Review of Basic Trigonometry

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Review of Basic Trigonometry
A mathematical thinking race!

• Get a pencil and paper to write your answers to
  practice problems as you move through each slide.
• Repeat as needed until you remember your trig
  basics.
• Look at the links on our class website to find
  more trig lessons and practice if needed.
• Trig is also reviewed in the each section of
  chapter one of your textbook. You MUST know your
  trig, to be successful in calculus!

2

• In geometry, you learned that one diagonal of a
  square forms what special triangles?

Answer 45 45 90 or isosceles right
triangles
Answer 30 60 90 triangles
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3. How many families of Pythagorean triples can
you remember from geometry? Dont list any
multiples in the same family!
Answer (3, 4, 5) (5, 12, 13) (7, 24, 25)
(8, 15, 17) (9, 40, 41) . . .thats enough
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6. Find the missing side in each triangle.
(Click to get answer and then click to get next
get triangle.)
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5
7. In geometry, you also learned the right
triangle definitions of sin, cos and tan. What
are those definitions?
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9. In trigonometry, you learned the definitions
of the reciprocals of sin, cos and tan. What are
those definitions?

• 10. Next, instead of degrees, you learned to
  measure angles and arcs in radians. What is the
  definition of radian?

11. Radians are easy to use in a UNIT CIRCLE
because what is the radius of a unit circle?
Answer 1
12. So what is the circumference (or total arc
length) around a UNIT CIRCLE?
Answer 2?
13. Radians arc length/radius, so how many
radians are there in one complete revolution
around the unit circle?
Answer 2? /1 2?
Answer ?
14. A semi-circle or 180o is how many radians?
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15. To convert 57o to radians what would you
multiply or divide by?
17. If ? 180o, then you should recognize common
conversions . Convert 30o to radians.
Answer ? /6
18. Convert 45o to radians.
Answer ? /4
19. Convert 60o to radians.
Answer ? /3
Answer ? /2
20. Convert 90o to radians.
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21. In STANDARD POSITION, we measure angles of
rotation from zero radians going
counter-clockwise. If there are 2? radians in
one entire revolution, then what is the measure
of each of the QUADRANT angles shown with colored
arcs below?
Blue angle in radians?
Answer ? /2
Green angle in radians?
Answer ?
Purple angle in radians?
Pink angle in radians?
Answer 2?
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22. Do you recognize the FOUR angles that would
form 45-45-90 triangles? BUT, now can you give
these four angles in RADIANS? (Coterminal angles
end up at the same places. We could add or
subtract multiples of 2? for more revolutions
that end up in these four places .)
Blue angle in radians?
Answer ? /4
Green angle in radians?
Purple angle in radians?
Pink angle in radians?
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23. Now visualize four TALL 30-60-90 triangles in
STANDARD POSITION. (Next slide well flip them
the SHORT way.) Can you name these four angles
in RADIANS?
Blue angle in radians?
Answer ? /3
Green angle in radians?
Purple angle in radians?
Pink angle in radians?
Answer
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24. Now visualize four SHORT 30-60-90 triangles
in STANDARD POSITION. Can you name these four
angles in RADIANS?
Blue angle in radians?
Answer ? /6
Green angle in radians?
Purple angle in radians?
Pink angle in radians?
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• Angles that measure more than 2? are more than
  one revolution the red angle below is a 2p
  revolution plus 5p/4 more, i.e., 13p/4.
• NEGATIVE angles are measured CLOCKWISE starting
  at zero. every additional 2? radians is one more
  revolution. Give the RADIAN measure of each
  angle below.

Blue angle in radians?
Answer - ?
Green angle in radians?
What is the measure of a counter-clockwise angle
of three complete revolutions that terminates at
the same place as zero radians?
Answer 6?
Add another counter-clockwise revolution to the
red angle?
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(No Transcript)
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29. Right triangle (SOHCAHTOA) definitions are
necessary whenever the hypotenuse is not equal to
one, but when the hyp 1, the UNIT CIRCLE
DEFINTIONS simply become the x y coordinates.
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30. Put it all together! Name the radian measure
(in the box), the (x, y) coordinates - which are
also the cos sin and the tangent at each of
the special angles. You need to recognize each
in a snap!
Click to see answers
( , ) tan
tan ( , )
( , ) tan
tan ( , )
( , ) tan
( , ) tan
tan ( , )
tan ( , )
( , ) tan
tan ( , )
( , ) tan
tan ( , )
( , ) tan
tan ( , )
( , ) tan
( , ) tan
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31. Now practice all six ratios in random order.
Visualize a unit circle, but dont waste time
drawing it. Practice until you can do these
quickly and confidently! Click to check your
answers.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
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32. Find the exact value of cos x in the diagram
below.
NOTE This is not a special ? we have memorized,
so to solve for x you would need a calculator to
approximate cos-1(3/5). (Result is approx. 53.1o)
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• Match each basic trig function with its graph.

Answers B E A D C F

• y cos x
• y sec x
• y sin x
• y csc x
• y tan x
• y cot x

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• SUMMARY (Use next slide to check your
  understanding)
• Trig Function Transformations
• y a sin b (x c) d
• a ? amplitude
• a lt 0 ? vertical reflection
• 2? / b ? period for sin, cos, csc sec
• ?/b ? period for tan cot
• - c ? horizontal translation
• (shifts in opposite direction of the sign )
• d ? vertical translation
• (shifts in same direction of the sign )

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35. What is the difference in the graphs of y
sin x and y - sin x?
Answer Second graph is a vertical reflection of
the first.
36. What is the difference in the graphs of y 3
cos x and y 2 cos x?
Answer amplitude (max min displacement) of
first is 3 its graph passes through the pt (0,3)
. The amplitude of the second is 2 it passes
through the pt (0, 2).
37. What is the difference in the graphs of y
sin x and y sin x - 5?
Answer Second graph is shifted five unit down.
First graph passes through the pt (0,0) and the
second passes through (0, -5) Note this is
very different from y sin (x 5) which is a
horiz shift to the right.
38. What is the difference in the graphs of y
cos 4x and y cos (x/2)?
Answer First has a shortened period of 2?/4
?/2 whereas the second has a lengthened
period of (2?)/(?/2) 4?
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• What is the difference in the graphs of
• y cos (x ?)and y cos (x - ? )?

Answer First graph is shifted ? units to the
left beginning at (- ?, 1) and the second graph
is shifted ? units right beginning at (?, 1), so
when you repeat cycles in both directions, the
two graphs are exactly THE SAME NO DIFFERENCE!

• What is the difference in the graphs of
• y sin x and y cos (x ?/2)?

Answer when you shift the cos graph ?/2 units
to the left, it lands on top of y sin x. The
two graphs are exactly THE SAME NO DIFFERENCE!
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41. What points do y sin x and y csc x share
in common? Why?
Answer (-?/2, -1), (?/2, 1), (3?/2, -1), (5?/2,
1), because the reciprocal of 1 stays at 1 and
the reciprocal of -1 stays at -1.
42. Where are the vertical asymptotes for y csc
x? Why?
Answer x -?, ?, 3?, 5?, because the
reciprocal of 0 is undefined and the reciprocals
of the very small fractional sine values close to
these locations become infinitely large values
that go towards 8 and - 8.
43. What hints might you give someone to graph y
-5 sec (2x ?) 1?

• Answer
• Make a dotted graph of y -5 cos (2x ?) 1
• Keep the max min pts fixed
• make vertical asymptotes through the (transformed
  locations of the) x-intercepts
• Flip over the cos curves into U-type curves.

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44. Where are the vertical asymptotes for y tan
x? Why?
Answer x - ?/2, x ?/2, x 3?/2, x 5?/2,
because tan y/x or sin/cos and this is
undefined whenever the x-coordinate is zero.
45. Where are the vertical asymptotes for y cot
x? Why?
Answer x - ?, x 0, x ?, x 2?, x 3?,
because cot x/y or cos/sin and this is
undefined whenever the y-coordinate is zero.
46. Once you know where the asymptotes are, what
is the other visual difference between the graphs
of y tan x and y cot x?
Answer y tan x goes up to the right and down
to the left (since tangents are pos in quadrant 1
between 0 and ?/2 and neg in quad 4 between -?/2
and 0) y cot x goes up to the left and down to
the right (since cotangents are also pos in
quadrant 1 between 0 and ?/2 and neg in quad 4
between -?/2 and 0)
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47. Again, sketch a graph of each function.
Click to see answers.

• y sin x
• y cos x
• y tan x
• y sin x
• y csc x
• y cot x

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48. Graph y -5 cos (x/2 p/4)
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49. What function is graphed below? (There are
many possible answers you only need to find
one.)
? /4
? /2
Hints
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• Were close to the end! What is another symbolic
  way to write
• y arc sin x and how is your answer read
  aloud?

Answer y sin-1 x which is read y is equal
to inverse sine of x
51. Do all functions have inverses?
Answer No, only functions that are one-to-one
meaning each unique x is paired with a unique y.
(No repeats on x or y, so the graph must pass
both the vertical and the horizontal line tests.)
52. Since y sin x has many repeated y values
(imagine a horizontal line passing through all
through those humps), how can there be an arc sin
or inverse sine function?
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53. Inverse functions are easy to graph if you
recall that f -1(x) is a reflection of f(x)
across what line?
Answer y x or the 45 0 diagonal line
54. This means that the points (-p/2, -1) and
(p/2, 1) on the graph of y sin x, are
reflected to (?, ?) and (?, ?) on y arc sin x?

Answer (-1, - p/2) and (1, p/2)
55. Sketch the graph of y sin -1 x Hint
sketch y sin x first reflect a few key pts
across y x. Click to see the answer (blue
graph).
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56. Visualize y cos x. What restriction on the
domain (close to the origin) will produce a
one-to-one section with no repeats on x or
y? Hint always go from a minimum to a maximum
height or max to min.
Answer 0 x p
57. This means that the points (0, 1) and (p, -1)
on the graph of y cos x, are reflected to (?,
?) and (?, ?) on y arc cos x?
Answer (1, 0) and (-1, p)
58. Sketch the graph of y cos -1 x Click to
see the answer (blue graph).
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59. Visualize y tan x. What happens to a
vertical asymptote, when it is reflected across y
x?
Answer it becomes a horizontal asymptote.
60. What restriction on the domain of y tan x
would produce a one-to-one section?

• Sketch the graph
• of y tan -1 x
• Click to see
• the answer
• (blue graph).

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62. To solve inverse trig functions, think in
reverse order what angle has that value as an
answer. That is, n arc sin ½ simply translates
to n is an angle that has a sin of ½ so we know
n p/6! Why do we know n cannot equal 5p/6?
Answer because the arc sin function is limited
to the one-to-one interval from -p/2 to p/2.
Answer -p/4 (You cannot answer 5p/4 or 7p/4
because the arc sin function is limited to the
one-to-one interval from -p/2 to p/2. )
Answer 5p/6
Always remember arc sin and arc tan functions
are limited to the one-to-one interval from -p/2
to p/2 and arc cos is limited to 0 to p .
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65. Now practice in random order. Visualize a
unit circle, but dont waste time drawing it.
Practice until you can do these quickly and
confidently! Click to check your answers.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
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1. What geometry equation must be true about x y
   in any right triangle like that shown below?

Answer x2 y2 1
(x,y)
67. What trig identity does this equation
become?
hyp 1
y opp
Answer sin2a cos2a 1 which is called the
Pythagorean Identity for obvious reason!
a
adj x
68. What trig identity does this equation
become if you divide through by cos2a?
Answer tan2a 1 sec2a Also called an
Pythagorean Identity
69. What trig identity would it have become if
you divided through by sin2a?
Answer 1 cot2a csc2a Also called an
Pythagorean Identity
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70. If you reflect angle a vertically, what
changes are made in the three trig ratios?
cos a
Answer cos (-a) sin (-a) tan (-a)
(x,y)
- sin a
- tan a
1
Called OPPOSITE ANGLE identities
71. What geometry term applies to the pair of
acute angles a b in a right triangle (like the
pink triangle below)?
a
Answer complementary Sum of 90o but we now
prefer to say b (p/2 a) radians
(x,-y)
72. What COFUNCTION trig identities relate the
ratios for COmplementary angles?
Answer sin a cos(p/2 a) csc a sec(p/2
a) cos a sin(p/2 a) sec a csc(p/2
a) tan a cot(p/2 a) cot a tan(p/2 a)
b?p/2-a
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73. Can you give the Double Angle Formulas?
(Memory works, but as long as you recognize this
and know where to find it quickly, youre
probably OK. This is even more true of others
like half-angle, sum difference, etc. which we
will seldom use.)
74. Can you give the Law of Sines Formula?
75. Can you give the Law of Cosines Formulas?
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77. Find ALL solutions of the equation sec2 t
2 tan t 0
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78. What term is used to represent any function
whose graph has repeated crests and troughs?
Answer sinusoidal
79. Use geometry to explain why all 26o angles
have the same sine value?
Answer All right triangles with a 26o angle are
similar to each other by AA theorem and we know
the ratios of corresponding sides of similar
polygons are equal . Thus, for all 26o angled rt
triangles, the ratio of opp/hyp will be the same!
Trigonometry is simply a study of ratios in
similar triangles.
80. One more time, what are the definitions of
the sin, cos and tan in any size right triangle
and their definitions in the unit circle when the
hypotenuse has a length of 1?
Answer Right triangles ? SOHCAHTOA Unit
circle? cos x sin y tan y/x
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THE END Congratulations!
Think trig is hard? Think again! Check out this
brief history of trig to see that folks have been
doing trig for thousands of years! You can learn
it too!

• Repeat as needed until you remember your trig
  basics.
• Look at the links on our class website to find
  more trig lessons and practice if needed.
• You MUST know your trig, to be successful in
  calculus!