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Algebra II/Trig Regents Review #11: Trig Equations and Graphs
Trig Identities:
Reciprocal Identities:
Quotient Identities:
sec  
1
cos 
tan  
sin 
cos 
csc  
1
sin 
cot  
cos 
sin 
cot  
1
tan 
Examples:
1.
(cos  )(tan  )
2. (sec  )(cot  )
3.
csc x
sec x
Example Regents Questions
1.
Prove that the equation shown below is an identity for all values for which the functions are defined:
csc   sin 2   cot   cos 
Pythagorean Identities:
Given the Pythagorean identities cos 2   sin 2   1 , you can derive four other identities:
Examples: Write each expression in terms as a single expression:
1.
2. tan   cot 
1  sin 2 
Example Regents Questions
1.
The expression
2
(1) cos 


2.
The expression
2
(1) 1 cos 

sin 2   cos 2 
is equivalent to
1  sin 2 
2
(3) sec 
2
(2) sin 


cos2 
 sin  is equivalent to
sin 
2
(2) cos 
2
(4) csc 

(3) sin 
(4) csc 
2
2
2
 3. Starting with sin 
A  cos2 A 1, derive 
the formula tan A 1 sec A.


Sum/Difference of two angles:
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B  sin A sin B
tan( A  B) 
Example: Find the exact value of
tan A  tan B
1  tan A tan B
sin105
Find the exact value of cos15
Example Regents Questions
1. The expression cos4xcos3x sin 4xsin 3x is equivalent to
(1) sin x

 2.


(3) cosx
cos2   cos2 is equivalent
The expression 
 to
2
(1) sin 

(2) sin 7x
3. If tan A 
2
(2) sin 
2
(3) cos  1
(4) cos 7x

2
(4) cos  1



2
5
and sinB =
and angles A and B are in Quadrant I, find the value of tan( A  B).
3
41

Double angle formulas: Use the following formulas for double angles
sin 2 A  2sin A cos A
cos 2 A  cos 2 A  sin 2 A
cos 2 A  2 cos 2 A  1
cos 2 A  1  2sin 2 A
tan 2 A 
2 tan A
1  tan 2 A
Examples:
If cos  
7
, find cos 2 .
25
If sin  
4
, find sin 2 .
5
Express as a single trig function:
sin 2
2 cos 
Example Regents Questions
1.



If sin A 
(1)
2.
2 5
3
If cos y 
2
where 0 o  A  90 o, what is the value of sin 2A?
3
(2)
2 5
9

(3)
4 5
9
4 


and angle y is an acute angle, what is the value of cos2y ?
5

(4) 
4 5
9
Finding the measure of an angle: When finding all values of theta when given the value of the trig function, follow these
steps:
1. Find the reference angle by pressing, 2nd -- trig function—value
 Always plug in a positive value
 The answer you find is a reference angle, not an answer!!
2. Use the sign of the given value to determine the quadrants in which your angle lies
3. Find the measure of the angles
Examples:
1.
sin  
4
5
2. cos   
9
10
Solving trig equations: The 4 steps for solving trig equations are:
1.
2.
3.
4.
5.
Isolate the trig equation
Find the reference angle
Determine the quadrants
Find the angle
Check
Examples:
1.
2cos  3  4
3. 4csc  5  3csc  4
2. 3tan   2  1
3. csc   
1
2
Solving Second Degree Trig Equations: You will be required to factor just as you would factor with x as the variable!!
Examples:
1.
sin 2   sin   1  1
3. 2cos  2sin  cos
6.
cos 2x  cos x
2. 2 cos 2   cos 
4. cos2  3sin   0
Graphing Trigonometric Functions:
y  sin x
y  cos x
y  tan x
y  a sin(bx  c)  d
y  a cos(bx  c)  d
Amplitude: a _________________________________________________________________________________
Frequency: b__________________________________________________________________________________
Period:
2
___________________________________________________________________________________
b
Phase Shift: c ________________________________________________________________________________
Reflection: ____________________________________________________________________________________
Vertical Shift: d________________________________________________________________________________
Examples Regents Questions
1.
What is the period of the function y 
(1)
1
2
(2)


1 x
sin    

2 3
1
3
(3)
2

3
(4) 6
2. Which graph represents one complete cycle of the equation y  sin 3x?





3.
A radio transmitter sends a radio wave from the top of a 50-foot tower. The wave is represented by the
accompanying graph. What is the equation of the radio wave?
(1) y  sin x
(2) y 1.5sin x
(3) y  sin1.5x
(4) y  2sin x
4. What is the period of the function f ( )  2cos3 ?


(1)



2
(2)
 3


3
(3)
2

(4) 2

5. Which equation is graphed in the diagram below?
  
x  8
30 
  
x  8
30 
(1) y  3cos
 
x  5
15 

(3) y  3cos
(2) y  3cos

 
x  5
15 
(4) y  3cos
6. The accompanying graph shows a trigonometric function. State the equation of this function.


7.
A radio wave has an amplitude of 3 and a wavelength (period) of
(a) Write an equation for this function.
 meters.
(b) On the accompanying grid, using the interval 0 to 2 , draw a possible sine curve for this wave that passes

through the origin.

1
2
8. (a) On the same set of axes, sketch the graphs of y  2cos x and y = -sinx in the interval 0  x  2 .
(b) Give the exact value(s) of the coordinates of the intersection points of the two graphs.

