Download 12-1 Define and Use Sequences and Series

13-1 Use Trigonometry with Right Triangles
Name:______________________
Objective: To use trigonometric functions to find lengths.
Algebra 2
* The six trigonometric functions: sine (sin), cosine (cos), tangent (tan),
cosecant (csc), secant (sec), cotangent (cot)
*Must know and use:
 Definitions of trig functions:
SHO CHA TAO (SOH CAH TOA)
 Pythagorean Theorem:
*RIGHT TRIANGLE DEFINITIONS OF TRIGONOMETRIC FUNCTIONS
Let  be an acute angle of a right triangle.
The six trigonometric functions of  are defined as follows:
sin  
csc  
cos  
sec  
tan  
cot  
The abbreviations opp, adj, and hyp are often used to represent the side lengths of the right triangle.
Note that the ratios in the second column are reciprocals of the ratios in the first column:
csc  
sec  
cot  
Ex. 1: Evaluate the six trigonometric functions of the angle 0.
Algebra 2 Ch. 13A Notes_Page 1
5
.
13
(make sure to draw a triangle to solve trig problems!)
Ex 2: Find the other trig functions when sin  
*Memorize special right triangles --- They are hugely important in this chapter!
30  60  90
(
)
45  45  90
(
)
*Trigonometric values for special angles

sin 
cos 
tan 
30
45
60
Ex. 3: (Use a calculator to solve a right triangle)
Solve ABC.
You Try: Solve ABC.
Algebra 2 Ch. 13A Notes_Page 2
Ex. 4: (Use an angle of elevation)
You are measuring the height of your school building. You stand 25 feet from the base of the school.
The angle of elevation from a point on the ground to the top of the school is 62°. Estimate the height
of the school to the nearest foot.
You Try: A kite makes an angle of 59° with the ground. If the string on the kite is 40 feet, how far
above the ground is the kite itself? Round to the nearest foot.
Ex. 5: You measure the angle of elevation from the ground to the top of a building as 22°. When you
move 3 meters closer to the building, the angle of elevation is 25°. How high is the building? Show
a labeled diagram and calculation below.
You Try: You measure the angle of elevation from the ground to the top of a building as 50°. When
you move 50 meters away from the building, the angle of elevation is 30°. How high is the building?
Algebra 2 Ch. 13A Notes_Page 3
13-2 Define General Angles and Use Radian Measure
Objective: To use general angles that may be measured in radians.
*Angles in Standard Position
In a coordinate plane, an angle can be formed by fixing one ray,
called the initial side, and rotating the other ray,
called the terminal side, about the vertex.
An angle is in standard position if its vertex is at origin
and its initial side lies on the positive x-axis.
 counter clockwise : (+) measure
 clockwise : (-) measure
Ex 1: Draw an angle with the given measure in standard position.
a. 405°
b.
-65°
* 1 radian angle: an angle whose arc length equals to the radius (about 57.3°)
* Any angle measure not labeled degree ( °) is radian!
Radian is usually given in terms of  .
* Unit Circle : a circle with radius of 1
* Degree/Radian Conversion Rules
1. To convert a degree measure to radians, multiply by
 radians
180
180
2. To convert a radian measure to degrees, multiply by
 radians
Ex 2: Convert.
(a) 315° to radians
You Try: Convert.
(a) 200° to radians
(b)

radians to degrees.
6
(b)

radians to degrees.
5
Algebra 2 Ch. 13A Notes_Page 4
* Coterminal angles are angles that have the same initial and terminal sides but have different
measures.
 Degrees: If α is the degree measure of an angle, then all angles measuring α + 360n˚, where n
is an integer, are coterminal with α.
 Radians: If α is the radian measure of an angle, then all angles measuring α + 2n  , where n is
an integer, are coterminal with α.
Ex 3: Find one positive angle and one negative angle that are coterminal with the given angle.
(a) 210°
(b)
11
4
You Try: Draw an angle with the given measure in standard position. Then find one smallest
positive coterminal angle and one smallest negative coterminal angle.
(If given in radians, answer in radians, too!)
(a)
5
2
(c) 485°
(b)
3
4
(d) 75°
Algebra 2 Ch. 13A Notes_Page 5
*Sector : A region of a circle that is bound by two radii and an arc of the circle
*Arc length and area of a sector
The arc length s and area A of a sector with radius r and
central angle  (measured in radians) are as follows.
Arc length: s = r
1
Area : A  r 2
2
Ex 4 : Find the arc length and area of a sector with a radius of 15 inches and a central angle of 60°.
You Try : Find the arc length and area of the sector with given radius and angle.
r = 5 ft,  = 75o
Ex 5 : Evaluate sec
7
without using a calculator.
3
You Try: Evaluate cot
2
using a calculator.
13
<Discussion> Which mode do you need to use? Radian or degree mode?
1. Evaluate sin 13°.
2. Evaluate sin
3
5
3. Find the area of the sector if r = 5 ft,  =

3
Algebra 2 Ch. 13A Notes_Page 6