Download Trigonometric Functions

Trigonometric Functions
1. Definition of radians and conversion between radians and degrees.
a. A radian= Measure of the angle when the arc subtended equals the radius
of the circle.
Hence, the length of the arc subtended by the angle divided by the radius of
ArcLength
the circle gives the measure of the angle in radians. It is  
Radius
0
2 radians  360
b.
 radians  180 0
0 radians  0 0 ,
2. Common angles in radians

2

6
 30 0 ,
 90 0 ,   180 0 ,

4
 45 0 ,

3
 60 0
3
 270 0 , 2  360 0
2
3. Unit circle. Definition of sine and cosine and their graphs.
a. Use the coordinates and the triangle determined to show sine and cosine.
Indicate the variation of sine and find the relationship with its
representation in the Cartesian plane. Show the meaning of a period. Make
sure the relation between what happens in the circle and the plane is
understood including the signs of both functions.
b. Fundamental trigonometric identity. sin 2 x  cos 2 x  1
sin(  )   sin(  )
c. Other relations
cos(  )  cos( )
4. Graphs of sine and cosine.
a. Basic relation between sine and cosine (analysis through translations)
sin( x  2 )  sin( x)
sin( x 
cos( x 

2
)  cos( x)

2
)  sin( x)
b. New graphs under transformations. Period and amplitude. Determine the
period and amplitude of the following functions:
y  sin( 3x   )
1
y  0.5 cos( x)  20
4
Trigonometric Functions 4/29/2017
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4. Find two equations for the following
graph. One in terms of sine and the other in
terms of cosine
2 sin( 4 x
)
12
11
10 10
9
8
0.79
1.18
1.57
x
5. Table with important angles:

0
sin x
0
cos x
1
tan x
0

6
1
2
3
2
3
3

4
2
2
2
2
1

3

2

3
2
2
3
2
1
2
1
0
-1
0
0
-1
0
1
Und.
0
Und.
0
3
1.96
2.36
6. Reference angle of an angle in standard position. It is the smallest angle it makes
with the x-axis. If  is the reference angle for  , sin(  )  sin(  ) except
probably for the sign.
II
I
β=α
Sin (β) = Sin (α)
Cos (β) = Cos (α)
β
α
Sin (β) = Sin (α)
Cos (β) = - Cos (α)
IV
III
β
α
Sin (β) = -Sin (α)
Cos (β) = -Cos (α)
Trigonometric Functions 4/29/2017
β
α
Sin (β) = -Sin (α)
Cos (β) = Cos (α)
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The reference angle is always between zero and  / 2 . The diagram above shows the
relationship between the value of the angle and its reference angle on each quadrant.
For each of the angles given below
o
o
o
Draw each angle
Indicate its reference angle
Determine the value of the function using the reference angle.
2

5
10
sin(
)
cos(  )
sin( )
sin(  )
3
6
4
3
o Find all solutions for each of the equations:
o sin( x)  1
o tan( x )  1
1
o cos( x) 
2
7. Definition of the other trig functions using triangles. Values for the common
angles. Signs in each quadrant using ASTC.
opp.
hip.
csc( ) 
Opp.
hip.
opp
adj
hip
cos( ) 
sec( ) 
θ
hip.
adj
opp.
adj
Adj.
tan( ) 
cot( ) 
adj.
opp.
Using these definitions we have the following relationships among trigonometric
functions:
Sin ( x)
1
1
Cos( x)
1
Tan( x) 
Csc( x) 
Sec( x) 
Cot ( x) 

Cos( x)
Sin ( x)
Cos( x)
Sin ( x) Tan( x)
sin(  ) 
Hip.
II
I
Students
Only sine (and
csc are positive)
All
Are positive
III
Taking
Only tan (and
cot are positive)
Trigonometric Functions 4/29/2017
IV
Now we can use the
reference angles to find the
value of all the trigonometric
functions. To help us doing
this, we will remember the
following rule ASTC (All
Students Take Calculus)
Calculus
Only cosine (and
sec are positive)
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EXERCISE: Find the value of the following trigonometric function, by first finding
the reference angle:
Tan( )
5
)
3
 8
Cot (
)
6
 22
Sec (
)
4
EXERCISE: In each case find an expression for each of the other trigonometric
functions of all the trigonometric functions given the following information:
o tan   x
3x
o tan  
2
o sin   2x
2x
o cos  
3
Csc (
8. Other important identities.
tan 2 x  1  sec 2 x
1  cot 2 x  csc 2 x
sin( x  y )  sin x cos y  sin y cos x
cos( x  y )  cos x cos y  sin x sin y
sin( 2 x)  2 sin x cos x
cos( 2 x)  cos 2 ( x)  sin 2 ( x)
 1  2 sin 2 ( x)
 2 cos 2 ( x)  1
1  cos( 2 x)
sin 2 ( x) 
2
1  cos( 2 x)
cos 2 ( x) 
2
EXERCISE: Use the identities above to simplify the given equations. Then solve
them:
o 2 cos 2 x  3 sin 2 x  3
o sin 2 x  cos 2 x
o cos 2  3 cos  2
9. Graphs of all the trig functions
Trigonometric Functions 4/29/2017
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10. Law of cosine
11. Law of sine
12. Inverse functions. Use definition to calculate some values.
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