1 trigonometry geometry mathematics trig and geometry cos sin 45 o 30 o 150 o 330 o iii iiiiv this...
DESCRIPTION
3 Trigonometry Geometry Mathematics Trig and Geometry Here is an example of how to use it… θ =36.87 o h =5 o =3 a =4 The value of the angle can also be determine by using any two of the sides. For example, Note: This is NOT drawn to scale!TRANSCRIPT
1
Trigonometry
Geometry
MathematicsTrig and Geometry
cos
sin
121
21
45o1
32
21
30o1
32
21 150o
330o
1 21
32
III
III IV
This is the unit circle…
It axes are
sine
and
cosine
All lines drawn here have a length of 1and an angle equal to the angle we are working with.
The height along the sin-axis is the sine of the angle.
The distance to the right on the cos-axis is the cosine of the angle.
Here are some other examples.
Note that the angle always goes from the positive cos-axis counterclockwise.
Also note that the cosine is negative if the line is drawn to the left on the cos-axis.
Why is the sine negative here?
We often speak of four quadrants.
The first quadrant has positive cosines and sines.
The second quadrant has negative cosines and positive sines.
The third quadrant has negative cosines and sines.
The fourth quadrant has positive cosines and negative sines.
2
Trigonometry
Geometry
MathematicsTrig and Geometry
Given a right triangle, the trigonometric functions for eithernon-right angle are given by the following…
θ
hypotenuse(h)opposite
(o)
adjacent(a)
ho
sin
ha
cos
ao
tan
oh
csc
ah
sec
oa
cot
ao1tan
The value of the angle can also be determine by using any two of the sides. For example,
3
Trigonometry
Geometry
MathematicsTrig and Geometry
Here is an example of how to use it…
θ=36.87o
h=5o=3
a=4
3sin 36.875
4cos 36.875
3tan 36.874
5csc 36.873
5sec 36.874
4cot 36.873
1 3tan 36.874
The value of the angle can also be determine by using any two of the sides. For example,
Note: This is NOT drawn to scale!
4
Trigonometry
Geometry
MathematicsTrig and Geometry
Here are some useful angle relations…
a
a
a b180ba a
aa
a
b
bb
b
180ba
b b
180ba
ab
c180 cba
AC
ac
cC
bB
aA
sinsinsin
B
b
a
a
5
Trigonometry
Geometry
MathematicsTrig and Geometry
For example…
a b
180ba
30 150a b Ift hen
6
Trigonometry
Geometry
MathematicsTrig and Geometry
Here are some basic geometric and trigonometric formulae which we will use often in this and the next class…
Circumference of a Circle rC 2
Area of a Circle 2rA
Surface Area of a Sphere 24 rA
Volume of a Sphere 3
34 rV
Surface Area of a Cylinder(not including end faces) rLA 2
Volume of a Cylinder LrV 2
Trigonometric Formulae
BABABA
BABABAsinsincoscoscossincoscossinsin
1cossin 22
Quadratic Formula
AACBBx
CBxAx
24
02
2