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