Download Trigonometry – Exact Value, Laws, and Vectors

Trigonometry – Exact Value, Laws, and Vectors
Conversions
Arc length and Area of a Sector
Arc length: s   r
 180 

  
  


 180 
rad  deg  
deg  rad
1 2
r
2
Only when the angle is in RADIANS!
Unit Circle
Area of a sector: Area 
Coterminal Angles
  360 n , n  
Reference Angles
 0,1
sin A
0


y
2
2


 y
2
2

Area of a Triangle
1
K  ab sin C
2
a
3
3
0y
Arc tan T an1 
Law of Sines
1 3
1
 ,

3  2 2 
  2 2


45 ,
,
4  2 2 
  3 1

30 ,
, 
6  2 2 

Circle with radius, r r cos , r sin  
Inverse Trigonometry
Arc sin  S in1 
60 ,
1, 0 
Degree-Minutes-Seconds
2nd APPS (Angle) ENTER
4: DMS
Arc cos  C os1 
3
undefined

b
sin B
[SAS]

c
sin C
[AAS, ASA, *SSA]
*used to find # of triangles
that can be formed
(0, 1, or 2)
Law of Cosines
a 2  b 2  c 2  2bc cos A [SAS, SSS]
-----------------------------------------------Vector Problems
construct a parallelogram and remember
properties of parallelograms
- opposite sides congruent
- opposite angles congruent
- consecutive angles supplementary
- diagonal creates alternate interior
angles which are congruent
- only the diagonals of a square or
rhombus bisect the angle
Conversions and Exact Value
1.
What is the radian measure of the smaller angle formed by the hands of a clock at
7 o’clock?

2
5
7
(1)
(2)
(3)
(4)
2
3
6
6
2.
Express 405º in radian measure.
3.
Find, to the nearest tenth of a degree, the angle whose measure is 2.5 radians.
4.
If sin A  0 and cot A  0 , in which quadrant does the terminal side of A lie?
5.
Find the exact value of each of the following.

sin 240
6.


cos 315


tan 150

csc  4 
If f  x   sin x  cos2x , then f   is
(1) 1
(2) 2
(3) 0
(4)
1
7.
 
If f  x   sin2 x  cos2 x , find f   .
4
8.
The accompanying diagram shows unit circle O, with radius OB  1 .
Which line segment has a length equivalent to cos ?
(1) AB
(2) CD
(3) OC
(4) OA
9. If is an angle in standard position and its terminal side passes through the point
, find the exact value of
.
10. Express
11. Express
as a function of a positive acute angle.
as a function of a positive acute angle.
12 Evaluate each of the following:
 2
a.
Cos 1 

 2 


d. sin Cos 1  1  =
 1 
b. Sin 1   
2

3
c. T an 1  

 3 


7 

e. cos  Arcsin

25 

Arc Length / Area of a Sector
13.
An arc of length 30 inches is drawn in a circle with radius 12 inches.
(a) Find, to the nearest integer, the degree measure of the arc.
(b) Find, to the nearest integer, the area of the sector formed by this arc.
14.
A sprinkler system is set up to water the sector shown in the accompanying
diagram, with angle ABC measuring 1 radian and radius AB  20 feet.
What is the length of arc AC, in feet?
(1) 63
(2) 31
(3) 20
15.
(4) 10
The accompanying diagram shows the path of a cart traveling on a circular track of
radius 2.40 meters. The cart starts at point A and stops at point B, moving in a
counterclockwise direction. What is the length of minor arc AB, over which the
cart traveled, to the nearest tenth of a meter?
16.
17.
In a circle with a radius of 4 centimeters, what is the number of radians in a
central angle that intercepts an arc of 24 centimeters?
The pendulum of a clock swings through an angle of 2.5 radians as its tip travels
through an arc of 50 centimeters. Find the length of the pendulum, in centimeters.
Area & Laws
18.
The three sides of a triangle have lengths 23, 25, and 40.
(a) Find, to the nearest degree, the measure of the largest angle of the triangle.
(b) Find, to the nearest integer, the area of the triangle.
19.
In KLM, KL  100, mK  40, and LM  80 . Explain why there are two possible
triangles with these measures.
20. In ABC , m A  30, a  12, and b  10. Explain why there is only one possible
triangle with these measures.
21.
In ABC , m A  75, m B  40 and b  35 . What is the measure of side c?
(1)
(3)
35 sin 40
sin 65
35 sin 40
sin 75
(2)
(4)
35 sin 75
sin 40
35 sin 65
sin 40
4
3
, sin B  , and a  16 . Find b.
5
4
22.
In triangle ABC, sin A 
23.
In ABC , m A  30, a  12, and b  10. Which type of triangle is ABC ?
(1) acute
24.
(2) isosceles
(3) obtuse
(4) right
In the accompanying diagram of triangle RST, m R  17 20', RT  40, and
mT  34 50'.
What is the length of RS to the nearest integer?
25.
The accompanying diagram shows the plans for a cell-phone tower that is to be
built near a busy highway. Find the height of the tower, to the nearest foot.
26.
A ship at sea heads directly toward a cliff on the shoreline. The accompanying
diagram shows the top of the cliff, D, sighted from two locations, A and B,
separated by distance S. If mDAC  30, mDBC  45 and S  30 feet, what is
the height of the cliff, to the nearest foot?
27.
An airplane traveling at a level altitude of 2050 feet sights the top of a 50-foot
tower at an angle of depression of 28 from point A. After continuing in level
flight to point B, the angle of depression to the same tower is 34 . Find, to the
nearest foot, the distance that the plane traveled from point A to point B.
28. To the nearest degree, what is the measure of the largest angle in a triangle with
sides measuring 10, 12, and 18 centimeters?
(1) 109
(2) 81
(3) 71
(4) 32
29.
Peter (P) and Jamie (J) have computer factories that are 132 miles apart. They
both ship their completed computer parts to Diane (D). Diane is 72 miles from
Peter and 84 miles from Jamie. Using points D, J, and P to form a triangle, find
m PDJ to the nearest ten minutes or nearest tenth of a degree.
Vectors
30.
Two forces of 40 pounds and 20 pounds, respectively, act simultaneously on an
object. The angle between the two forces is 40°.
(a) Find the magnitude of the resultant, to the nearest tenth of a pound.
(b) Find the measure of the angle, to the nearest degree, between the resultant
and the larger force.
31.
Two forces act on a body to produce a resultant force of 70 pounds. One of the
forces is 50 pounds and forms an angle of 67 40' with the resultant force. Find,
to the nearest pound, the magnitude of the other force.
32.
One force of 20 pounds and one force of 15 pounds act on a body at the same point
so that the resultant force is 19 pounds. Find, to the nearest degree, the angle
between the two original forces.
33.
A jet is flying at a speed of 526 miles per hour. The pilot encounters turbulence
due to a 50-mile-per-hour wind blowing at an angle of 47°, as shown in the
accompanying diagram.
`
Find the resultant speed of the jet, to the nearest tenth of a mile per hour. Use
this answer to find the measure of the angle between the resultant force and the
wind vector, to the nearest tenth of a degree.
Answers – Trig Day 1
1.
2.
9
4
6.
(1) 1
3.
143.2
4.
Quadrant III
7.
1
8.
10.
tan 50 
11.
 sin 10 
13.
(a) 143
(b) 180
17.
20 cm
14.
(3) 20
15.
6.9 meters
12.
135 ,  30 ,
24
30 , 0,
25
16.
6 radians
18.
(a) 113
(b) 265 units sq.
21.
(4)
25.
88 feet
29.
115 20' or 115.4
22.
b  15
26.
41 feet
30.
(a) 56.8 pounds
(b) 13
19.
Using Law of Sines
there are 2 possible
angles and both are
possible.
23.
(3) obtuse
27.
796 feet
31.
69 pounds
20.
Using Law of Sines
there are 2 possible
angles, but only one
is possible.
24.
29
28.
(1) 109
32.
116
(3)
5.
5
6
3
2
,
,
2
2
3

, undefined
3
9.

13
2
33.
561.3 mph
43.3
(4) OA