Trigonometry Review for Calculus
Day 2: Graphs, Inverses, and Identities
Starting point: Functions
Important properties:
1. Functions as input-output machines or transformations: Functions take an input and
produce a single output for that input. The set of all inputs is called the domain and the
set of all outputs is called the range.
2. Functions often capture how two quantities compare in size or co-vary. For example,
as we analyze a function using a graph or table, we might note that one quantity is always
six times as large as the other quantity, or that as one quantity increases, the other
quantity decreases.
Trig Functions:
As input-output machines: put in an angle measure, get out a ratio of sides on a triangle. This
is what we did last time when we constructed triangles and calculated ratios.
As comparisons of size: the comparison of the size of the angle and the ratio of sides is too
complex to fit any of the common comparisons we are familiar with, so we often ignore
this aspect of the function.
As covariation: the covariation between angle measure and ratio of sides is very predictable
and interesting, so this is studied in detail.
Covariation between angle measure and ratio of sides
Big Idea #1: We can study the covariation between angle measure and ratio of sides by
examining quantities associated with the unit circle as the angle “swept out” by the radius
changes.
Big Idea #2: We can create a graph of the covariation in the xy-plane by first following the
change of the angle on the x-axis; second, following the change of the ratio on the y-axis; and
lastly, coordinating the two changes in the xy-plane.
Sine Function
-2π
-π
Cosine Function
π
2π
-2π
-π
π
2π
Tangent Function
-2π
-π
Secant Function
π
2π
-2π
-π
π
2π
Inverse Trig Functions
Big Idea #3: We can ask the question, given a particular value for a trig function, what was the
angle that produced that value?
Example 1: If sinθ = 1/2, what is θ?
Big Idea #4: We have to limit the range, or the angles we will allow to be the output, of the
inverse operations to get a function.
Function
Symbolic Name
Inverse Sine
sin-1x or arcsinx
Domain
Range
Example 2: Find sin(cos-1(4/5)).
Inverse Cosine
cos-1x or arccosx
Inverse Tangent
tan-1x or arctanx
Must-Know Trig Identities for Calculus
1. cos2 θ + sin 2 θ = 1 or cos2 θ = 1 − sin 2 θ or sin 2 θ = 1 − cos 2 θ
Reason it works:
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2
2
2
2
2
2
2. tan θ +1 = sec θ or tan θ = sec θ −1 (Another just like it: cot θ +1 = csc θ )
Reason it works:
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3. Double angle formulas
cos(2θ ) = cos 2 θ − sin 2 θ = 2cos 2 θ −1 = 1 − 2sin 2 θ
sin(2θ ) = 2sin θ cos θ
Reason it works: Blah, blah, blah, special case of the angle addition formulas, blah, blah,
blah…. Just memorize it.
4. Half angle formulas
⎛ α ⎞ 1+ cos α
cos2 ⎜ ⎟ =
⎝ 2 ⎠
2
⎛ α ⎞ 1 − cos α
sin 2 ⎜ ⎟ =
⎝ 2 ⎠
2
Reason it works:
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Next steps:
1. Draw a picture of the graphs of the six trigonometric functions by reasoning about the unit
circle. Don’t peek at these notes if possible.
2. Memorize the general shapes of the sine, cosine, and tangent functions.
3. Pick a few common values for sine (e.g., 1/2, -√3/2, 1) and then find all of the angles that
give that particular sine value. Next, find the inverse sine of that value (should be only one
angle, right?). Do that for cosine and tangent, too.
4. Memorize the must-know identities. Make more flash cards.
5. Review your trig flash cards every day for a week. Then do it once or twice a week for the
rest of the semester.
6. Send me an email (dsiebert@mathed.byu.edu) if there was something you particularly liked
about this two-day review or if you have suggestions for making it better.
7. Have a great semester in calculus!