Last WebQuiz Review and General Course Summary
Some key formulas/pictures to remember:
• Notation for inverse trigs: trig−1 is also called arctrig or atrig.
• Domain and range of inverse trigs: See class notes from 6.6A and the accompanying handout online with
graphs.
– For practice, draw the graphs of sin, cos, tan with the parts shaded that are inverted! Why does this
explain the quadrants the arctrigs use?
– When a > 0 and θ is acute, trig(θ) = a ⇔ θ = trig−1 (a). This might not be the case for negative a
values...
• Law of Sines: Any two opposing pairs in a triangle have the same sine(angle) / opposite ratio. Here’s all
three ratios:
sin(β)
sin(γ)
sin(α)
=
=
a
b
c
This is used when you know an opposing pair and want to complete another, such as in ASA or AAS
scenarios.
– The SSA scenario: If you get sin(α) = a, make sure α is even defined! When αR = arcsin(a), does
α = αR (acute) or α = 180◦ − αR (obtuse) make more sense?
– Remember that smaller angles face smaller sides.
• Law of Cosines: There are three versions of this which all feature an opposing side squared on the left
and a cosine of an angle on the right. For instance, one version is
c2 = a2 + b2 − 2ab cos(γ)
This lets you get c from an SAS scenario (with a, b, γ known) or get γ from SSS.
• Bearings: Bearings denote rotation from a vertical axis, such as N10◦ E or S20◦ W.
– Bearings and reference angles are complements: e.g. N10◦ E has an 80◦ reference angle.
Summaries to keep in mind:
• All inverse trigs use Quad I for their positive values.
– If a > 0, trig−1 (a) is a reference angle.
– For the negatives, only cos−1 uses II; the others use negative numbers for IV. (Why?)
• For trig(trig−1 a) cancels, is a in domain of trig−1 ? For trig−1 (trigθ), is θ is range of trig−1 ?
– For the latter, if θ isn’t in range, get the sign of trig(θ) and also θR .
• Label inverse trigs with angle names, like θ = trig−1 (x). How do you then get a quadrant and a reference
triangle for θ?
• LOS helps when you know more angles, and LOC helps when you know more sides.
– Also, don’t forget the angles add to 180◦ in a triangle.
– Watch for complementary angles (adding to 90◦ ) or supplementary angles (adding to 180◦ ), and
occasionally alternate interior angles play a role.
• You may need to draw multiple triangles and find sides or angles they share. For instance, drawing a
height can produce a right triangle up against an oblique triangle.
• Work-saving tips:
– LABEL YOUR STEPS ON THE DIAGRAM! It will help you stay organized and save time.
– Leaving unrounded answers in the calculator can be better than recopying unsimplfied answers.
Selected WebQuiz 11 Problems with Hints
#4: “Find the exact values of (a) tan(arctan(−20/7)) and (b) arctan(tan(73π/40))”.
With (a), what’s the domain of arctan? With (b), 73π/40 is not in the range of arctan; it’s between 3π/2
and 2π (i.e. between 60π/40 and 80π/40). Normally, Quadrant IV is fine for arctan, but which range does
arctan ACTUALLY use for Quadrant IV?
#5: “Use inverse trig functions to find the solutions to sin(x)(10 sin(x) + 1) = 6 in the interval [0, 2π).”
This factoring DOES NOT give us roots, because the right side is not 0. Instead, expand this to a quadratic
in sin(x). You should get two values for sin(x) of opposite sign. Let’s say a is the positive solution and b is the
negative solution.
To solve sin(x) = a, the reference angle is xR = arcsin(a). Which quadrants have sin > 0?
To solve sin(x) = b, the reference angle needs a POSITIVE value in the arctrig, so xR = arcsin(−b). (Note
that b < 0, so −b > 0.) Which quadrants have sin < 0?
For this problem, it helps to keep a and b’s values in the calculator rather than having to write them out
unsimplified each time.
#7 variant 1: “A regular pentagon is inscribed in a circle of radius 9. Find the PERIMETER of the
pentagon.”
We’ve done this kind of problem before where we found AREA. The key was that we broke the pentagon
into 5 triangles which each used the SAS area formula. This time, for perimeter, you want to find the side
length of the pentagon (the one triangle side that isn’t a circle radius) and multiply it by 5.
What law do we use for finding the third side of a triangle in an SAS situation?
#7 variant 2: “The box in the figure has dimensions 9in x 5in x 4in (so a = 9, b = 5, c = 4). Approximate the DEGREE measure of the angle θ between a
diagonal in the base and a diagonal of the 5in x 4in
face.”
First and foremost, θ is an angle in a triangle; you really want to draw all three sides of it. You’ll notice
that each √
side is a diagonal
of some√rectangular face. Thus, each of their lengths can be found by Pythagoras!
√
They are a2 + b2 , a2 + c2 , and b2 + c2 (though you need to think about which is which!).
This means you know SSS for this triangle, which means we should use LOC. Write the LOC version that
features θ’s opposite pair. From there, get cos(θ) on its own, and lastly use an inverse cosine. Make sure your
answer is in degrees... the WA default is radians, so you may have to multiply by 180/π!
#8: “A road makes a 22◦ angle with the horizontal. From a point P on the road, the elevation angle at
the point A is 57◦ . At the same instant, from another
point Q located 140 meters farther up the road, the
angle of elevation is 63◦ . Find the distance from P to
the airplane A.”
We’ve seen a somewhat similar problem in the last class featuring a slanted road. The main trick there was
that we needed to compute the difference between elevation angles to actually get the angles in our triangle.
Thus, if you look at ∆AP Q, the angle at P is not 57◦ ; it’s 57◦ − 22◦ , since it measures the angle between the
plane and the road. Similarly, the angle on the OUTSIDE of Q is 63◦ − 22◦ = 41◦ .
However, how do we get the angle that’s INSIDE the triangle at Q? Note that it and the 41◦ angle form
a straight line together, so they are supplements! This means the inside angle is 180◦ − 41◦ = 139◦ . You now
have an ASA scenario which uses LOS.
#9: “From a point A, a forest ranger sights a fire
in the direction S29◦ W. From a point B, 5 miles due
west of A, another ranger sights the same fire in the
direction S46◦ E. Find the distance d of the fire from
the point A.”
Draw the two bearings from A and from B. Notice that the bearings occur on the OUTSIDE of the triangle
∆ABC (if C is the location of the fire). Look at the picture; how do you get the angles on the INSIDE of the
triangle from those bearings? After you do that, you have an ASA scenario again.
#10: “A jogger runs at a constant speed of one mile every 8 minutes in the direction S42◦ E for 20 minutes
and then in the direction N19◦ E for the next 24 minutes. Find the straight-line distance from the endpoint to
the starting point of the jogger’s course.”
First and foremost, let’s note that there are three points on this course! Say the jogger starts at A, turns
around at B, and ends at C. The question gives bearings from A to B and from B to C, along with speed and
time. From speed and time, find the lengths of the sides corresponding to the two jogs, which are c = AB and
a = BC. We want to know the length b = AC.
If you draw the bearings on A and B, you’ll see A’s bearing is OUTSIDE the triangle ∆ABC. We can’t do
the same trick as #9 to fix this, since we can’t assume that the side from A to C is horizontal. Instead, try
drawing the bearing from B to A along with the bearing from A to B; you should see alternate interior angles!
How does that give you the angle at B, and hence give you an SAS triangle?
See the last page for some final course remarks.
Some Last Remarks about the Course
Along the way, you saw a lot of mathematics very quickly in this course. You saw several types of functions,
such as linear, quadratic, exponential, or trigonometric (along with info about the circumstances in which they
occur). You saw several types of shapes, such as circles, polygons, and waves. You saw applications of these
formulas to a variety of situations.
Most importantly, though, this course gives you skills and problem-solving strategies. It’s one thing
to be able to follow along with someone else’s solution, but it’s another thing entirely to be able to figure out
that solution process on your own by noticing key words, setting up variable names, and intelligently applying
the results we developed. Recognizing how to take the tools we have, figuring out which ones are relevant, and
setting up a plan of attack is something that comes from practice, but you also need to learn these results with
appropriate context in mind.
Skills used in writing equations or expressions:
• Be precise about terminology. What’s the difference between an expression and an equation? What’s a
term, and what’s a factor? What’s the coefficient of a term? What does “constant” mean? What’s the
difference between a power and an exponential?
– You need to know these distinctions in order to know which laws make sense in which situations!
• Identify which variable(s) is/are unknown and which are known. Look for phrases like “in terms of” or
“as a function of” to tell you which variables should be in your answer expression.
– It may help to put question marks or check marks over variables to remind you which ones you know
and which you want.
• Substitute for known quantities in appropriate positions. Sometimes, you have to match up similar parts
of two equations.
• Recognize the form of an unknown. Is the equation linear in x? Quadratic in sin x? Exponential in x?
• Decompose an expression into layers using order of operations. Which layer is the outermost one of the
problem? Which is the innermost?
Skills for solving equations:
• Do not mix up the insides and outsides of functions. For instance, 2 sin(x) and sin(2x) are very different.
You can’t move something from the inside to the outside, or vice versa, without some identity.
• When your unknown is in only one place, cancel the outermost layer on each step. This is especially
important with inverses!
• When your unknown is in multiple places, we often bring the variables together on the same side of the
equation. Do you have identities which can combine your parts? (e.g. quadratic formula, log laws, trig
identities)
• More specialized tactics:
– With zero on one side: Can you factor? Remember to set each factor to zero to get separate, smaller
problems.
– With fractions in an equation: Can you multiply a common denominator to both sides to get rid of
fractions?
– Similarly, if you have fractions inside other fractions, can you rewrite the fraction more simply?
(For instance, you can multiply to both the top and bottom, or you can multiply by reciprocal of
denominator...)
• With complex equations, handle one layer at a time. (That’s what substitutions with “u” or “θ” do...
why would somebody do this variable change?)
Word-problem strategies:
• Write the entire equation first! Then replace the values you know, and solve for the ones you need. (Do
NOT assume you’re always solving for the left side of an equation.)
• To eliminate a variable from an answer, use some other equation to solve for it!
• You probably should label more on a diagram than you think. You can always make a separate lesscluttered diagram if you need to.
• If needed, break up shapes into more familiar parts.
• Sometimes it’s easier to remove parts you don’t want instead of adding parts you do want.
• Consider units: do your calculations work with the right types of values? (e.g. radians versus degrees,
area versus volume, miles per hour instead of meters per second, etc.)
Good test-taking strategies:
• Skim over the question when done: did you answer what the question wants? Is your answer plausible?
• Only simplify if it helps you, not for the answer’s sake (unless the question says otherwise). A good
unsimplified answer has a form where you can read it and figure out what the solution did!
• Outline steps of long problems, especially on a diagram. Find ways to reuse old parts without rewriting
everything!
Study strategies:
• When you study formulas, also study why and how you use them.
• First skim over examples and problems, asking yourself “What tools do I need to use to do this?” and
“Do other approaches work? Why/why not?”.
– Only do a few problems in detail from any section. You do not want to overload yourself or get
yourself to only recognize fixed problem types... you want to understand, not just memorize.
– You cannot memorize every problem type on the final ; we’re looking to see if you know how to
recognize valid concepts. In fact, consider your Test 2-3 “midterm” to be a warning of what
can happen if you memorize too broadly rather than understanding the general idea!
• Ask yourself questions. Failing that, get someone else to ask you questions and vice versa.
– Consider variants of problems! e.g. What if someone asked for area instead of perimeter? What
if the equation had cosine instead of sine? If you can figure out how to make changes to problems,
then you understand better which steps matter for which parts of the problem.
• If you made mistakes, try to figure out why you made the mistakes and what you could learn from it.
Some Random Review Exercises
These problems do not review everything we did. They should remind you, though, of a lot of major ideas.
Try to answer all the questions following each exercise.
From Unit 1: “A rectangular box has a square base with side length x and a total volume of 100 cubic
meters. If the sides of the box cost $2 per square meter, and the box has no top lid, find the total cost of the
material as a function of x.”
Ask yourself: Which formulas are you going to need to do this? How many variables will you create? Which
variables will be eliminated from the final answer? If different parts of the box had different costs, how would
this change the work?
From Unit 2: “A ball is tossed from a height of 5 feet above the ground, and it hits its highest point of 20
feet after 3 seconds. Find when the ball lands.”
Ask yourself: Why do I use a quadratic function here? What form of a quadratic is appropriate here? Where
do I put in the given information? How do I know which solution for time is correct? If they instead gave me
the equation and asked for the highest point, how could I find that?
From Unit 3: “If an account earning continuously compounded interest doubles its value after 10 years,
determine how long the account takes to triple its value.”
Ask yourself: What’s the formula for continuous interest (as opposed to a fixed number of compoundings)?
How do I work with an unspecified initial value? Why won’t that initial value matter in the calculations? What
point do I plug in first, and what do I find as a result? How does that help me with the next point?
From Unit 4: “If sin(t) = 5/6 and π/2 < t < π, find the values of cos(t) and cos(t + π).”
Ask yourself: What quadrant does t belong to? How do I draw a reference triangle in that quadrant? How
does SOHCAHTOA tell me cos(t)? How do I know whether my answer is positive or negative? Where does
the angle t + π lie in relation to t? What does that mean for the cosine? (Remember coordinates on the unit
circle!)
From Unit 5: “A wave has height y = 10 cos(π/5t) feet after t minutes; note that its period is 10 minutes.
At which times in the first half-hour does the wave reach a height of 5 feet?”
Ask yourself: Which equation am I solving? What interval do I have for my solutions? How does the
reference angle help me get ALL solutions for θ? (And what is θ?) How do I get solutions for t? How many
solutions should I be expecting? (Think about how many periods elapse in a half hour...)
Good luck! Try the Practice Final on WebAssign!