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Trig Functions of angles
Note that by what we know
of 'similar triangles' from
geometry, that the ratios of the
sides in the blue and red
trangles must be the same.
there are six possible ratios,
hence six trig functions!
Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent are the names of the
ratios and the definitions (and abbreviations!) are given below: (pause video to note
them!)
sin( θ ) =
cos( θ ) =
tan( θ ) =
opposite
hypotenuse
adjacent
hypotenuse
opposite
adjacent
csc ( θ ) =
hypotenuse
sec ( θ ) =
hypotenuse
cot( θ ) =
adjacent
opposite
adjacent
opposite
Two things to note for memory's sake:
Always write them in this order as each is opposite the 'upside down version' of itself
Easy to memorize LHS by SOH-CAH-TOA
Some simple trigonometric identities are apparent from the 'upside down'
functions opposite each other:
sin( θ ) =
cos( θ ) =
tan( θ ) =
1
csc( θ )
1
sec ( θ )
1
cot( θ )
Note that the letters sin, cos,tan etc have no meaning without an angle. Sine is not defined at
all, in fact but sin of theta is defined.
For today's lesson, assume that the argument (angle, if you want) for each function is
acute. 0o <θ<90o
Ex given that tan( θ ) =
3
4
, find the other five trig functions of theta.
(Pause and think about how to do this!)
sin( θ ) =
cos( θ ) =
tan( θ ) =
opposite
hypotenuse
adjacent
hypotenuse
3
4
csc ( θ ) =
hypotenuse
sec ( θ ) =
hypotenuse
cot( θ ) =
4
opposite
adjacent
3
(Pause here again and think of a way to find the third side.)
Pythagorean theorem!! a2+b2=h2
Hypotenuse can be found if we know opp and adj sides!
2
2
2
h = 3 + 4 = 25
sin( θ ) =
cos( θ ) =
tan( θ ) =
3
5
4
5
3
4
h =5
csc ( θ ) =
5
sec ( θ ) =
5
cot( θ ) =
4
3
4
3
Ex if sin( θ ) =
1
2
find the other five trig functions of θ. (pause and try it first!)
sin( θ ) =
2
cos( θ ) =
tan( θ ) =
csc ( θ ) =
1
3
2
2
1
sec ( θ ) =
3
cot( θ ) =
1
3
2
3
1
During the semester, some angles are quite commonly used.
We do NOT have to actually memorize the functions of the
angles, but can get all from memory by knwing the definitions of
the six trig functions and memorizing the following picture of two
special right triangles
π
6
o
= 30
π = 1

6 2
csc 
π = 3

6 2
sec 
π = 1

6
3
cot
sin
cos
tan
π = 2

6 1
π = 2

6
3
π = 3

6 1
π
3
o
= 60
π = 3

3 2
csc 
π = 2

3
3
π = 1

3 2
sec 
π = 3

3 1
cot
sin
π = 2

3 1
cos
π = 1

3
3
tan
π
4
o
= 45
π = 1

4
2
csc 
π = 2

4 1
π = 1

4
2
sec 
π = 1

4 1
cot
sin
π = 2

4 1
cos
π =1

4
tan
Ex
Suppose I am 90 feet from the base of the ACM clock tower,
the angle of elevation to the top of the tower is 30o. How high is the tower?
( o) = 90x
tan 30
1
3
=
x
90
x=
90
3
= 30 3 feet tall
Calculator has 'mode' button allowing you to change from degrees to radian
mode.Older models also has 'grads'. It will give you decimal approximations of
the values of all six trig functions of a given angle, but you must be cautious
about the mode your calcluator is in.
Ex set your calculator to degrees and find sin(60)
π
The set it to radians and compare to sin , they should give the same result.
3
Check it against
3
.
2
(pause here and do this!)
Note that you merely have a decimal approximation of an irrational number. Use
exact numbers whenever possible to lessen roundoff errors.
So how would you find sec(60o) with your calculator? (Pause and think!)
Notation for powers of trig functions: sin(θ)*sin(θ) usually written sin2(θ)
This software doesn't understand that properly, so sin( θ ) if I'm
typing using this.
2
( 2)
sin θ
Problem is with this pair of different things
sin(θ2) and sin2(θ). One squares the angle, the other squares the sine of the angle.
So do not write sin θ2 ! (Even though I must!)
8 Fundamental Identities
csc ( θ ) =
sec ( θ ) =
1
sin( θ )
opp
tan( θ ) =
opp
adj
=
hyp
=
adj
1
cos( θ )
sin( θ )
cot( θ ) =
cot( θ ) =
cos( θ )
1
tan( θ )
1
tan( θ )
=
cos( θ )
sin( θ )
hyp
Pythagorean identities:
sin2 θ +cos 2 θ = 1
1 + cot 2θ = csc2θ
tan2 θ+1=sec 2 θ
All 8 are here:
csc ( θ ) =
1
sin( θ )
tan( θ ) =
sin( θ )
cos( θ )
sin ( θ ) + cos ( θ ) = 1
2
2
sec ( θ ) =
cot( θ ) =
1
cos( θ )
1
tan( θ )
cos( θ )
sin( θ )
sec ( θ ) − tan ( θ ) = 1
2
cot( θ ) =
2
csc ( θ ) − cot ( θ ) = 1
2
2
Ex prove the identities
sin( θ ) ⋅ cot( θ ) = cos( θ )
1
sin( θ ) ⋅
cos( θ )
sin( θ )
cos( θ )
cos ( 2θ ) − sin ( 2θ ) = 2cos ( 2θ ) − 1
2
2
2
2
cos ⋅ ( 2 ⋅ θ ) − sin ⋅ ( 2 ⋅ θ ) + cos ( 2θ ) − cos ( 2θ )
2
2
2
2
2cos ⋅ ( 2 ⋅ θ ) − sin ⋅ ( 2 ⋅ θ ) − cos ( 2θ )
2
2
2
2 ⋅ cos ⋅ ( 2 ⋅ θ ) −  sin ⋅ ( 2 ⋅ θ ) + cos ⋅ ( 2 ⋅ θ ) 
2
2
2cos ( 2θ ) − 1
2
3
sin( θ ) + cos( θ )
cos( θ )
sin( θ )
cos( θ )
sin( θ )
cos( θ )
+
= 1 + tan( θ )
cos( θ )
cos( θ )
+1
tan( θ ) + 1
1 + tan( θ )
2
4
Write all six trig functions in terms of sin(θ)
sin( θ ) = sin( θ )
cos( θ ) = ± ⋅
tan( θ ) = ± ⋅
csc ( θ ) =
2(
1 − sin
θ)
sin( θ )
1 − sin ( θ )
2
cos ( θ ) + sin ( θ ) = 1
1
2
sin( θ )
sec ( θ ) = ± ⋅
2
1
1 − sin ( θ )
2
cot( θ ) = ± ⋅
1 − sin ( θ )
2
sin( θ )
Trigonometric Functions of ANY Angles
Ex Place the angle 2π/3 in standard position, and find values for all 6 trig functions
 2π  = 3

 3  2
csc 
 2π  = 2

 3 
3
 2π  = − 1

2
 3 
sec 
sin
 2π  = −2

 3 
cos
 2π  = − 3

 3
tan
 2π  = − 1

 3
3
cot
Ex Find the exact values of all six trig functions if (2,-1) is on the terminal side of an angle
in standard position.
sin( θ ) = −
1
csc ( θ ) = − 5
5
cos( θ ) =
sec ( θ ) =
2
5
tan( θ ) = −
Ex
1. find all six trig functions of an angle of π
1
2
5
2
cot( θ ) = −2
sin( π ) = 0
cos( π ) = −1
tan( π ) = 0
csc ( π ) = ∞
sec ( π ) − 1
cot( π ) = ∞