Download 2. - Farmingdale School District

MATH 4H
TRIGONOMETRY 2
HOMEWORK
NAME__________________________
DATE___________________________
HW # 50:
Inverse Trigonometric Functions (Packet pp. 5 – 6) – ALL
HW # 51:
Finish Evaluating Using Inverse Trig Functions (Packet p. 7)
Solving Linear Equations (Packet p. 8) – ALL
HW # 52:
Solving First Degree Equations (Packet p. 10) – ALL
HW # 53:
Second Degree Trigonometric Equations (Packet p. 12) - # 1, 4, 5
HW # 54:
Quadratic Trig Equations (Packet p. 13) - # 2, 3, 4, 6
HW # 55:
Trigonometric Identities (Packet p. 15) - # 1, 3, 8
Proving Trigonometric Identities (Packet p. 16) - # 4
HW # 56:
Trigonometric Identities (Packet p. 15) - # 4 – 7
Sum and Difference Identities (Packet p. 18) - # 1, 11
HW # 57:
Using Pythagorean Identities to Solve Quadratic Equations (Packet p. 19) - # 3, 5, 7
HW # 58:
Trigonometry Trig 2 Test Review (Packet pp. 21 – 22)
Study for Test!!!
Inverse Trigonometric Graphs
On the axes below, sketch a graph of
sin
on the interval 2 2 .
sin
x
y
On the axes below, sketch a graph of
.
State the range and domian of this function. ________________________________
y
x
1
On the axes below, sketch a graph of
cos
on the interval 2 2 .
cos
x
y
On the axes below, sketch a graph of
.
State the range and domian of this function. ________________________________
y
x
2
On the axes below, sketch a graph of
tan
on the interval 2 2 .
tan
x
y
On the axes below, sketch a graph of
.
State the range and domian of this function. ________________________________
y
x
3
Domain Restrictions on Inverse Trigonometric Functions
4
Inverse Trigonometric Functions
1)
2)
5
6
Evaluating Using Inverse Trig Functions
1. If
cos
2. If
sin
√
, what is the measure of angle ?
, find
∠ .
3. What is the smallest positive value of
4. If
tan
6. If
cos
(1) 30°
1 , find
(2) 60°
7. The value of
cos
√
(3)
sin
11. cos
(2)
13.
sin
tan 1
17. sin
sin 1
18. If 3 sin
√
(3) 30°
is
(4) 60°
12. cos
14. 2
16. sin
tan 1
sin 1
sin
cos 1
1, then which is an expression for
(1)
sin 3
(2)
sin
19. If 5 sin
tan 1 ?
tan 1
15. tan
∠ .
(4) 150°
√
sin
, find
(4)
, the value of
cot
(1)
√
, what is the value of tan ?
9. What is the value of sin
10. If
sin
measures
(3) 120°
(2)
8. If
5. If
1 is
sin
(1) 1
∠ .
, then
cos ?
that satisfies
(3)
3
3, express
in inverse trigonometric form?
sin
(4)
sin
as an inverse of a trigonometric equation.
7
Solving Linear Equations
1. 2
1
0
3. 4
1
2
5. 3
2
7. 3
√3
9.
11. 3
12
5
1
2
2
7
2. 3
√3
0
4. 5
1
5
6.
√2
2√2
8. 5
3
11
10.
√2
9
12. 4
8
3
1
5
√
2
2
Linear Trigonometric Equations
Find the exact solution set of each equation if °
1. 2
1
0
3. 4
1
2
5. 3
2
1
2
7
Find the exact values for
7. 3
9.
11. 3
12
5
2
2. 3
√3
0
4. 5
1
5
√2
6.
in the interval
√3
11
9
°.
2√2
.
8. 5
3
10.
√2
12. 4
9
3
1
5
√
2
2
Solving First Degree Equations
1-5, solve for  in the interval 0    360 .
(Express your answer to the nearest degree)
1. 2sin   3  0
R:
Q:
A:
To Find an Angle
in Quadrant:
I
A
In Degrees
(Reference Angle of θ)
II
S
180  
III
T
180  
IV
C
360  
2. 3cos   1  1
3. 3tan   2  tan 
4. 2(sin   1)  sin   3
5. 10(cos   1)  30

6-7, solve for  in the interval 0    2 .
6. sec   6  3sec
7. 3(sin   2)  1  5sin 
10
Solving First Degree Equations Continued
Directions:
In 1-3, solve for θ in the interval 0    360 .
(Express your answer to the nearest degree)
In 4-6, solve for θ in the interval 0    2 .

*Hint Multiply Answer by
180
4. 2(sin   2)  2
1. 2 tan   3  5
2. 4(csc   2)  csc   14
5. 3sec  
2
(3sec   3)
3
6. 4 cos   3  3

3
3. 6  cot  
  5cot   2 3
2 

11
Second Degree Trigonometric Equations
Remember: If you have difficulties factoring, you can use the quadratic formula.
The Quadratic Formula
b  b 2  4ac
x
2a
Solve for θ on the interval 0    360 .
2. tan  (tan   1)  tan   3
1. 2 cos   cos   1
2
Solve for θ on the interval 0    2 .
3. cos  
5.
1
cos 
2
3
0
12
4. 2
3
1
0
6. 8
2
1
0
Quadratic Trig Equations
Find the exact solution set of each equation if °
1.
3
0
2. 2
3. 2
6. 2
1
0
Find the exact values for
4.
°.
1
0
in the interval
.
5.
0
13
0
Second Degree Trigonometric Equations
2
  cos   0 when 90    180 .
1.
Solve for all values of cos
2.
2
What is the value of θ in the interval 0  x  360 that satisfies the equation cos x  3sin x  1 ?
3.
Solve the equation sin x  2 cos x  2 on the interval 0  x  90 .
4.
Find, to the nearest degree, all values of θ in the interval 0    180 that satisfy the equation
4 tan 2   3 tan   2  0 .
5.
Find all values of x in the interval 0  x  360 that satisfy the equation 2 sin 2 x  sin x  2  3
Express your answers to the nearest degree.
2
6. Solve the equation tan x sin x  2 tan x in the interval 0
2
14
2 .
Trigonometric Identities
Pythagorean Identities
Double Angles
1
sin 2
cos 2
cos 2
cos 2
1
1
2 sin cos
2
1
1
2
2
Notes:
1. If 2. If
3. If 1
1
1, then ______________________________________
and ______________________________________
, then
_____________________________________
, then
_______________________________________
Express in simplest form:
1)
2)
3)
4)
5)
6)
7)
sin 2
9)
1  cos 2
8)
15
Proving Trigonometric Identities
Ex. Prove each identity
1. sin 2 sec  2 sin 
2. sin  
sin 2
2 cos 
3.
sin 2
 tan 
1  cos 2
4. sin 2 csc   2 cos 
5.
cos 2
cot 
 sin  
sin 
sec 
6. tan   cot  
16
2
sin 2
7. sin 2 
2 tan 
1  tan 2 
2
8. sin 2 sec   2 tan 
2
9. cos   sin    1  sin 2
2
10. cos   sin    1  sin 2
2 sin 2 
 cot   sec  csc 
11.
sin 2
12.
17
cos 2
 sin   csc   sin 
sin 
Sum and Difference Identities
sin
sin cos
cos sin
cos
cos cos
sin sin
sin
sin cos
cos sin
cos
cos cos
sin sin
Use the angle sum identity to find the exact value of each.
1) cos 105°
2) sin 195°
3) cos 195°
4) cos 165°
5) cos 285°
6) cos 255°
Use the angle difference identity to find the exact value of each.
11) cos 75°
12) cos −15°
13) tan 75°
14) cos 15°
15) tan −105°
16) sin 105°
18
Using Pythagorean Identities to Solve Quadratic Equations
Solve the following equations;
2
1 2
∈
,
2
2) 2 cos x  3 sin x  3  0
0
2
3) 3 cos x  5 sin x  4
2
5) 2
7)
2
9) 3
2
4
4) 2
cos
6) 2
3
8) 2
3
10)
19
3
2
1
3
2
0
Using Double Angle Identities to Solve Quadratic Equations
1 2
3)
2
5)
2
7)
2
9)
2
3
1
2) 3
0
0
2
2
4)
2
0
6)
2
0
2
8)
1
10)
20
2
0
2
Trigonometry Test2 Review 1. Graph 2. Graph 3. Graph 4. Which of the following are not an inverse trigonometric functions? 5. Use the angle sum identity to find the exact value of each. a) 75° b) 105° 6. Use the angle difference identity to find the exact value of each. 75° a) 15° b) 7. Simplify the following expressions: a) 2
b) 21
8. Prove the identity Solve the following equations for x in the interval 0
2 9. 4
2 2
10. 2
3 0 11. 2
1 0 13. 2
3
3
12. 2
0 15. 2
2
0 14. 22
1
0 0 23
Trigonometric Identities
Reciprocal Identities
sin
cos
csc
sec
sin csc
1
1
cos
cos sec
Pythagorean Identities
1
Ratio Identities
tan
tan
cot
cot
tan cot
1
1
csc
Double Angles
2 sin cos
cos 2
cos 2
2
cos 2
1
1
Cofunctions Identities
1
sin 2
cos
sin
cos 90°
cos
sin 90°
tan
cot 90°
sec 90°
sec
csc 90°
cot
tan 90°
1
2
Sum & Difference Identities
sin
sin cos
cos sin
cos
cos cos
sin sin
sin
sin cos
cos sin
cos
cos cos
sin sin
24