three views of trigonometric functionsjrfaudree.github.io/m251s20/m251_review_day_3-s.pdf · three...
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REVIEW DAY 3: TRIGONOMETRY REVIEW
Three Views of Trigonometric Functions• graphs in the xy-plane
• sides of a right triangle
• points on the unit circle
The Graphs
On the axes below, graph at least two cycles of f(x) = sinx, f(x) = cosx, and f(x) = tanx. Label all x- andy-intercepts, any asymptotes, and all maximums and minimums.
UAF Calculus 1 1 Day 3 Trigonometry
y=smxYtcosx
.
.int#iIYItIExI?i..IiEEEtIEF.EE.YEEi
.
1 1 1
1
11
1
111
.. µ;µ,:
thnx1
.- -
1 1 1if:#1, 1
k¥2 ×=I x=3I2 2
The Triangle Defintion
Sketch a right triangle with side a adjacent to an angle ✓, o opposite of the angle ✓ and hypotenuse h. Define each
of the six trigonometric functions in terms of that triangle.
a) sin ✓ b) cos ✓ c) tan ✓ d) sec ✓ e) csc ✓ f) cot ✓
The Unit Circle Approach
Using a 45-45-90 triangle and a 30-60-90 triangle find the coordinates of ALL of the points on the unit circle.
UAF Calculus 1 2 Day 3 Trigonometry
h
0,0a
=f =ag =Qa = 's .
=s÷no = ttano
=L =1=
a-a 0 0
ftz ,Bk) Also
are;fP t.ME?a#'⇒ he:* .
90° o
1350 60450 450=4' rad
y180 600=51 rad
th 07 \a ,o)
9o°= Esrad( ÷ ,r⇒ - - ( rz ,
- E),
EESET ^t@rr⇒ :
frog ,tµ ( (GEIE) conversion
?
( 0,
- 1) 3600=2 # rad
So ...
is;÷gorTso¥
Each of the problems below can be solved using one of the approaches above: graphs, triangles, or unit circle.
When you solve each problem, think about which method is the best one.
1. An isosceles triangle has a height of 10 ft and its base is 8 feet long. Determine the sine, cosine and tangent
of the base angle ↵.
↵↵
2. Without a calculator evaluate:
(a) sin( 2⇡3 ) (b) cos( 5⇡4 ) (c) tan(�⇡4 )
3. Solve for x.
(a) cosx = 1
(b) sinx = 1
(c) tanx = 0
(d) sinx = 1/2 (Find all solutions in [0, 2⇡].)
UAF Calculus 1 3 Day 3 Trigonometry
qq.to#ym6sma=E=:#
Cosca )=£=÷ngtaho=Yo= §
EEFD=E = # = - I
•
I¥¥2 5¥ #¥
2
r# *t•
EEt⇒• fire, -E)
y=ws×X= ZTK
,K integer
X= . . . -497,27, ...
HUEY x=. .
.IR#,o,2t,4t,6t,
...
( from graph ! )
X=2HK+¥ ,K integer ÷i¥ X= If , 5¥
OR
* ... -3¥ ,¥ .EE 's " ⇐i¥t*¥¥To
4. Find the domain of f(x) = csc(x/2).
5. Solve the equation 2� 2 cos(x) = 0.
6. Find the domain of g(x) =psin(x� 1)� 1.
UAF Calculus 1 4 Day 3 Trigonometry
gtaph y=sinL×k )
t.ve= gn÷(×⇒
.zµµ,q←,,6,T(
We need to find where sin (E) =O
dywait"[ We Know sinlo )=o when 0=+1 . K,
K integer .
Fephi " "T
So we find X where ×z=HK or X=2TK
AhswI : Domain f 4) is all real numbers except X= ZIK,
kintegr
. . . ( 2 'T ,o)U( 0,
ZTDU ( IT ,4 'T )U .. .
+
2wsX= -2
Cos X= -1
µg X= IT or ZTHF or YFTF . .
- l -- -• aus : ×=2IKtt for Kintyet
OR
X= .. . - I
,I
,3 'T
, 55 . . .
We need sincx . 1) - 1 zo or sink - D 3 1
( But S1hHt< 1 ( ! D) So we need sink - D= 1.
We know sin 0=1
eprit when 0= TYZHIK .
* ( So we need X -1=51 + Ztk or ×= Iztl +2 'TK,
K integer
y=smCx - A)
graphically ; :t.at; ZTH" ⇐they, IyH2 'T
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