Transcript 11.1 - Basic Trigonometry Identities

WARM-UP
Prove: sin2 x + cos2 x = 1
This is one of 3 Pythagorean Identities that
we will be using in Ch. 11. The other 2 are:
1 + tan2 x = sec2 x
1 + cot2 x = csc2 x
11.1 - Basic Trigonometry Identities
Objective:
to be able to verify basic trig identities
You must know and memorize the following.
Pythagorean Identities:
Tangent/Cotangent Identities:
1 + cot2 x = csc2 x
sin x
cos x
cos x
cot x 
sin x
Reciprocal Identities:
Cofunction Identities:
sin2
x+
cos2
tan x 
x=1
1 + tan2 x = sec2 x
1
csc x
1
cos x 
sec x
sin x 
tan x 
1
cot x
1
sin x
1
sec x 
cos x
csc x 
cot x 
1
tan x


sin   x   cos x
2



cos  x   sin x
2



t an  x   cot x
2

sin2 x = (sin x)2


csc   x   sec x
2



sec  x   csc x
2



cot  x   t an x
2

Summary of
Double-Angle Formulas
sin 2  2 sin  cos
cos 2  cos   sin 
2
2
cos 2  1  2 sin 
2
cos 2  2 cos   1
2 tan 
tan 2 
2
1  tan 
2
All Students Take Calculus.
Quad I
Quad II
Quad III
cos(A)<0
sin(A)>0
tan(A)<0
sec(A)<0
csc(A)>0
cot(A)<0
cos(A)>0
sin(A)>0
tan(A)>0
sec(A)>0
csc(A)>0
cot(A)>0
cos(A)<0
sin(A)<0
tan(A)>0
sec(A)<0
csc(A)<0
cot(A)>0
cos(A)>0
sin(A)<0
tan(A)<0
sec(A)>0
csc(A)<0
cot(A)<0
Quad IV
 1 3
 2, 2 


 1 1 
,


2
2




3
  2 , 21 


(-1, 0)

3 1
  2 , 2



(0, 1)
1 3
 2, 2 


 1 1 
9 0
 
2

120 
,
60



3 
3
2
2

2
135  3
45  
4
4
 3 1
,

6
0  2  360
1 80  
210  7 
6
6
270
3
2
4

240  4
 1
3
  2 , 2 


3
315  7 
300  5
(0, -1)
(1, 0)
330  11

225  5
 1

, 1 

2
 2
 2 2


30  
6

150  5

4
3
1
3
 2 , 2 


 3 1
 2 , 2 


 1

, 1 

2
 2
Reference Angles
Quad I
Quad II
θ’ = 180° – θ
θ’ = θ
θ’ = π – θ
θ’ = θ – 180°
θ’ = 360° – θ
θ’ = θ – π
θ’ = 2π – θ
Quad III
Quad IV
We can prove the trigonometric identities for specific angles.
Ex1) 1 + tan2 45°  sec2 45°
Ex2)
(sin 30°)( sec 30°)(cot 30°)  1
We can prove the trigonometric identities by using the trigonometric ratios.
Ex3) (tan x) (cos x)  sin x
tan A
Ex5)
 sin A
secA
Ex4) (sin x) (csc x)  1
Prove each using the trigonometric identities.
Ex6) (1 – cos x)(1 + cos x) 
Ex8)
cos2 x  1
2
cos x
  tan2 x
sin2
x
Ex7) 1 +
1
2
tan x
Ex9) sin x  cot
 csc2 x
x  sec x  1
Can you prove trig identities for specific angles? Using trig ratios? Or, using trig
identities?
Assignment: ws11.1
11.2a Trigonometric Identities
Objective: To use trigonometric identities and factoring to do basic trig proofs.
Ex1)
sin2 x  49
sin2 x  1 4sin x  49
Helpful Hints:

sin x  7
sin x  7
cos2 x  2 cosx  4
Ex2)

2
cosx  2
cos x  4
•
Factor and cancel
•
Start with the more complicated
side and manipulate it to equal the
other side.
•
Convert to sines and cosines.
•
Do you need a common
denominator?
•
YOU MAY NOT CROSS THE
ARROW!!!!
cos3 x  8
Prove each identity.
sec x
Ex3)
 sec2 x
cosx
1  cos2 x
Ex4)
 sin x
sin x
Ex5) csc x  sin x + (cos x)(cot x)
Write each in terms of sine. (What does this mean?)
cosx
Ex6)

sec x
sin x
Ex7)

csc x
Write each in terms of cosine. (What does this mean?)
sin2 θ
Ex8)
 cos θ
cos θ
1
1
Ex9)

sec θ  tan θ sec θ  tan θ
Can you use the trigonometric identities to work a trig proof?
Assign WS 11.2a
11.2a Solutions
sin 
7.
1  sin 2 
sin 3   1
8.
sin 
2
9.
sin x
2
10.
2
sin x
11. cos 
1
12.
cos4 
2
11.2b – More Trigonometric Identities
Objective: To continue trigonometric proofs using trig identities.
sin t
1  cost
Ex1)

1  cost
sin t
Ex2) (cot2  )(sec2 ) 1 + cot2 
Ex3) cos x(csc x + tan x)  cot x + sin x
1  tan θ
Ex4)
 sec θ
sin θ  cos θ
Ex5)
cos2 θ  9
cos2 θ  6 cosθ  9

cosθ  3
cosθ  3
Ex6) sec – csc 
sin   cos
sin  cos
Have you memorized your trig identities? Are you ready for an IDENTITY QUIZ?
Assignment: Worksheet 11.2b
WARM-UP
1. Given a triangle with a=5, b=7, and c=9.
Find all of its angles.
2. Given a triangle with A=60, c=12, and
b=42. Find the remaining side and
angles.
WARM-UP
The expressions sin (A + B) and cos (A + B) occur
frequently enough in math that it is necessary to find
expressions equivalent to them that involve sines and
cosines of single angles. So….
Does
sin (A + B) = Sin A + Sin B
Try letting A = 30 and B = 60
11.3 Sum and Difference Formulas
Objective: To use the sum and difference formulas for sine and cosine.
sin ( + ) = sin  cos  + sin  cos 
sin ( - ) = sin  cos  - sin  cos 
45
45
60
30
1. This can be used to find the sin 105.
2. Calculate the exact value of sin 375.
HOW?
cos ( + ) = cos  cos  - sin  sin 
cos ( - ) = cos  cos  + sin  sin 
Note the similarities and differences to the sine properties.
3. This can be used to find the cos 285.
HOW?
4. Calculate the exact value of cos 345.
5. Pr ove : sin(α  β)  sinα  β  2 sinβ cosα
6. Pr ove: sin2     cos
7. Prove: cos  2   sin 
Write each expression as the sine or cosine of a single angle.
cos 80 cos 20 + sin80 sin 20 
sin 30 cos 15 + sin15 cos30 
cos 12 cos x - sin12 sin x 
Do you understand the difference between the sum and difference properties
for sine and cosine difference? Assignment: ws 11.3
11.5a - Solving Trigonometric Equations
Objective: To solve trigonometric equations involving special angles.
What does it meant to solve over 0 < x < 360 ?
What does it meant to solve over 0 < x < 2 ?
Recall: You need the values of your special angles.
Do you have your unit circle?
Can you reproduce your special triangles?
Do you remember how to determine the
values of your axis angles?
45
45
60
30
Solve over the interval 0 < x < 360.
Ex1) 2 sin x  1  0
Ex2) 2 cos2 x  1  0
Ex3) 5 3 cot x  5  0
Ex4) 4 sin x cos x  6 cos x
Solve over the interval 0 < x < 2.
Ex5) tan2 x  3  0
Ex6) tan 4 x  4 tan2 x  3  0
Just a few more!!! Solve these over the interval 0 < x < 360 .
What happens when the angle doesn’t = x????
Ex7) tan2 2x  3
Ex8) cos2 2x  1 cos2x
2
Can you solve trig equations? Do you know/remember how to pick the appropriate quadrant
for each answer?
Assign Worksheet 11.5a
11.5b More Equations
Objective: To solve trigonometric equations that do not have
special angle answers.
These are similar to the problems from 11.5a, except you will
need your calculator to solve these. You will also need to know
how to find angles in each of the four quadrants.
Ex1: 5 cos2 x – 15 cos x + 3 = 0
Ex2: 49sin2 x – 1 = 0
Ex3: sin 3x sec x = 3 sin 3x
Ex4: 4csc2 x – 8cscx = 5
Try this one!
Ex5: 2cos2 x + 4 cos x – 1 = 0
Just so you don’t forget!
Ex6: sin 4x = ½
Assign WS 11.5b And…. Start studying for your Ch 11 test! Look over your
proof quiz too!
Chapter 11 Review
What have we covered?
Proving identities using specific angles, trigonometric
ratios and trigonometric identities. (Basically the
first quiz)
Trigonometric Identities (see note packet)
 Sum and difference properties for sine and cosine.
Solving trigonometric equations. You will have a unit
circle for this test.
How do you know what quadrant you should choose for
your answers? How do you determine answers for
angles other than x? (sin 2x = 1)
This is the last test! 
11.1 - Basic Trigonometry Identities
Objective:
to be able to verify basic trig identities
You must know and memorize the following.
Pythagorean Identities:
Tangent/Cotangent Identities:
1 + cot2 x = csc2 x
sin x
cos x
cos x
cot x 
sin x
Reciprocal Identities:
Cofunction Identities:
sin2
x+
cos2
tan x 
x=1
1 + tan2 x = sec2 x
1
csc x
1
cos x 
sec x
sin x 
tan x 
1
cot x
1
sin x
1
sec x 
cos x
csc x 
cot x 
1
tan x


sin   x   cos x
2



cos  x   sin x
2



t an  x   cot x
2

sin2 x = (sin x)2


csc   x   sec x
2



sec  x   csc x
2



cot  x   t an x
2

The Unit
1 3
(0, 1)
 2, 2 
Circle


 1 3
 2, 2 


 1 1 
,


2 2



3
  2 , 21 


(-1, 0)

3 1
  2 , 2



 1 1 
9 0
2

 
120 
,
60



3 
3
2
2


 3

2

135 
45 
4
4
 3 1
,

6
0  2  360
1 80  
210  7 
6
330  11

6
270
3
2
225  5
 1

, 1 

2
 2
 2 2


30  
150  5
6

4

240  4
 1
3
  2 , 2 


3
315  7 
300  5
(0, -1)
4
3
1
3
 2 , 2 


(1, 0)
 3 1
 2 , 2 


 1

, 1 

2
 2