Transcript Document

Review of
Basic
Trigonometry
• 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!
Carol Hardtke – MUHS
Aug 2009
1.
In geometry, you learned that one diagonal of a square forms
what special triangles?
Answer:
45 – 45 – 90 or
“isosceles right triangles”
2.
In geometry, you learned that an altitude of an equilateral
triangle forms what special triangles?
Answer:
30 – 60 – 90 triangles
Carol Hardtke – MUHS
Aug 2009
3.
How many families of Pythagorean triples can you remember from
geometry? Don’t list any multiples in the same family!
Answer:
(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) . . .that’s enough
4.
In geometry, what variables represented the three sides of a
45-45-90 family?
x 2
5.
Answer:
x
(x, x, x 2 )
x
In geometry, what variables represented the three sides of
a 30-60-90 family?
Answer:
2x x 3 2x
x
x
Carol Hardtke – MUHS
Aug 2009
(x, x 3 , 2x)
6.
Find the missing side in each triangle. (Click to get answer and then click to
get next get triangle.)
a
72
7
c
b
7 3 14
d 14 3
73
30
e 21
60
7
7
1
7
30
45
f7
7 2

2
2
h1
3
3 or
2
2
1
45
k
1
2

2
2
Carol Hardtke – MUHS
Aug 2009
1
g2
j 1  2
2
2
7.
In geometry, you also learned the right triangle definitions of
sin, cos and tan. What are those definitions?
Answer: SOHCAHTOA
adj leg
opp
leg
sin =
cos =
hyp
hyp
8.
tan =
opp leg
adj leg
What equation would you use to find the needed angle of
elevation if you want to install a 24.2 ft escalator to reach a
height of 11.5 ft? Solve your equation.
24.2
xo
11.5
Answer:
11.5
sin x =
24.2
On calculator :
sin
-1
 
11.5
24.2
?
Don't forget this is DEGREE MODE :
sin
Carol Hardtke – MUHS
Aug 2009
-1
 
11.5
24.2
 28.4o
9. In trigonometry, you learned the definitions of the reciprocals of
sin, cos and tan. What are those definitions?
Answer:
hyp
1
csc =
or
sin
opp leg
sec =
hyp
1
or
cos
adj leg
cot =
adj leg
1
or
tan
opp leg
10. Next, instead of degrees, you learned to measure angles
and arcs in radians. What is the definition of radian?
Answer:
arc length
s
radian = =
r
radius
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 Answer: 2
length) around a UNIT CIRCLE?
13. Radians = arc length/radius, so how many Answer: 2 /1 = 2
radians are there in one complete
revolution around the unit circle?
14. A semi-circle or 180o is how many radians? Answer:
Carol Hardtke – MUHS
Aug 2009

15. To convert 57o to radians what would you multiply or divide by?
Answer: 57 o 

180
o
 0.994
16. To convert 7 π radians to degrees, what would you
5
multiply or divide by?
Answer:
7
180o
π
= 252O
5
π
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
20. Convert 90o to radians.
Answer:  /2
Carol Hardtke – MUHS
Aug 2009
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?
Answer: 3π
2
Pink angle in radians?
Answer:
Carol Hardtke – MUHS
Aug 2009
2
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?
Answer: 3 π
4
Purple angle in radians?
Answer: 5 π
4
Pink angle in radians?
Answer: 7 π
4
Carol Hardtke – MUHS
Aug 2009
23. Now visualize four TALL 30-60-90 triangles in STANDARD
POSITION. (Next slide we’ll flip them the SHORT way.) Can you name
these four angles in RADIANS?
Blue angle in radians?
Answer:  /3
Green angle in radians?
Answer: 2π
3
Purple angle in radians?
Answer: 4 π
3
Pink angle in radians?
Answer: 5 π
3
Carol Hardtke – MUHS
Aug 2009
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?
Answer: 5 π
6
Purple angle in radians?
Answer: 7 π
6
Pink angle in radians?
Answer: 11 π
6
Carol Hardtke – MUHS
Aug 2009
25. Angles that measure more than 2 are more than one revolution:
the red angle below is a 2π revolution plus 5π/4 more, i.e., 13π/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?
Answer:  3π
2
What is the measure of a counterclockwise 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?
13
8
21


Answer: 4
4
4

Carol Hardtke – MUHS
Aug 2009


26. COTERMINAL angles end up at the same terminal side like

4
,
- 74 and 94
or 0 and 2. Which of these angles is coterminal
to 5 ?
6
- -
7
7
,
or
6
6
6

Answer:
-
7
6
27. If n represents any integer, which expression below can be used
to give all angles coterminal to  ?

4
+ 2 ,

4
+ 2n , or

4
4
+ n Answer:

4
+ 2n
Recall n = all integers so this covers all positive
and negative rotations at same time.
28. Which expression below represents ALL angles that terminate at
any of the four quadrant axes? (Let n = any integer)

2
+ 2n , 2n , or
Carol Hardtke – MUHS
Aug 2009
n
2
Answer:
n
2
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.
(x,y)
hyp =
sin a =
cos a =
1
y = opp
tan a =
a
adj =
x
y
=y
1
x
=x
1
y
x

In the diagram shown, if a = radians,
3
find each:
1
3
(x, y) = ( ? , ? )  ,

2
2


3
2
1
cos a =
2
sin a =
tan a =
Carol Hardtke – MUHS
Aug 2009
3
csc a = 2  2 3
3
3
sec a =
2
cot a =
1
3

3
3
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==undef
( (0 ,, 1 ) )tan
tantan
= -= 3


(  - 1, , 3 )
 2 2 
 ( 2, 2) 
tan
=
tan = -1  - , 

2
2 
3


tantan
= -= 3 (  -, 3 , 1) 
3

2
2  5
tan
( ,,0 ) )
tan
==
0 ( -1
tantan
=
tan
3
=3
(
2
4
tantan
= =3 (- 1 ,,-
3
2

Aug 2009
 2

6
0 &2

)

11
4
4
3
3
2
cos
=1
sin
,,  tan
) tan== 3

2
3
( 1 ,, 0 ) tan = 0
6
 3
1
3
,
tan
=


( 2 , 2 ) tan = 3
7


4


5
( 2 ,, - 2) tan
= = -1
 tan
3
 2
2 

( 1 , 3 ) tan =
 ,  tan = 2
2


, ) tan
) tan= =undef
(0( , -1
Carol Hardtke – MUHS
(
4



=tan
1 =- (2 , -, 2 )
2 
 2
 2
3
3
6
7

3
1
6
 - , , -) 
2
5
 2

2

1
3 
,
=
( ,  )tan
tan =
2
2



2
2 

( , , ) tan
=
 tan
3
2
2



 3 1
3
31. Now practice all six ratios in random order. Visualize a unit circle,
but don’t waste time drawing it. Practice until you can do these
quickly and confidently! Click to check your answers.
1.
sin 2
3
4.
cot 5
6
7.
cos  - 7  =
=
= - 3
2

10.
13.
2.
3
2

0
cos 3  = - 2
4
2
3
-1
sin  =
Carol Hardtke – MUHS
2
Aug 2009
undef
6.
4
5. sec  =
2
8.
-1
3.
tan 3  =
tan 13  = 3
6
3
14.


4
tan  - 4  =

3

3
 6
2
csc 4  = - 2 3
3
3
sec 5  =
-2
12.
-3
15.
11. csc  - 3   =


9.
cos  -   =
3
2
csc 11  = -2
6
cot 11  =
2
0
32. Find the exact value of cos x in the diagram below.
(-3, 4)
4
Answer:
This makes a 3-4-5 ∆. Since
hyp  0, we use SOHCAHTOA definitions
for the reference angle inside the triangle.
5
cos x = - 3
x
5
-3
Reference
Angle
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)
33. Find the exact value of cos x AND the value of x in radians.
 1
3

-2 , 2 



Answer:
You should recognize these
coordinates from a 30-60-90 ∆ with hyp = 1
(In unit circle, cos is simply the x-coordinate.)
x
cos x =
1
- and x = 2 
2
3
Reference Angle must be 60o
Carol Hardtke – MUHS
Aug 2009
34. Match each basic trig function with its graph.
Answers:
A.
B.
B 1. y = cos x
E 2. y = sec x
A 3. y = sin x
D 4. y = csc x
C.
D.
E.
F.
C 5. y = tan x
F 6. y = cot x
Carol Hardtke – MUHS
Aug 2009
SUMMARY: (Use next slide to check your understanding)
Trig Function Transformations
y = ± a sin b (x ± c) ± d
• |a|  amplitude
• a < 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 )
Carol Hardtke – MUHS
Aug 2009
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
Carol Hardtke – MUHS
Aug 2009
39.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!
40.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!
Carol Hardtke – MUHS
Aug 2009
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 ∞ and - ∞.
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.
Carol Hardtke – MUHS
Aug 2009
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)
Carol Hardtke – MUHS
Aug 2009
47. Again, sketch a graph of each function. Click to see answers.
Answers:
y = sin x
y = cos x
y = tan x
y = csc x
y = sec x
y = cot x
1. y = sin x
2. y = cos x
3. y = tan x
4. y = sin x
5. y = csc x
6. y = cot x
Carol Hardtke – MUHS
Aug 2009
48. Graph y = -5 cos (x/2 – π/4)
Answer: Factor b to rewrite as y = -5 cos ½ (x - π/2)
Amp = 5
Vertical reflection
Period: 21  4
5
4
3
2
Phase start: π/2
Phase end: 9π/2
2
1
 /2
2
Fix your graph
if needed, then
click to see
answer.
-1
-2
-3
-4
-5
Carol Hardtke – MUHS
Aug 2009
3/2
4
9 /2
5 /2
6
8
10
7 /2
12
14
49. What function is graphed below? (There are many possible
answers – you only need to find one.)
Possible Answers Include:
1. y =3sin2x +1


π
π
2. y =3cos2  x-  +1 =3cos  2x-  +1
2
4



π
3. y =-3sin2  x-  +1 =-3sin 2x- π  +1
2

4
3
2
1
 /4
-2
 /2
Hints
-1
-2
Carol Hardtke – MUHS
Aug 2009
2
4
50. We’re 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?
 x
Answer: we restrict the domain to 2
2
(from a minimum to a maximum sine value)
where no x and no y will be repeated.
Carol Hardtke – MUHS
Aug 2009
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 (-π/2, -1) and (π/2, 1) on the graph
of y = sin x, are reflected to (?, ?) and (?, ?) on y = arc sin x?
Answer: (-1, - π/2) and (1, π/2)
A: (1.00, 1.57)
55. Sketch the graph of y = sin -1 x
1.5
gx 1=
Hint: sketch y = sin x first &
reflect a few key pts across y = x.
0.5
Click to see the answer (blue graph).
-1
-1
-1.5
B: (-1.00, -1.57)
Aug 2009
fx = sin x
1
-0.5
Carol Hardtke – MUHS
hx = x
sin-1x
2
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 ≤ π
57. This means that the points (0, 1) and (π, -1) on the graph
of y = cos x, are reflected to (?, ?) and (?, ?) on y = arc cos x?
Answer: (1, 0) and (-1, π)
B: (-1.00, 3.14)
58. Sketch the graph of y = cos -1 x
3
gx =2.5cos-1x
hx = x
Click to see the answer (blue graph).
2
1.5
1
0.5
1 0.00)
A: (1.00,
-1
-0.5
-1
Carol Hardtke – MUHS
Aug 2009
2
fx = cosx
3
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-toone section?


Answer: - 2  x  2
(same as for arc sin)
3
fx = tanx
61. Sketch the graph
qx =
2
of y = tan -1 x
Click to see
the answer
(blue graph).
hx = x
4
1
-4
-2
2
4
6
-1
rx =
-3
ty =
Aug 2009
2
gx = tan-1x
-2
Carol Hardtke – MUHS

-
2
-4
sy =

2
-
2
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 = π/6! Why do
we know n cannot equal 5π/6?
Answer: because the arc sin function is limited to
the one-to-one interval from -π/2 to π/2.
63. Solve for y, given
y  sin 1  

Answer: -π/4
2

2 
(You cannot answer 5π/4 or 7π/4 because
the arc sin function is limited to the one-toone interval from -π/2 to π/2. )

3

1
r

cos

64. Solve for r, given


Answer: 5π/6

2 
*Always remember arc sin and arc tan functions are
limited to the one-to-one interval from -π/2 to π/2 and
arc cos is limited to 0 to π .
Carol Hardtke – MUHS
Aug 2009
65. Now practice in random order. Visualize a unit circle, but don’t
waste time drawing it. Practice until you can do these quickly and
confidently! Click to check your answers.
1.
4.
sin-1
-1 =
cot 5
6
10.
csc 4
3
13.
=-3
8.

=
2 3
=3
-1 =
Aug 2009

tan-1 


0
arc cos
Carol Hardtke – MUHS
2
2.
5.
 5 
cos  - 
 2 
7.

-

cos-1 


3   
3  6
3  = 
2 
6
tan 13  = 3
11.
6
arc tan
14.
3.
sec  = undef
2
6.

-
arc tan
tan  - 4 
 3 
=
-3
4
sin  =
-1
csc 11  =
-2
3
9.
2
3
 0 = 0
-1 =
12.
6
15.
arc cot  
3  =


-
6
66.What geometry equation must be true about x & y in
any right triangle like that shown below?
2 + y2 = 1
Answer:
x
(x,y)
67. What “trig identity” does this
hyp = 1
equation become?
y = opp
a
adj =
x
Answer: sin2a + cos2a= 1 which is
called the Pythagorean Identity
for obvious reason!
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
Carol Hardtke – MUHS
Aug 2009
70. If you reflect angle “a” vertically, what changes are
made in the three trig ratios?
Answer:
(x,y)
1
cos (-a) = cos a
sin (-a) = - sin a
tan (-a) = - tan a
*Called “OPPOSITE ANGLE identities”
a
-a
(x,-y)
71. What geometry term applies to the pair
of acute angles a & b in a right triangle
(like the pink triangle below)?
Answer: “complementary” Sum of 90o but we
now prefer to say b = (π/2 – a) radians
72. What “COFUNCTION trig identities” relate
the ratios for COmplementary angles?
Answer:
b bπ/2-a
y
1
a
x
Carol Hardtke – MUHS
Aug 2009
sin a = cos(π/2 – a)
csc a = sec(π/2 – a)
cos a = sin(π/2 – a)
sec a = csc(π/2 – a)
tan a = cot(π/2 – a)
cot a = tan(π/2 – a)
73. Can you give the Double Angle Formulas? (Memory works, but as
long as you recognize this and know where to find it quickly, you’re
probably OK. This is even more true of others like half-angle, sum &
difference, etc. which we will seldom use.)
Answer: sin(2x) = 2sinxcosx
cos(2x) = cos2 x - sin2 x
= 2cos2 x - 1
= 1- 2sin2 x
74. Can you give the Law of Sines Formula?
Answer:
2tanx
tan(2x) =
1- tan2 x
a
b
c


sina sinb sinc
75. Can you give the Law of Cosines Formulas?
Answer: c2 = a2 + b2 - 2abcos c
a2 + b2 - c 2
cos c =
2ab
Carol Hardtke – MUHS
Aug 2009
76. Find the solutions of the equation in [0, 2π):
2 sin 3 = Solution:
2 sin 3 + 3  = 0
sin 3 = 3 =
77. Find ALL solutions of the equation:
sec2 t
– 2 tan t = 0
Solution:
=
3
3
2
 2 4 5
3
,
,
3
3
Aug 2009
3
 2 4 5
9
,
9
,
9
,
9
sec2 t - 2 tan t = 0
tan2 t + 1 - 2 tan t = 0
(Pyth ident.)
(tan t - 1)2 = 0
tan t - 1 = 0
tan t = 1
3  5 9 13
t=, ,
,
,
....
4 4 4 4 4
 n
t = +  (n is any int)
4 2
Carol Hardtke – MUHS
,
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
Carol Hardtke – MUHS
Aug 2009
THE END
Congratulations!
• 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!
Carol Hardtke – MUHS
Aug 2009