Download 12-1 Define and Use Sequences and Series

13-3 Evaluate Trigonometric Functions of Any Angle
Objective: To evaluate trigonometric functions
Name:___________________
of any angle.
* VOCABULARY
Quadrantal angle:
Reference angle:
*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. The six trigonometric functions of  are defined as follows:
sin  
csc  
cos  
sec  
tan  
cot  
Ex 1: (Evaluate trigonometric functions given a point)
Let (12, 5) be a point on the terminal side of an angle  in standard position. Evaluate the six
trigonometric functions of . (always find r first!)
You Try: Let (6, 8) be a point on the terminal side of an angle  in standard position. Evaluate the six
trigonometric functions of .
Alg 2 Adv_ Ch 13B Notes- Page 1
*The Unit Circle
The circle x2 + y2 = 1, which has center (0, 0) and radius 1,
is called the unit circle.
sin  =
cos  =
*Use the unit circle to find the trig functions of any quadrantal angle!
Ex 2: Use the unit circle to evaluate
the six trigonometric functions of  = 450°.
You Try: Evaluate the six trig functions of
 = 360°.
*Reference Angle relationships
Let  be an angle in standard position. The reference angle for  is the acute angle ’ formed by the
terminal side of  and the x-axis. The relationship between  and ’ is shown below for nonquadrantal

angles  such that 90° <  < 360° (
<  < 2).
2
Quadrant II
Degrees: (90° <  < 180°)
Radians: (

<  < )
2
Quadrant III
Quadrant IV
Degrees: (180° <  < 270°)
Radians: (<  <
Degrees: (270° <  < 360°)
3
)
2
Radians: (
Ex 3: Find the reference angle ’ for (a)  = 165° and (b)  
3
<  <2 )
2
7
4
Alg 2 Adv_ Ch 13B Notes- Page 2
You Try : Draw the angle and find the reference angle for (a)  = 150° and (b)   
8
3
* Evaluating Trigonometric Functions
Use these steps to evaluate a trigonometric function for any angle :
STEP 1 Find the reference angle  ' .
STEP 2 Evaluate the trigonometric functions for
'.
STEP 3 Determine the sign of the trigonometric function value
from the quadrant in which  lies.
Ex 4 : (Use reference angles to evaluate functions)
10
Evaluate (a) cos(225° )
(b) cot
3
You Try : Evaluate without using a calculator.
3
(a) sec 300°
(b) sin
4
.
(c) cos (-240) °
**We are doing tons of these problems, so make sure to ask questions if you don’t understand!
Alg 2 Adv_ Ch 13B Notes- Page 3
Ex 5 : When sin  
3
in Quadrant II, find the other five trig functions of the angle.
5
You Try: When cos   
5
in Quadrant III, find the other five trig functions of the angle.
13
* Trigonometric Identity (This must be memorized!)
tan  
sin 
cos 
and
cot  
cos 
sin 
<Proof>
You Try: When sin   
3
4
and cos    , find the other four trig functions of the angle.
5
5
Alg 2 Adv_ Ch 13B Notes- Page 4
13.4 Evaluate Inverse Trigonometric Functions
Objective: To find
Name:___________________
angles given values of trigonometric functions.
*Please read textbook p.875.
Inverse sine:
If 1  a  1, then the inverse sine of a is an angle ,
written  = sin1 a, where sin   a and


(or 90    90 )
  
2
2
Inverse cosine:
If 1  a  1, then the inverse cosine of a is an angle ,
written  = cos1 a, where cos  a and
0     (or 0    180 )
Inverse tangent:
If a is any real number, then the inverse tangent of a is an angle ,
written  = tan1 a, where tan   a and


  
(or 90    90 )
2
2
Ex 1 : Evaluate the expression in both radians and degrees.
a. cos1 3
b. tan1 3
3
c.
tan1 2.6
Alg 2 Adv_ Ch 13B Notes- Page 5
You Try : Evaluate the expression in both radians and degrees.
a. sin 1
3
2
Ex 2 : Solve the equation cos  
b. cos1 (0.5)
c. tan1(1)
2
where 270° <  < 360°.
5
Ex 3 : Find the measure of the angle  in the triangle shown.
You Try : Solve the equation sin  = 0.7 where 90 <  < 270.
You Try : Find the measure of the angle  in the triangle shown.
You Try: Compare and evaluate using a calculator.
a. tan 2
c.
b. tan 2
d.
csc 0.7
sin1 0.7
Alg 2 Adv_ Ch 13B Notes- Page 6
*Be Careful!: cot  tan 1 
1
---Since cot 
 (tan  ) 1  tan 1 
tan 
*Know the trig function of 0 and 90
0
0
1
0
sin 
cos 
tan 
90
1
0

Ex 4 : Evaluate the expressions.
a. tan 1 2
b. cot 2
c. sin 1 1
Ex 5 : Evaluate the expression without using a calculator.
5
tan(sin 1 )
13
You Try : Evaluate the expressions without using a calculator.
1
a. sin(sin 1 )
2
b. cos(sin 1
3
)
2
Alg 2 Adv_ Ch 13B Notes- Page 7
13.5 Apply the Law of Sines
Objective: To solve triangles
Name:________________________
that have no right angle.
* The Law of Sines can be used to solve any triangles when AAS, ASA or SSA case.
---The law of sines can be written in either of the following forms
sin A sin B sin C


a
b
c
a
b
c


sin A sin B sin C
* Compare: (SOH CAH TOA) and ( a 2  b 2  c 2 )  apply to solve right triangles only
Law of Sin  apply to solve any triangles
Ex 1: Solve ΔABC with A = 28°, B = 102°, and a = 8.
You Try: Solve ΔABC with C = 14, B = 117, and b=21
Alg 2 Adv_ Ch 13B Notes- Page 8
*POSSIBLE TRIANGLES IN THE SSA CASE
Consider a triangle in which you are given a, b, and A. By fixing side b and angle A, you can sketch
the possible positions of side a to figure out how many triangles can be formed. In the diagrams
below, note that h = b sin A.
A is obtuse.
A is acute.
ab
_______________
h > a.
_____________
a>b
_______________
h<a<b
______________
h=a
_______________
a>b
_______________
Ex 2: Solve ΔABC with A = 94°, a = 18, and c = 13.
Ex 3: Solve ΔABC with A = 77°, a = 6.1, and b = 9.
Alg 2 Adv_ Ch 13B Notes- Page 9
You Try: Solve ΔABC .
1. A = 56, a = 24, b = 16
2.
B = 122, b = 5, a = 8
Ex 4: Solve ΔABC with A = 30°, a = 10, and b = 15.
---First make a sketch. Because b sin A = _____________ = _______ , and 7.5 < 10 < 15 (h < a < b),
two triangles can be formed.
Triangle 1
Triangle 2
Use the law of sines to find the possible measures of B.
Alg 2 Adv_ Ch 13B Notes- Page 10
*Area of a Triangle (any SAS case)
The area of any triangle is given by one half the product of the lengths of two sides times the sine of
their included angle. For ΔABC shown, there are three ways to calculate the area:
Area = __________________ or ___________________ or __________________
Ex 5: A piece of land is bordered by three roads as shown. Find the area of the land.
You Try: Suppose the side lengths in Example 5 are 4.6 miles and 2.8 miles. Find the area.
You Try: Solve ΔABC with A = 35°, a = 11, and b = 14.
Alg 2 Adv_ Ch 13B Notes- Page 11
13.6 Apply the Law of Cosines
Objective: To solve
Name:________________________
triangles using the law of cosines.
* The Law of cosines can be used to solve any triangles when SAS or SSS case.
(recall: Law of sines-ASA, AAS, or SSA case)
---The law of cosines can be written in the following forms
a2 = b2 + c2  2bc cos A
b2 = a2 + c2  2ac cos B
c2 = __________________

Use law of cos when you cannot use law of sin.
(= when you don’t know an angle and the opposite side)

Use law of cos only once to solve a triangle.
(You could use it more than once but law of sin is easier than law of cos.)

When SAS, find a couple (an angle and the opposite side) with law of cos
and then solve for the smaller side first using law of sin.

When SSS, always start solving for the longest side using law of cos.
Ex 1: Solve  ABC with a = 7, c = 12, and B = 26°.
When you know all
three sides and one
angle, you can use
the law of cosines
or the law of sines
to find the measure
of a second angle.
Alg 2 Adv_ Ch 13B Notes- Page 12
You Try: Solve  ABC with b = 15, c = 13, and A = 77°
Ex 2: Solve  ABC with a = 7, b = 13, and c = 9.
In Example 2,
the largest angle
is found first to
make sure that
the other two
angles are acute.
This way, when
you use the law
of sines to find
another angle
measure,you will
know that it is
between 0° and
90°.
Alg 2 Adv_ Ch 13B Notes- Page 13
*You can solve for cos A, cos B or cos C from the law of cosine.
b2  c 2  a 2
 cos A 
2bc
2
a  c2  b2
 cos B 
2ac
2
a  b2  c2
 cos C 
2ab
---If you can memorize them, you can save time when SSS case.
But if it’s too much for you, then you can resolve them each time.
You Try: Solve ΔABC with a = 23, b = 18, and c = 20
* HERON'S AREA FORMULA
The area of a triangle with sides of length a, b, and c is
1
Area  s(s  a)(s  b)(s  c) where s  (a  b  c)
2
Ex 3: A triangular path around an exhibit at the zoo is shown. Find the area of the exhibit.
You Try: Find the area of  ABC.
Alg 2 Adv_ Ch 13B Notes- Page 14