full graphs of sine will fit where one used to. Since the period of sine is 2π, now two full graphs will occupy the period instead of one. Had the coefficient been 3, three graphs would squeeze into the same space. The 3/2 gives the amplitude of the sine wave. The rest of the translations work the same as they did earlier in the chapter.