Download Unit 13 Notes: Trigonometric Ratios and Functions

Algebra 2
Name_____________________________
Hour_______
Unit 8 Notes
Trigonometric Ratios and Functions
1
Algebra 2
8.1 Use Trigonometry with Right Triangles
Vocabulary:

The six trigonometric functions that consist of a right triangle’s side lengths are:

Sine (sin)= __________________=________

Cosine (cos)= __________________=________

Tangent (tan) = __________________=________

Cosecant (csc): ___________ = _____________=_______________=_______.

Secant (sec): ___________ = ________________= ______________=_______.

Cotangent (cot): ___________ = ________________=____________=_______.
Finding Values:
Let θ be an acute angle of a right triangle. Find the value of the other five trigonometric
functions of θ.
1. sin θ =
5
6
2. cot θ =
4
5
2
Algebra 2
Solving Triangles:
Solving a right triangle means to….
3. A = 350 , c = 16
5.
A= 42o, b=5
4. B = 750 , a = 15
6. B= 21o, b=17.5
Summary
3
Algebra 2
8.2 Use Trigonometry with Special Right Triangles
Special Triangles:
Complete the chart below with the trigonometric values for special angles.
45-45-90 Triangle

sin 
30-60-90 Triangle
cos 
tan 
csc 
30o
45o
60o
4
sec 
cot 
Algebra 2
8.3 Define General Angles and Use Radian Measures
Vocabulary:

An angle can be formed by fixing one ray, called the _____________________, and
rotating the other ray, called the _____________________, about the vertex.

The position of an angle whose vertex is at the origin and its initial lies on the positive xaxis, is called _______________________________.

Angles in standard position whose terminal sides coincide (“land in the same spot”), are
called _______________________________. Can be found by adding or subtracting
multiples of _________.
Drawing Angles:
Draw an angle with the given measure in standard position.
1. 2400
2. -500
3. 405o
5
Algebra 2
Find Coterminal Angles IN DEGREES.
Find one positive angle and one negative angle that are coterminal with the given angle.
4. 700
5. - 4500
RADIANS
Angles can also be measured in radians. To define a radian, consider a circle with radius r
centered at the origin as shown. One radian is the measure of an angle in standard position
whose terminal side intercepts an arc of length r.
Converting Between Degrees and Radians
Convert degrees to radians
  radians
Multiply degree measure by
180 0
6. Convert to radians 3150
Convert radians to degrees
Multiply radian measure by
7. Convert to degrees
6

6
180 0
  radians
8. Convert to degrees
3
7
Algebra 2

Angles in standard position whose terminal sides coincide (“land in the same spot”), are
called _______________________________. IN RADIANS, these can be found by
adding or subtracting multiples of _________.
Find Coterminal Angles in RADIANS.
Find one positive angle and one negative angle that are coterminal with the given angle.
9.
5
6
10.

4
Complete the following problems.
10. Find one positive coterminal angle
of 2000
9. Convert to radian measure 400
Summary
7
Algebra 2
8.4 Evaluate Trigonometric Functions of any Angle
Vocabulary:

The circle x 2  y 2  1 , which has center (0, 0) and radius 1 is called a
_______________________________.
General Definitions of Trigonometric Functions:
Let  be an angle in standard position and let (x, y) be the point where the terminal side of 
intersects the circle x2+y2=r2 and a triangle is formed drawing a vertical line from that point to the
nearest x-axis. The six trigonometric functions are defined as follows:
sin  
csc 
cos 
sec 
cos 
cot  
Using a Point
Use the given point on the terminal side of an angle in standard position to evaluate the six
trigonometric functions.
1. (8, 15)
First find the value of r:
r  x2  y2
r = ________, x = _________, y = _________
sin   __________
csc  __________
cos  __________
sec  __________
tan   __________
cot   __________
8
Algebra 2
2. (-2, -3)
First find the value of r:
r  x2  y2
r = ________, x = _________, y = _________
sin   __________
csc  __________
cos  __________
sec  __________
tan   __________
cot   __________
So on the unit circle we can see that….
If r=1 on Unit circle
sin  
csc 
cos 
sec 
cos 
cot  
LET’S FILL IN THE UNIT CIRCLE IN OUR NOTES.
9
Algebra 2
From your unit circle, find the following:
3. sin 45o
7. tan
3
2
8. cos
9
4
4. cot 120o
5. tan 90o
6. sec
9. csc 300 o
5
4
10.sin 
11. tan 480o
10
Algebra 2
The Unit Circle
2
2
x +y =1(radius is 1 unit)
2π=Circumference
(0,0) center
45
60
30
45
11
Algebra 2
Complete using the unit circle

Radian
 Degree
sin 
cos
tan 

6

4

3

2
2
3
3
4
5
6

7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
12
csc 
sec 
cot 