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Pre Calculus
Worksheet: Fundamental Identities Day 1
Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated
strategy AND those strategies before.
Strategy I: Changing to sines and cosines
csc θ cot θ
1.
sec θ
Strategy II: Pythagorean or Co-Function Identities
2. ( sin 2 x + cos 2 x ) − ( csc 2 x − cot 2 x )
3. sin θ − tan θ cos θ + cos ( π2 − θ )
Strategy III: Factoring the GCF
4.
cot 3 w + cot w
cos w
Strategy IV: Even-Odd Identities
5. cot( − x) cot ( π2 − x )
6. tan( −θ ) ( csc 2 θ − 1)
We can use Identities to simplify trigonometric expression, but there is also a visual explanation for why we can
simplify. Before we get into proving identities, try this exploration to further your understanding…
7. Use your graphing calculator, set in radian mode, to complete the following:
a. Graph the function
=
y sin 3 x + cos 2 x sin x .
Note: you will need to enter the exponents using parenthesis around the trig functions, such as ( sin( x) ) .
3
b. Select Zoom 7: ZTrig for a good window.
c. Explain what you see and sketch a graph.
d. Write an Identity for the expression you graphed and what it equals. PROVE your identity!
For the remainder of this lesson, we will PROVE identities. Remember to indicate where you are starting and to
show all steps that lead you to the other side. If you are stuck, think through Strategies I through IV and keep in
mind what you are trying to prove!!
Verify (Prove) each identity.
8. tan2 x = sin2 x + sin2 x tan2 x
10.
tan x csc x
=1
sec x
12. cos θ = cot θ sin2θ cscθ
9.
11.
sin 2 x + cos2 x
2
= cos x
2
sec x
1 − 2 cos 2 x
( sin x )( tan x cos x − cot x cos x ) =
π

− x =
−1
2

13. tan( − x) tan 
16. cos θ =
18.
( sec θ − tan θ )( sec θ + tan θ )
sec 2 w − tan 2 ( − w )
cos 2 ( − x ) + sin 2 x
sec θ
=1
17.
sin 2 y + tan 2 y + cos 2 y
sec y
19. csc w =
3
= sec y
cot 3 w + cot w
cos w
Warm Up: Fundamental Identities Day 2
Perform the operation without a calculator…
5 3
−
20.
8 4
5 3
−
21.
8 7
x x−7
+
22.
5
3
x
4
23.
x
1−
5
1−
Pre Calculus
Worksheet: Fundamental Identities Day 2
Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated
strategy and those strategies before.
Strategy V: Combining Fractions with LCD
1.
1
sin θ
2
−
cos 2 θ
sin x
2.
sin θ
2
Strategy VI: Factoring with Difference of Squares
3.
1 − cos x
+
1 − cos x
sin x
Strategy VII: Factoring trinomials with box
sec 2 x − 1
4. sin 4 x − sin 2 x cos 2 x − 2 cos 4 x
1 − sec x
In general we don’t want to leave a trigonometric expression with a fraction in it. Sometimes, however, we have no
choice (you may have noticed we left a fraction on example 10 of the notes). If we do have to leave a fraction in
our expression, we want to make it a “nice” fraction. For trigonometry, this means we prefer the fraction only
have one trig. function and we prefer any addition or subtractions to be left in the numerator. To make this occur,
we multiply by a special form of “1” similar to what we did with division involving “i”or division of radicals…
For example,
3 − 4i
2+i

2−i
2−i
=
2 − 11i
5
or
1
5
Multiply by “1”

5
5
=
5
5
. Let’s try it with trigonometric expressions.
Strategy VIII: Multiply by “1” ASK if you are stuck!!!
5.
cos x
6.
1 − sin x
tan x
sec x − 1
Verify (Prove) each identity. Use extra paper if needed!!
7. 2 sin 2 θ − 1 = sin 4 θ − cos 4 θ
2
9. csc
θ
=
cos 2 θ
+ csc θ sin θ
sin 2 θ
8. sec x − sin x = cot x
sin x
10.
cos x
1 − 3cos u − 4 cos 2 u
2
sin u
=
1 − 4 cos u
1 − cos u
11.
cos 2 θ − csc 2 θ + sin 2 θ
sin θ sec θ
13. tan θ =
= − cot 3 θ
1 + tan θ
1 + cot θ
12. =
2 csc β
14.
1
sec w − 1
sin β
1 − cos β
−
+
1
sec w + 1
sin β
1 + cos β
= 2 cot 2 w
Warm Up Lesson 5.3:
Tell whether each statement is True or False. Then, given an example to justify your answer.
x+3 3
2
x+ y
=
1. ( x + y ) =x 2 + y 2
2. x + y =
3.
x+4 4
4.
log (=
xy ) log x ⋅ log y
=
° ) sin 30° + sin 60°
5. sin ( 30° + 60
Pre Calculus
Worksheet 5.3 Introduction
1. A refresher as to why the sum/difference rules don’t work the way many people want them to:
a) Find sin 30  60 , and then find sin 30  sin 60 . Are they the same?
b) Find cos 120  60 , and then find cos 120  cos 60 . Are they the same?
c) Find tan 60  30 , and then find tan 60  tan 30 . Are they the same?
Now let’s get a little deeper into where the sum and difference identities come from…
Cosine of Differences: cos ( a − b )
a) Angles a and b are drawn below on one unit circle. Assume a > b and label these angles on the left
picture below. Let θ= a − b . Notice you could have drawn angle θ in the first quadrant and it is the same size
as θ from part (a). Label θ in the unit circle on the right.
C (cos θ, sin θ)
B (cos b, sin b)
D (1, 0)
A (cos a, sin a)
b) Draw the chord created by the points where angle θ intersects each unit circle. Their coordinates are given.
c) Do you agree these two chords are =? Then, their lengths according to the distance formula must be the
same. This is the foundation of our derivation. The rest of the derivation contains Identities and Algebra.
AB = CD
( cos a − cos b ) + ( sin a − sin b=
)
2
2
( cos θ − 1) + ( sin θ − 0 )
2
2
− b ) cos a cos b + sin a sin b , we can find 3 more identities.
Using the Cosine of a Difference cos ( a=
2. Using cos ( a − b ) , let’s find cos ( a + b ) .
a) From Algebra, subtraction is defined as adding the opposite. Use this definition to rewrite cos ( a + b ) as
the difference of two angles.
− b ) cos a cos b + sin a sin b to what you wrote in part (a). Then, simplify using Evenb) Now apply cos ( a=
Odd Identities.
3. Using cos ( a − b ) , let’s find sin ( a + b ) .
 π − θ  to

2

rewrite the sine of an angle as a cosine function. Apply this identity to sin ( a + b ) . Let θ= a + b .
a) From Fundamental Identities Day 1, recall we can use the Co-Function Identity:=
sin θ cos 
b) Next, distribute your negative to write your expression as the cosine of the difference of two angles.
You will need to regroup.
− b ) cos a cos b + sin a sin b to what you wrote in part (b). Then, simplify using
c) Now apply cos ( a=
 π − θ  again.

2

=
sin θ cos 
4. Using sin ( a + b ) , let’s find sin ( a − b ) .
a) Again, subtraction is defined as adding the opposite. Use this definition to rewrite sin ( a − b ) as the sum
of two angles.
+ b ) sin a cos b + cos a sin b to what you wrote in part (a). Then, simplify using Evenb) Now apply sin ( a=
Odd Identities.
Pre Calculus
Worksheet 5.3
1. Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible.
 π  cos  π  − cos  π  sin  π 
  
   
5 7
5 7
a) sin 
tan
c)
() ()
() ()
π
− tan
2
1 + tan
tan
2
tan19° + tan 41°
1 − tan19° tan 41°
π
3
π
b)
π
d) cos 26° cos 94° − sin 26° sin 94°
3
2. Use a sum or difference identity to find the exact value for each function.
a) cos ( 75° )
 π 

 12 
c) cos  −
b) sin (195° )
 11π 

 12 
d) tan 
3. Simplify the following expressions as much as possible:
a) sin  x  6  
4. Prove the following identities
a) sin  x  y   sin  x  y   2 sin x cos y
b) cos  x  y   cos  x  y   2 cos x cos y
c) tan  x     tan   x   2 tan x
b) tan   4  
5. Use the function shown to answer the following questions.
a) Write a sine function that fits the graph.
b) Write a cosine function that fits the graph.
c) Use identities to PROVE your answers from part a and b are the same.
6. Use the function shown to answer the following questions.
a) Write a tangent function that fits the graph.
b) Notice this graph could also be the graph of the parent _____________________ function.
c) Use identities to PROVE your answer from part a is the same as the parent function in part b.
Questions 7-8 review concepts covered in Unit 5 and extend these concepts to Unit 6…
7. Use the given information to evaluate the identity. HINT: you do NOT need to find the angles first!!
a) If sin A =
b) If tan x =
3
5
when A is in Quadrant I and if cos B =
5
12
5
13
when x is in Quadrant III and if cos y =
when B is in Quadrant IV, evaluate cos ( A − B ) .
−3
5
when y is in Quadrant II, evaluate sin ( x + y ) .
8. Write each trigonometric expression as an algebraic expression.
a) sin arcsin x  arccos x 
b) cos sin 1 x  tan 1 2 x 
In preparation for lesson 5.4…
9. Prove the following identities…HINT: Start with the left side.
a) sin 2 x  2 sin x cos x
b) cos 2 x  cos 2 x  sin 2 x
PreCalculus
Worksheet 5.4
For questions 1 and 2, write as the function of one angle. Simplify, if possible, without using a calculator.
π 
π 
 cos  
6
6
2
1. 1 − 2sin (15° )
2. 2sin 
For questions 3 – 5, suppose sin A =
3. cos (2A)
3
and A is an angle in the first quadrant, find each value.
5
4. tan (2A)
For questions 6 – 8, if tan y =
6. sin (2y)
5. sin (2A)
5
and y is an angle in the third quadrant, find each value.
12
7. tan (2y)
8. cos (2y)
For questions 9 – 14, prove each identity. Use a separate sheet of paper if necessary.
9. sin  2 A 
2 tan  A
1  tan 2  A
10. sin  2 x   2 cot  x  sin 2  x 
11. cot  x  
sin  2 x 
1  cos  2 x 
12. sin  2 x   cot  x   tan  x   2
13. csc  x  sec  x   2 csc  2 x 
14. cos  4 x   1  8sin 2  x  cos 2  x 
15. sin 3u   3cos 2 u sin u  sin 3 u
16. cos 3 x   cos x  2 cos  2 x  cos x
PreCalculus
PreRequisites for Solving Trig Equations
Graph the equation y = sin x and y  1 2 on your calculator.
1. How many times do these two graphs intersect?
2. On the interval  0, 2  , how many solutions does the equation sin x  12 have? Find them in terms of  .
3. Find the solutions to the following equations on the interval  0, 2  … a sketch of the parent function may help you
determine how many solutions each equation has.
a) cos x 
2
2
d) sin x  
3
2
g) tan x  0
b) sin x  1
c) tan x  1
e) cos x   12
f) cos x  0
h) sin x 
i) tan x   3
2
2
Factor the following trigonometric expressions.
4. cos 2 x  2 cos x  1
5. 1  2 sin x  sin 2 x
6. sin x  sin x cos x
7. 4  cos 2 x
8. sec 2 x  sec x
9. 2 cos 2 x  5 cos x  7
Pre Calculus
Worksheet: Solving Trigonometric Equations
For questions 1-6, solve each equation on the interval [0, 2π).
1. 2 cos x + 5 =
4
2. 2sin x tan x − 2sin x =
0
3. cos 2 x + sin x =
1
4. 2sin 2 x − 5sin x + 2 =
0
5. 2 cos 2 x + cos x =
0
2
6. 2 cos
=
x cos x + 1
7. Solve for x on the domain [0, 2π ) : cos x =
1
2
8. Solve for x on the domain (−∞, ∞) : cos x =
9. Explain the difference in your solutions for questions 7-8.
1
2
Find all solutions to each equation.
10. 4 cos x − 3 =
2 cos x
11. sin 2 x − 2sin x =
3
12. sin 2 x − 3sin x + 2 =
0
13. cos 2 x = 1
14. 2 cos 2 x − 4 =
7 cos x
15. sin x − sin x cos x =
0
16. 3sin 2 x − cos 2 x =
0
17. sin 
 3π

1
− x =
 2
