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Chapter 3 - Trigonometric Functions If I turn my car around to face the other direction, we say I have turned it 180 degrees. Degrees are one of the units we use to measure an angle. We base the degree system off the fact that a circle has 360°. So, if we’re turning if our angle is one-quarter of a full circle, we say it has
measure oo 90)360(4
1= .
However, this is not the way most sciences measure angles. Just as we have Fahrenheit and Celsius measures of temperature, there is more than one way to measure an angle. The preferred method in the sciences is to work in radians. Almost all of your work in math from this point forward will work in radians. To find the measure of an angle using radians, we use the fact that a circle has 2π radians. So, if we are talking about an angle that measures a quarter of a circle, it’s radian measure would be
2)2(
4
1 !! = . To convert between degrees and radians, we can use the following:
1 radian = !"
#$%
&
'
180 degrees
1 degree = !"
#$%
&
180
' radians
Problems
1) Convert the following degrees to radians.
Degrees
30 45 60 75 90 105 120 135 150 180
Radians
Degrees
210 225 240 270 315 330 360
Radians
Trigonometric functions for acute angles To begin our conversation about trigonometric functions, remember the Pythagorean theorem guarantees for any right triangle with sides of length a, b and c, the equation a2+b2=c2 will be true. Also, remember two famous triangles: the 45-45-90 triangle (the triangle with angle measures 45°, 45° and 90°) and the 30-60-90 triangle (the triangle with angle measures 30°, 60° and 90°). Any triangle with the angle measures of 45-45-90 will have sides with lengths proportional to 1, 1 and 2 . Any triangle with angles measuring 30°, 60° and 90° will have sides with lengths proportional to 1, 2 and 3 . These triangles with their radian measures are shown below. Since the ratio of the sides of any two similar triangles are the same, we have some functions to describe these ratios. The sine function represents the ratio of the side opposite θ to the hypoteneuse, the cosine function is the ration of the side adjacent to θ to the hypotenuse and the tangent function is the ratio of the side opposite θ to the side adjacent to θ. These formulas can be remembered as “SOHCAHTOA”.
adj
opp
hyp
adj
hyp
opp
=
=
=
!
!
!
tan
cos
sin
Example 1: Find 6
sin! .
Solution: We locate the triangle with an angle measuring 6
! . Since sin measures hyp
opp we look
at the side opposite 6
! and get 1 and the hypotenuse is 2. Thus, 2
1
6sin =
! .
1
4
!
2
4
!
1
3
2
3
!
6
!
1
An easy way to remember the values for sine and cosine for these basic acute angles is to draw a table. List the basic angles in increasing order as follows: θ 0 π/6 π/4 π/3 π /2 sin θ cos θ Then, fill in all the blanks with a fraction with a 2 in the denominator. θ 0 π/6 π/4 π/3 π /2 sin θ /2 /2 /2 /2 /2 cos θ /2 /2 /2 /2 /2 Working your way across the row for sin, fill in with the numerators with 0,1, 2 , 3 , 2. θ 0 π/6 π/4 π/3 π /2 sin θ 0/2 1/2 2 /2 3 /2 2 /2 cos θ /2 /2 /2 /2 /2 Last, fill in the cosine row from right to left with the same numbers: 0,1, 2 , 3 , 2. θ 0 π/6 π/4 π/3 π /2 sin θ 0/2 1/2 2 /2 3 /2 2 /2 cos θ 2/2 3 /2 2 /2 1 /2 0/2 Problems
2) Fill in the following chart using your knowledge of the triangles. Do not use the above chart.
! 0
6
! 4
! 3
! 2
!
!sin !cos
!tan
Trigonometric Functions for Other Angles The above method will help us evaluate the trigonometric functions for acute angles. To evaluate obtuse or negative angles, we can apply this information to the circle with radius r. If we let (x,y) be any point on the circle then the trig functions can be defined as:
x
y
r
x
r
y
=
=
=
!
!
!
tan
cos
sin
Note that this gives the same result as the triangles above, it’s just a different way to look at it. Example 2: Find cos(0). Solution: We must find the x-coordinate and y-coordinate that correspond to the angle of 0. If we assume we’re working on a circle of radius 1, then the angle with measure 0 will intersect the circle at the point (1,0). Thus x=1 and y=1 so cos(0)=1. Obtuse angles To evaluate the trigonometric functions for obtuse angles, we use the idea of reference angles. The reference angle of θ is the measure of the angle between the terminal side of θ and the x-
axis. So, for example, the reference angle of 6
7! is 6
! because the measure from the x-axis to
6
7! is 6
! .
!
r
x
y
(x,y)
6
!
6
7!
Evaluating a trig function at any angle can be accomplished by evaluating that function at it’s reference angle and then adjusting the sign if necessary. Since sine depends on y, it is positive where y is positive: in the first and second quadrants. Since cosine depends on x, it is positive where x is positive: in the first and fourth quadrant. Since tangent depends on both x and y, it is positive when both x and y have the same sign: in the first and fourth quadrants. This can be remembered as “All Students Take Calculus”.
Example 3: Find !"
#$%
&
6
7sin
' .
Solution: To evaluate !"
#$%
&
6
7sin
' , we first find the reference angle of 6
7! which is 6
! . So,
instead of finding !"
#$%
&
6
7sin
' we find 2
1
6sin =!
"
#$%
&' . We then check to be sure the sign is correct.
Since 6
7! is in the third quadrant, we know it’s sine value should be negative, so
2
1
6
7sin !="
#
$%&
' ( .
Example 4: Find !"
#$%
&
4
3tan
' .
Solution: The reference angle of 4
3! is 4
! . 14
tan =!"
#$%
&' and 4
3! is in the second quadrant
where tangent is negative. Thus, 14
3tan !="
#
$%&
' ( .
T Tangent
S Sine
A All
C Cosine
Problems
3) Fill in the following chart of values using the unit circle, and your knowledge of reference angles.
! 3
! 3
2! !
3
4! 3
5! 6
5! 6
7! 6
11! 2
3! 4
3! 4
5! 4
7! !2
!sin !cos !tan Other Trigonometric Functions
Secant, cosecant and cotangent are related sine, cosine and tangent as follows:
!!
!!
!!
tan
1cot
cos
1sec
sin
1csc
=
=
=
One way to remember which function goes with which is that there is exactly one “co” in each pair.
Example 5: Evaluate !"
#$%
&
4
5sec
' .
Solution: To find !"
#$%
&
4
5sec
' , we first find !"
#$%
&
4
5cos
' .
2
1
4
5cos !="
#
$%&
' (
!"
#$%
&
4
5sec
' =!"
#$%
&
4
5cos
1
'= 2
2
1
1!=
!
Problems
4) Fill in the following chart using your knowledge of the triangles. Do not use the above chart.
!
6
! 4
! 3
!
!csc !sec
!cot
5) Fill in the following chart of values using the unit circle, and your knowledge of reference angles.
! 3
! 3
2! 3
4! 3
5! 6
5! 6
7! 6
11! 4
3! 4
5! 4
7!
!csc !sec !cot
Graphs of the trigonometric functions The graphs of the basic trig functions are given below.
Inverse trigonometric functions Sometimes we want to work a trigonometric function backwards. So, given a ratio of sides, we want to find the angle. To do this, we use the inverse trig functions. Remember that for a function to have an inverse it must pass the horizontal line test. So, in order to make our trig functions invertible we restrict their domains.
sin-1x=y if and only if siny=x and –π/2<y< π /2 cos-1x=y if and only if cosy=x and 0<y< π
tan-1x=y if and only if tany=x and –π/2<y< π /2
Example 6: Find tan-1(1). Solution: This asks what angle gives a tangent value of 1? π/4
Problems
1) Find the following:
a) 2
1sin
1! g) 2
3cos
1! m) 3
3tan
1!
b) 0sin1! h)
2
2cos
1! n) 1tan1!
c) 2
2sin
1! i) !!"
#$$%
&''
2
2cos
1 o) 0tan1!
d) 2
3sin
1! j) !!"
#$$%
&''
2
3cos
1 p) )1(tan 1!
!
e) !"
#$%
&''
2
1sin
1 k) 0cos1! q) )3(tan 1
!!
f) !!"
#$$%
&''
2
3sin
1 l) 1cos1! r) !
!"
#$$%
&''
3
3tan
1
g) Trigonometric Identities In working with trigonometric expressions, there are times when we want to move from one
expression to one that is easier to work with. For example, we might have xsec
1 in an equation
and we can agree that cosx is a much simpler way to express this fraction. Here are some of the most common identities you’ll be using: Pythagorean Identities sin2x+cos2x=1 1+tan2x=sec2x 1+cot2x=csc2x The last two from above can be remembered using the sentence “I tan in a second.” Even and Odd Identities sin(-x) = -sinx cos(-x) = cosx Since we can take the negative sign and write it in front of the sine function, we say sin is odd. Since the putting a negative angle into the cosine function is no different from the positive angle we say cosine is even. Angle addition formulas Please remember that sin(a+b)≠sin(a)+sin(b). Instead you must work it out using the following formulas: sin(a+b)=sin(a)cos(b) + cos(a)sin(b) cos(a+b)=cos(a)cos(b) – sin(a)sin(b)
Example 7: Find )43
sin(!!
+ .
Solution: )43
sin(!!
+ = )4
sin()3
cos()4
cos()3
sin(!!!!
+ = 4
26
2
2
2
1
2
2
2
3 +=!
!"
#$$%
&+!!"
#$$%
&
Example 8: Simplify )cos(
sin1 2
x
x
!
!
Solution: )cos(
sin1 2
x
x
!
! = xx
x
x
xcos
)cos(
cos
)cos(
cos 22
==!
Problems
2) Verify each identity. a) xxx cotcsccos =
b) xx
xxtan
cot
seccos=
c) r
r
rr sec
sin1
costan =
++
d) tt
t
t
tsec2
cos
sin1
sin1
cos=
!+
!
3) Find the exact value of each expression.
a) !"
#$%
&+64
sin''
b) !"
#$%
&+64
3cos
''
c) !"
#$%
&+36
sin''
d) cos !"
#$%
&+43
4 ''
e) )75cos( o f) )105sin( o