Download Trig Review - Jack Nilan

Review of Right Triangle Trigonometry
Name:
UNIT 7:Trigonometry I
Do Now:
1. For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the
length is 12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9.
Notes:
Sin  =
Cos  =
Tan  =
1. Use the diagram below to answer the evaluate a-f.
F
a) Sin D
b) Cos D
c) Tan D
5
4
d) Sin F
e) Cos F
f) Tan F
D
3
E
2. In PQR , mP  90 , PR  5 and QR  13 . Find 3. In CAT , mA  90 , AT  8 and CT  10 . Find
sinC.
cosR.
4. Find x to the nearest tenth.
5. Find x to the nearest tenth.
12
40
x
x
32
50
pg. 0
6. Find x to the nearest degree:
3
7. A 10-foot tree is supported in a vertical position by
a rope running from the top of the tree to a stake in the
ground. If the angle that the rope makes with the
ground is 63o, find the length of the supporting rope to
the nearest foot.
x
4
In # 8 – 15, determine the quadrant in which an angle of the given measure lies.
8. 140o
9. -430o
10. 210o
11. 97o
12. 315o
13. -168o
14. -200o
15. 600o
In #16– 23, draw an angle of the given measure, indicating the direction of the rotation by the arrow. Find the
angle coterminal with an angle of the given measure.
16. 100o
17. -50o
18. 210o
19. -300o
20. 470o
21. -800o
22. -200o
23. 60o
24. Which pair of angles are coterminal?
(1) 30o and -30o
(2) 150o and -210o
(3) 200o and 160o
(4) -100o and -80o
pg. 1
Homework on Review of Right Triangle Trigonometry
1. Which ratio represents tan A in the diagram below?
(1)
(3)
(2)
(4)
2. In the diagram below of right triangle KTW,
,
, and
What is the measure of
(1) 33o
.
, to the nearest degree?
(2) 34o
(3) 35o
(4) 36o
3. Which pair of angles are coterminal?
(1) 40o, 140o
(2) -30o, -150o
(3) 310o, -50o
(4) 120o, 240o
In # 4 – 7, determine the quadrant in which the angle terminates.
4. -85o
5. 468o
6. -1000o
7. 219o
8. A ladder 10 feet in length leans against a tree, making an angle of 28o with the ground. How far from the
tree was the ladder placed, to the nearest hundredth?
9. Put the equation of the circle in center-radius form.
Determine the center and radius:
10. If f ( x)  x 2  2 x  11 & g ( x)  3x  2 ,
find f ( g ( x)) .
x 2  y 2  6 x  16 y  2  0
pg. 2
The Unit Circle, Signs of the Quadrants, and Quadrantal Angles
Do Now:
The conjugate of
is
(1)
(2)
(3)
(4)
Notes:
Signs of the Quadrants
Unit Circle:
Center =
Radius =
P (x, y) =
In #1-4, name the quadrants in which an angle of measure  could lie when:
1. cos  0
2. sin   0
3. tan   0
4. sin   0
In #5 – 10, name the quadrant in which an angle of measure  could lie when:
5. sin   0 and cos  0
6. sin   0 and cos  0
8. cos  0 and sin   0
9. cos  
3  10
and tan   0
5
7. tan  
5
and cos   0
6
10. sin   0 and tan   0
pg. 3
11. If P is a point that lies on the unit circle
and P(.6,  .8) , then find:
12. In the figure, a unit circle is drawn and
mCOA   . Name the line segment whose
length is:
a) sin 
y
a) sin 
B
A
b) cos
O
P
C
x
c) tan 
b) cos
c) tan 
Complete the following table, giving the functional value of each quadrantal angle. If a function is not defined
for an angle measure, write “undefined.”
A
Sin A
0o
90 o
180 o
270 o
360 o
Cos A
Tan A
In #13 - 18, find the exact numerical value of the expressions given below.
13. cos  270

16. sin 90 o  cos 0 o
tan 180 o
sin 90 o
14. (cos 0 o ) 2  tan 180o
15.
17. cos180 o  sin 270 o
18. sin  720   cos  180

pg. 4
HW on Unit Circle, Signs of Quadrants, Quadrantal Angles
1. If
is negative and
(1) I
2. If
is negative, in which quadrant does the terminal side of
(2) II
(3) III
and
(1) I
3. If
4. If
(1) I
5. If
(1) I and II
6. If
(1) I and II
(4) IV
lies in which quadrants?
(2) II and III
and
terminate?
(3) III
, then angle
(1) I and II
(4) IV
, in which quadrant does
(2) II
lie?
(3) I and III
(4) III and IV
, in which quadrant does angle x lie?
(2) II
(3) III
, in which quadrants may angle
(2) II and III
, then angle
(4) IV
terminate?
(3) I and III
(4) III and IV
may terminate in Quadrant
(2) II and III
(3) I and III
(4) III and IV
1
3
7. If P  , 
 lies on the terminal side of  of a unit circle, find:
2 
2
a) the quadrant  terminates in
b) the value of sin 
c) the value of cos
d) the value of tan 
pg. 5
8. An angle whose measure is -214o is coterminal with all of the following except
(1) -574o
(2) 34o
(3) 146o
(4) 506o
 5 12 
9. If  is an angle in standard position and P  ,  is a point on the unit circle on the terminal side of  ,
 13 13 
what is the value of tan  ?
(1) 
12
5
(2) 
5
13
(3) 
5
12
(4)
12
13
10. Find the solution set to the inequality and graph it on the number line below:
10  2 x  4  4
11. Solve for x and express your answer in simplest radical form:
4
3

7
x x 1
pg. 6
Special Angles of 30o, 60o, and 45o
Do Now:
1. If cos  
3 2
, in which quadrants can  terminate?
6
2. Simplify:
3
2 5
Notes:
60
2
45
2
1
1
45
30
1
3
Fill in the table below:

Sin 
30o
45o
60o
Cos 
Tan 
Find the exact numerical value of the expression.
1. sin 30 o  cos 60 o
2. (sin 60 o )(cos 60 o )
3. (tan 60 o ) 2
4. sin 45 o  cos 45 o
pg. 7
5.
cos 60 0 (cos 0 0  cos180 0 )
9. (cos 45o ) 2
6. 2  sin 30
 cos 30 
10. (tan 60 o )(tan 30 o )
13. If  is the measure of an acute angle and sin  
(1) cos  
3
2
(2) tan   1
7. (tan 2 45 )(sin 45 )
8. tan 45o  2 cos 60 o
11. (cos 60 o  2 tan 45o ) 2
12. (tan 60 0 ) 2  sin 180 0
3
, then:
2
(3) cos  
1
2
(4) tan  
3
3
14. If   30 o , which expression has the largest numerical value?
(1) sin 
(2) cos 
(3) tan 
(4) (cos  )(tan  )
(3) 45 o
(4) 60 o
15. If sin   3 cos  , then  can equal:
(1) 0 o
(2) 30 o
pg. 8
Special Angles Homework
In #1 – 6, find the exact value of each expression:
1. tan 30º + cos 60º
2. (tan 45º)(cos 45º)
3. cos 60º + sin 30º - tan 45º
4. sin2 45º • cos 45º
5. (cos 60º)2 +(sin 30º)2
6. (sin 0 0 )(tan 30 0 )  cos 0 0
7. If sin   0 and tan   0 , what might be the measure of  ?
(1) 81o
(2) 128o
(3) 207o
(4) 313o
8. Which of the following is a false statement?
(1)
(2)
(3)
(4)
Tan  is undefined whenever cos  equals zero.
Tan  equals zero whenever sin  equals zero.
Sin  can equal cos  in Quadrant I or Quadrant III of the unit circle.
Sin  can equal cos  only in Quadrant I of the unit circle.
9. If sin   3 cos  , then  can equal
(1) 0 0
(2) 30 0
(3) 45 0
(4) 60 0
10. The value of 2(sin 30 0 )(cos 30 0 ) is equal to the value of
(1) sin 60 0
(2) cos 60 0
1
a
11. Simplify:
1
1 2
a
(3) sin 90 0
1
12. Solve for x:
(4) tan 30 0
x2  x
pg. 9
Reference Angles
Do Now:
1. Which angle is coterminal to -400o?
(1) 40o
(2) -40o
(3) 140o
(4) -220o
Notes:

What is a reference angle??
In #1-8 write the reference angle for each given angle.
1. 100o
2. 340o
3. 248o
4. 98o
5. 200o
6. 45
7. 745o
8. 460
In 9 – 16, express the given function as a function of a positive acute angle.
9. sin100
10. cos150
11. tan 300
12. cos190
13. tan145
14. sin 650
15. cos(200 )
16. tan(80 )
19. tan(120 )
20. cos 570
In 17 – 20, find the exact function value.
17. cos 315
18. sin(135 )
pg. 10
HW on Reference Angles
In # 1 – 4, find the EXACT value:
1. sin120
3. tan  225
2. cos 315

4.
 cos 390  sin  270 
2
In # 5 – 8, express each function as a function with a positive acute angle.
6. tan  50
5. sin 320

7. cos 560
8. sin 400
 1
3
9. If P   , 
 lies on the terminal side of  on a unit circle, what can  be?
2
2


(1) 60o
(2) 120o
(3) 210o
(4) 240o
(3) 228o
(4) -228o
10. What is the reference angle for -132o?
(1) -48o
(2) 48o
11. If f ( x)  sin 2 x  cos x , evaluate:
a) f 180

b) f  30
12. Is r directly or inversely proportional to s?
13. If z1  2  5i and z2  5  2i , find and
graph z1  z2 . Evaluate z1  z2
Find x.
r
s

4
6
12
2
x
3
pg. 11
Converting to and from Radian Measure!!
Do Now:
1. For the angle -500o, find:
a) the reference angle
2. Given the diagram below, circle O is a
unit circle. Find the line segment that
represents:
b) an angle co-terminal to it
y
a) sin 
B
A
b) cos
O
P
C
x
c) tan 
CONVERTING:
In 1-8, convert each degree measure radian measure in terms of  .
1. 90
2. 45
3. 60
4. 30
5. 120
6. 150
7. 75
8. 160
In 9 – 16, find the degree measure of an angle of the given radian measure.
9.

3
13. 
10. 
5
6
14. 

9
11.

10
15. 2.35 radians
12. 

2
16. 3.14 radians
pg. 12
HW on Converting to and from Radian Measure!!
In #1 – 5, convert the following angles to radian measure.
1. 270
2. 180
3. 240
4. 620
5. 50
In #6 – 10, convert the following angles to degree measure.
6. 2
7. 
7
4
8.
9. 
5
3
10.

2
11. Convert 5.21 radians to degrees. Round your answer to the nearest thousandth.
12. What is the domain of y 
(1) 3  x  3
1
9  x2
?
(2) 3  x  3
(3) x  3 or x  3
(4) x  3 or x  3
13. If one of the roots to the equation x 2  kx  6  0 is 3, find k.
(1) 1
(2) -1
(3) -2
(4) 5
14. If f ( x)  x  2 and g ( x)   x 2  3 , find the value of f  g (1)  .
15. What type of roots will the equation x 2  4 x  8 have?
(1) real, rational, and equal
(2) real, rational, and unequal
(3) real and irrational
(4) imaginary
16. Solve for x by completing the square. Place your answer in simplest radical form.
2 x 2  12 x  6  0
pg. 13
Evaluating Trigonometric Functions!!
Do Now
1. If P(0.6, 0.8) lies on the terminal side of  on the unit circle, find:
c) cos
b) tan 
a) sin 
2. Simplify: 2 3 16a7bc24
3. Simplify and express with positive exponents:
25a 6bc 1
15ab8c 3
Evaluating in Radians and Degrees!
In 1-8, find the exact value of the trigonometric function.
s

1. cos
2. tan  45 
3
5. sin  225

6. tan
5
6
3. sin
2
3
7. cos  225
4. cos

4
3
 
8. tan   
 2
9. If a function f is defined as f ( x)  cos 3x find the numerical value of:
 
a) f  
2
b) f ( )
c) f (45 )
d) f (30 )
In 10 – 12, find the numerical value of f ( ) for the given function f.
x
10. f ( x)  tan  
3
11. f ( x)  sin x  cos 2 x
12. f ( x)  sin x cos 2 x
pg. 14
HW on Evaluating Trigonometric Functions in Radians and Degrees!
In #1 – 4, find the exact value of each.
2. cos  30
1. tan3
In 5-7, find the numerical value of f  60

3. sin
5
3

 for the given function f.
6. f ( x)  sin 12 x  cos5x
5. f ( x)  sin 2 x
4. sin  510
7. f ( x)  tan x tan 2 x

8. An angle whose measure is 3 has the same terminal side as an angle whose measure is
(1)
2
3
(2)
5
3
(3) 

(4) 
3
5
3
9. Which expression is undefined?
(1) sin

2
(2) cos

2
(3) tan

2
(4) tan

4
10. If sin x > 0 and cos x < 0, x must be the measure of an angle in Quadrant
(1) I or II
(2) II or IV
(3) I or IV
(4) II or III
In #11-13, write each expression as a function of a positive acute angle (reference angle).
11. sin138 o
12. cos490 o
 3 
13. tan 

 4 
14. Brianna lets out 300 feet of string while she is flying her kite. Assuming that the string is stretched taut and
makes an angle of 48 with the ground, find to the nearest foot the height of the kite above the ground.
pg. 15