Download review notes for chap 10

Math 623
STUDY SHEET – CHAPTER 10 TRIGONOMETRY
10-1 & 10-2:
1)
The Trigonometric Ratios
If given a RIGHT TRIANGLE then:
sin(  ) 
2)
3)
4)
Name_________________________
opp
,
hyp
cos( ) 
adj
,
hyp
tan( ) 
opp
ady
To remember the ratios, remember: Soh Cah Toa
Soh Cah Toa works ONLY for RIGHT TRIANGLES
Use sin-1, cos-1 and tan-1 buttons on your calculator if you are asked to find the angle
measure.
10-3: Important Properties of Sines and Cosines
1)
Complements Theorem:
sin   cos(90   )
cos  sin( 90   )
and
2)
The Pythagorean Identity
(sin  )2  (cos )2  1
3)
Exact Value Theorem (memorize these or be able to derive them from 45-45-90
triangles or 30-60-90 triangles)
sin 30  cos 60 
sin 45  cos 45 
1
2
1
2
3
sin 60  cos 30 
2
10-4 & 10-5:
1)
2)
3)
4)
6)
2
2
Unit Circle
The Unit Circle is a circle of radius 1 centered at the origin.
Positive Angles in the coordinate plane are measured counter-clockwise.
Negative Angles in the coordinate plane are measured clockwise.
If the point (1,0) is rotated around the unit circle by an angle of  , then its new
coordinates are:
x  cos
5)

and
y  sin 
Values of Cosine are positive only in quadrants I and IV. (this is where x’s are positive)
Values of Sine are positive only in quadrants II and III. (this is where y’s are positive)
If asked to find the exact values of sine and cosine:
i)
Draw the angle
ii)
Find the reference angle
iii)
iv)
Use memorized exact values or 30-60-90 or 45-45-90 triangles.
Determine if the sine or cosine values should be positive or negative
10-6: LAW OF COSINES
a 2  b2  c 2  2bc(cos )
1)
2)
Use Law of Cosines to find an angle when given three sides. (SSS)
Use Law of Cosines to find a side when given 2 sides and the angle opposite the side you
need to find. (SAS)
10-7: LAW OF SINES
sin A sin B sin C


a
b
c
1)
2)
If the Law of Cosines doesn’t work, then you can most likely use the Law of Sines
If using the Law of Sines to find an angle then you MUST also consider
180 – (angle you find with the Law of Cosines).
10-8: Solving sin   k
1)
There are ALWAYS two values of  such that sin   k . Use your calculator (sin-1) to
find one angle. Then use 180 – (angle you found) to find the other angle.
10-9: Sine and Cosine Functions and their Properties
1)
Be able to identify a graph as either y = cos  or y = sin  . Know when the graphs reach
values of 1, 0, and –1.
2)
Properties of Sine and Cosine:
i.
The DOMAIN (x-values) of both functions is ALL REAL NUMBERS
The RANGE (y-values) of both functions is  1  y  1 .
ii.
Both functions are called periodic because they repeat after a certain value of “x”.
Each have a period of 360° (period of a function is how long it takes to repeat)
iii.
The cosine curve is a horizontal translation of the sine curve.
(MAKE SURE YOU KNOW HOW TO APPLY THE INFO ON THIS SHEET!)