Download Trigonometry - Net Start Class

Trigonometry
A Pre-calculus Preview
Ch 13 (1,2,4) Notes
Vocabulary: Trigonometric function, sine, cosine, tangent, cosecant, secant, cotangent, standard position, initial side,
terminal side, angle of rotation, coterminal angles, reference angles, inverse sine function, inverse cosine function,
inverse tangent function
A Trigonometric function is a function whose rule is given by a trigonometric ratio. A Trigonometric ratio
compares the lengths of two sides of a right triangle. The Greek letter theta  is often used to represent the acute
angles of the right triangle. The values of trigonometric ratios depend on  .
13.1 Notes (Right Angle Trigonometry)
Trigonometric Functions and Reciprocal Functions (use the Pythagorean Theorem to find missing side of right
triangle a 2  b 2  c 2 remember c is the hypotenuse which is across from the right angle)
sine  =
opposite
hypotenuse
cosine  =
cosecant  =
hypotenuse
opposite
secant  =
hypotenuse
adjacent
adjacent
hypotenuse
Example:
5
3
sin  =
3
5
csc  =
3
5
cos  =
4
5
sec  =
4
5
tan  =
3
4

tangent  =
opposite
adjacent
cotangent  =
4
adjacent
opposite
cot  =
3
4
Special Right Triangles
You will frequently need to determine the value of trigonometric ratios for 30, 60, and 45 angles. Recall from
geometry that the ratio of side lengths for a 30  60  90 is 1: 3 : 2 and for a 45  45  90 the ratio is 1:1: 2 .
sin 30 =
1
2
sin 60 =
3
2
1
2
60
30
45
cos 30 =
3
2
cos 60 =
tan 30 =
1
3

3
3
tan 60 =
45
sin 45 =
1
2

2
2
tan 45 =
90
cos 45 =
1
2

2
2
90
* The angle of elevation and depression are
alternate interior angles therefore have
the same measure.
3
 3
1
1
1
1
13.2 Notes (Angles of Rotation)
An angle is in standard position when its vertex is at the origin and one ray is on the positive x-axis. The initial side of
the angle is the ray on the x-axis and the other ray is called the terminal side. An angle of rotation is formed by
rotating the terminal side and keeping the initial side in place.
Coterminal angles are angles in standard position with the same terminal side. The reference angle is the positive
acute angle formed by the terminal side of  and the x-axis.
Coterminal Angles
Reference Angle
To determine the value of the trigonometric functions for an angle  in standard position you begin by selecting
a point P with coordinates (x,y) on the terminal side of the angle. The distance r from point P to the origin is
given by
x2  y2 .
sine  =
O y

H r
cosine  =
A x

H r
tangent  =
O y

A x
The Must Know Chart!!!!
angle
cos (x)
sin (y)
tan (y/x)
30 or

6
3
2
1
2
3
3
45 or

4
2
2
2
2
1
60 or

3
3
2
3
1
2
13.4 Notes (Inverses of Trigonometric Functions)
To find the measurement of angles given the value of the trigonometric function you must use an inverse
trigonometric relation. * Inverse trig functions are also called arcsine, arccosine, and arctangent.
Function (to find value)
Inverse Relation (to find angle)
sin   a
sin 1 a  
cos  a
cos 1 a  
tan   a
tan 1 a  
*Calculators must be set to degrees
Example: A group of hikers wants to walk from a lake to an unusual rock formation. The formation is 1 mile east and
0.75 mile north of the lake. To the nearest degree, in what direction should the hikers head from the lake to reach the
rock formation?
Rock
Step 1: Draw a diagram. The hikers’ direction should be based
on  , the measure of an acute angle of a right triangle.
0.75 mi
Step 2: Find the value of  .
tan  
θ
Lake
opp. .75

 0.75
adj.
1
1 mi
  Tan 1 0.75
  N 37E
The hikers should head north 37 east.
Inverses of trigonometric functions are not functions themselves. In order to define inverses as functions you
must restrict the domains. When this restriction is used it is noted by using a capital letter.
Inverse Trigonometric Functions
Symbol
Domain
Range
Sin 1a
1,1
90,90 or
  
  2 , 2 
Cos 1a
1,1
0,180
0,  
Tan 1a
, 
90,90 or
or
Quadrant Restriction
  
  2 , 2 
Homework: *don’t forget to rationalize the denominators and simplify radicals
13.1 Day 1: page 933-34 (13-18, 21-23)
13.1 Day 2: page 933-34 (8, 9, 19, 20, 25, 26, 27, 30-32)
13.2 Day 1: page 939 (26-41)
13.2 Day 2: page 939 (42-49)
13.4 Day 1: page 953-54 (11, 25, 30, 31, 33, 40)