Download Trig Refresher

Trigonometry
Refresher
0011 0010 1010 1101 0001 0100 1011
By Erin Hill
[email protected]
1
2
4
What it’s used for:
0011 0010 1010 1101 0001 0100 1011
•
•
•
•
•
Trigonometry is best applied to:
Geometry
Astronomy
Physics
Advanced mathematics
Almost any engineering field
1
2
4
How far to your friend’s house
from your house?
0011 0010 1010 1101 0001 0100 1011
Friend’s
House
?
a
Your
House
b
Grocery
Store
1
2
4
Triangles and
Pythagorean Theorem:
0011 0010 1010 1101 0001 0100 1011
Pythagorean theorem:
c 2  a 2  b2
This rule works
only when using a
right triangle.
(Special case of
the law of cosines).
1
c  a 2  b2
a
b
2
4
How far to your friend’s house
from your house?
0011 0010 1010 1101 0001 0100 1011
Friend’s
House
1
?
Your
House
θ
b
Grocery
Store
2
4
Triangles and
Pythagorean Theorem:
0011 0010 1010 1101 0001 0100 1011
Only know the
length of one side
and the value of
one angle.
c=?
a=?
1
θ
2
4
b
Use trigonometry to solve for either
of the unknown sides (again only for
a right triangle)
Unit Circle
• “b” = length of the
0011 0010 1010 1101 0001 0100 1011
right triangle on the
x-axis (x-value)
• “a” = height of the
right triangle on the
y-axis (y-value)
• “c” = distance
between the origin
and our point (b,a).
(c = 1 for unit circle).
y
(b,a)
1
c
θ
1
b
a
2
4
x
Trig Functions: SOHCAHTOA
y
0011 0010 1010 1101 0001 0100 1011
sin(θ) = opp/hyp = a/c
cos(θ) = adj/hyp = b/c
c=1
1
θ
b = cos(θ)
tan(θ) = opp/adj = a/b
a = opposite side to angle, θ.
b = adjacent side to angle, θ.
c = hypotenuse side to angle, θ.
tan(θ) = sin(θ)/cos(θ)
2
a = sin(θ)
4
x
Similar Triangles
0011 0010 1010 1101 0001 0100 1011
tan(θ) = a/b
tan(θ) = c/d
a/b = c/d
c
a
θ
b
d
1
2
4
Same angle: ratio of sides must be the
same!
Angles in degrees:
• There are 360
degrees in a circle.
y
0011 0010 1010 1101 0001 0100 1011
90
60
45
135
30
• Some of the
common angles are
shown to the right.
• All positive angles
start from the xaxis.
1
180
225, -135 
2
0, 360
x
4
315, -45
Angles in radians
(standard unit):
• There are 2
radians in a circle.
y
0011 0010 1010 1101 0001 0100 1011
/2
/3
/4
3/4
• Some of the
common angles are
shown to the right.
/6
1

• All angles start
from the x-axis.
5/4
2
4
0, 2
x
Angles in radians
(standard unit):
y
0011 0010 1010 1101 0001 0100 1011
• 2 = 360   = 180
• Calculator has radian
and degree mode, so
check before you
calculate!
/2
/3
/4
3/4
/6
1

5/4
2
4
0, 2
x
Convert these:
0011 0010 1010 1101 0001 0100 1011
From degrees to radians:
From radians to degrees:
• 25
• /7
• -140
• 4
1
2
4
Convert these:
0011 0010 1010 1101 0001 0100 1011
From degrees to radians:
From radians to degrees:
• 25  /180 = 5/36
• /7  180/  25.7
1
2
4
• -140  /180 = -7/9 • 4  180/  229.2
• or: -140 + 360 = 220 
220   /180 = 11/9
Try these:
0011 0010 1010 1101 0001 0100 1011
c=?
a=?
30
3
1
2
4
Try these:
0011 0010 1010 1101 0001 0100 1011
cos(30) = 3/c
c*cos(30) = 3
c = 3/cos(30) = 3.46
c = 3.46
1
2
a = 1.73
tan(30) = a/3
a = 3tan(30) = 1.73
30
3
4
All Star Trig Class
y
0011
1010
1101 and
0001tangent
0100 1011
All:0010
sine,
cosine,
all
(0,1)
have positive values in this
quadrant.
Star: only sine has positive
values in this quadrant.
II
Star
Trig
III
1
All
(-1,0)
Trig: only tangent has positive
values in this quadrant.
Class: only cosine has positive
values in this quadrant.
I
2
4
(0,-1)
Class
IV
x
(1,0)
Calculating Angle
Use Inverse functions to “undo” the trig functions:
0011 0010 1010 1101 0001 0100 1011
• sin(θ) = 0.5
• cos(θ) = 0.5
• tan(θ) = 0.5
sin-1(c) = arcsin(c) = asin(c)
• sin-1(sin(θ))= sin-1(0.5)
θ = sin-1(0.5) = 30
1
2
• cos-1(cos(θ)) = cos-1(0.5)
θ = cos-1(0.5) = 60
4
• tan-1(tan(θ)) = tan-1(0.5)
θ = tan-1(0.5) = 26.57
Trig Functions: Periodic
θ =0010
0 
cos(θ)
1 and
sin(θ)
0011
1010
1101=0001
0100
1011=
y
0: point on circle is on x-axis.
(0,1)
θ = 90  cos(θ) = 0 and sin(θ)
= 1: point on circle is on y-axis.
θ = 180  cos(θ) = -1 and
sin(θ) = 0: point on circle is on
(-x)-axis.
θ = 270  cos(θ) = 0 and
sin(θ) = -1: point on circle is on
(-y)-axis.
1
c=1
θ
(-1,0)
(b,a)
2
a = sin(θ)
4
b = cos(θ)
x
(1,0)
(0,-1)
Sine and cosine increase and decrease as you go around the circle
Trig Functions: Periodic
0011 0010 1010 1101 0001 0100 1011
sin(θ)
cos(θ)
1
1
0.5
0.5
θ
0
0
0
pi/2
pi
3pi/2
0
2pi
-0.5
-0.5
-1
-1
pi/2
pi
1
3pi/2
2
4
θ
2pi
y-axis Shift
0011 0010 1010 1101 0001 0100 1011
3 + cos(t)
General form:
A + Bcos(ωt + φ)
4
3.5
1
3
2.5
• A: Shifts the function
up (+) or down (-)
along the y-axis.
2
1.5
1
0.5
0
-0.5
0
pi/2
-1
Dotted line represents the original cos(t) function.
2
4
pi
3pi/2
2pi
t
Amplitude
0011 0010 1010 1101 0001 0100 1011
0.5cos(t)
General form:
A + Bcos(ωt + φ)
• B (Amplitude): Shrinks
(small #) or stretches
(large #) along the yaxis.
1
0.5
0
0
-0.5
-1
pi/2
pi
1
3pi/2
2
4
2pi
t
Frequency
0011 0010 1010 1101 0001 0100 1011
General form:
A + Bcos(ωt + φ)
• ω (frequency): Shrinks
(large #) or stretches
(small #) along the xaxis. (1/s = Hz)
• T (period) = 2/ω:
Time it takes for the
wave to complete one
full cycle. (s)
cos(0.5t)
1
0.5
0
0
-0.5
-1
pi/2
pi
1
3pi/2
2
4
2pi
t
Phase Shift
0011 0010 1010 1101 0001 0100 1011
General form:
A + Bcos(ωt + φ)
• φ: Shifts right (-) or left
(+) along the x-axis.
cos(t+pi/4)
1
0.5
0
0
-0.5
-1
pi/2
pi
1
3pi/2
2
4
2pi
t
If you shift cosine by φ = -/2, your function becomes sine!
Common Trig Mistakes
0011 0010 1010 1101 0001 0100 1011
•
θ = x/sin
1
• sin(90) = 0.894
2
4
Why are these wrong?
sin(θ)
1
0011 0010 1010 1101 0001 0100 1011
0.5
θ
0
0
-0.5
-1
pi/2
pi
3pi/2
1
2pi
2
4
0011 0010 1010 1101 0001 0100 1011
b
a

c
•
sin(θ) = b/c
1
2
4
What’s wrong with this?
•
sin(θ2) = sin2(θ)
and
0011 0010 1010 1101 0001 0100 1011
•
sin-1(θ) = 1/sin(θ)
1
2
4
What’s wrong with this picture?
Trigonometric Identities
y
0011
0010
1010 1101 you
0001have
0100 a
1011
More
commonly,
(0,r)
circle of radius, r.
x2 + y2 = r2
r2cos2(θ) + r2sin2(θ)
(x,y)
=
1
r
r2
θ
(-r,0)
x = rcos(θ)
cos2(θ) + sin2(θ) = 1
divide by cos2(θ):
1 + sin2(θ)/cos2(θ): = 1/cos2(θ)
1 + tan2(θ): = sec2(θ)
(0,-r)
2
y = rsin(θ)
x
(r,0)
4
sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), cot(θ) = 1/tan(θ)
For more trig help…
0011 0010 1010 1101 0001 0100 1011
Come to the QSC!
UW2-030
www.uwb.edu/qsc
425.352.3170
[email protected]
1
2
4