B
C
sin A   ______
csc A   ______
cos A   ______
sec A   ______
tan A   ______
cot A   ______
--------------------------------------------------
B
C
A
sin B   ______
csc B   ______
cos B   ______
sec B   ______
tan B   ______
cot B   ______
A
Trigonometric Functions
sine
Pythagorean Triples
A=
cosine
A=
tangent
A=
-----------------------------------------------Reciprocal Trig Functions
cosecant
secant
cotangent
A=
A=
A=
1
Finding a Side of a Right Triangle
B
9
C
x
15
x
A
40
C
A
12
B
x
C
B
1
2
6
A
C
x
B
2
A
---------------------------------------------------------------------------------------------------Draw and label right triangle ABC, with the following ratios.
sin A  
9
41
tan A  
17
15
cos A  
5
8
csc A  
10
4
2
Practice / Homework
Referring to the diagram, give the value of each trigonometric ratio.
1.
sin  P   ________
2.
sin Q   ________
3.
cos  P   ________
4.
tan  P   ________
Q
3
R
4
P
5.
cot Q   ________
6.
sec  P   ________
7.
csc  P   ________
8.
tan Q   ________
5.
cot Q   ________
6.
sec  P   ________
7.
csc  P   ________
8.
tan Q   ________
Give the value of each trigonometric ratio.
1.
sin  P   ________
2.
sin Q   ________
3.
cos  P   ________
4.
tan  P   ________
Q
13
R
12
P
Give the value of each trigonometric ratio in simplest radical form.
1.
sin  P   ________
2.
sin Q   ________
3.
cos  P   ________
4.
tan  P   ________
Q
3
R
2
P
5.
cot Q   ________
6.
sec  P   ________
7.
csc  P   ________
8.
tan Q   ________
3
Referring to the diagram, give the value of each trigonometric ratio.
1.
sin  P   ________
2.
sin Q   ________
3.
cos  P   ________
4.
cos Q   ________
Q
1
R
P
3
5.
tan  P   ________
6.
tan Q   ________
7.
csc  P   ________
8.
cot Q   ________
Referring to the diagram, give the value of each trigonometric ratio.
Q
9. sin  P   ________
13. tan  P   ________
2
1
10.
sin Q   ________
11.
cos  P   ________
12.
cos Q   ________
R
14.
tan Q   ________
15.
csc  P   ________
16.
cot Q   ________
P
Draw and label right triangle, ABC, with the following ratios.
17.
sin A  
15
17
18.
tan A  
6
4
4
Finding an Acute Angle of a Right Triangle
Draw and label a right triangle whose sin   
1
. Find the measure of  .
2
We can find  , by using our inverse keys.
sin 1  
cos 1  
Draw and label a right triangle whose cos   
tenth.
Draw and label a right triangle whose tan   
degree.
tan 1  
5
. Find the measure of  , to the nearest
13
40
. Find the measure of  , to the nearest
30
5
Practice
Use your calculator to find  , 0    90 . Round your answers to the nearest degree. Draw
and label the triangle. Find the missing side to the nearest tenth.
cos   
1
3
  ________
sin   
5
8
  ________
tan   
8
13
  ________
sin   
2
5
  ________
tan   
11
10
  ________
cos   
6
8
  ________
tan   
3
7
  ________
cos   
1
4
  ________
sin   
4
10
  ________
6
SOH CAH TOA
sin A  
opposite
hypotenuse
cos A  
adjacent
hypotenuse
tan A  
opposite
adjacent
FIND THE MISSING ANGLE AND ROUND ALL ANGLES TO THE NEAREST DEGREE.
B
C
A
1. If BC = 15 and AC = 12, find
A.
2. If AB = 33 and BC = 8 , find
3. If BA = 15 and BC = 7, find
A.
4. If AB = 42 and AC = 20, find
5. If BC = 40 and AC = 28, find
B.
A.
B.
7
8
Finding a Missing Side of a Right Triangle using Trigonometric Ratios
Examples
FIND THE MISSING SIDE(S). ROUND ANSWERS TO THE NEAREST TENTH.
B
C
A
1. AB = 10,
B  43 Find CA.
2. AC = 18,
A  18 . Find BC.
3. AB = 35,
B  56 . Find BC.
4. AC = 23,
B  34 . Find AB.
5. AC = 22,
B  72 Find BC.
6. AC = 12,
A  39 . Find AB.
9
10
Applications of Right Triangle Trigonometry
Right Triangle Word Problems
1.
2.
3.
4.
5.
Read the problem carefully.
Draw and label the triangle.
Set up the equation.
Solve the equation.
Write a therefore statement.
Example 1
A boy who is flying a kite lets out 300 feet of string which makes an angle of 52 with the
ground. Assuming that the string is stretched taut, find, to the nearest foot, how high the
kite is above ground.
Example 2
Find, to the nearest degree, the angle which the sun’s rays make with the ground when a
flagpole 40 feet high casts a shadow 30 feet long.
Example 3
An airplane rises at an angle of 14 with the ground. Find, to the nearest 10 feet, the distance
it has flown when it has covered a horizontal distance of 1500 feet.
11
Example 4
The top of a 40-foot ladder which is leaning against a wall reaches the wall at a point 36 feet
from the ground. Find, to the nearest degree, the angle which the ladder makes with the wall.
Example 5
In an isosceles triangle ABC, AC and CB are each 15 inches. Angle A and angle B are both 55 .
Find the length of AB, to the nearest inch.
Example 6
In rectangle ABCD, diagonal AC, which is 20 inches in length, makes an angle of 35 with the
base AB.
a. Find AB, the base of the rectangle, to the nearest tenth of an inch.
b. Find CB, the altitude of the rectangle, to the nearest tenth of an inch.
12
Practice / Homework
1.
The taut string of a kite makes an angle with the ground of 60 degrees. The
length of the string is 400 feet. What is the height of the kite, to the nearest tenth?
2.
A ladder, 500 cm long, leans against a building. If the angle between the ground and
the ladder is 57 degrees, how far from the wall is the bottom of the ladder? Round the
answer to the nearest tenth.
3.
An isosceles triangle has sides length 5, 5, 6. Find the measure, to the nearest
degree, of each angle of the triangle. (Hint: Draw the altitude.)
4.
The sides of a rectangle are 25 cm and 8 cm. What is the measure, to the nearest
degree, of the angle formed by the short side and a diagonal of the rectangle?
5.
The lengths of a pair of adjacent sides of a rectangle are 14 and 22. Find, to the
nearest degree, the angle a diagonal makes with the shorter side.
13
6.
A kite is flying 115 ft above the ground. The length of the string to the kite is 150 ft,
measured from the ground. Find the angle, to the nearest degree, that the string
makes with the ground.
7.
An observation tower is 75 m high. A support wire is attached to the tower 20 m from
the top. If the support wire and the ground form an angle of 46 degrees, what is the
length of the support wire, to the nearest tenth..
8.
At a point 30 feet from the base of a tree, the angle formed with the ground looking to
the top measures 53 . Find, to the nearest foot, the height of the tree.
9.
The base of a rectangle measures 8 feet and the altitude measures 5 feet. Find to the
nearest degree, the measure of the angle that the diagonal makes with the base.
10.
In an isosceles triangle the vertex angle measures 64 degrees and each leg measures 10
inches, find, to the nearest tenth of an inch, the length of the altitude to the base.
14
Warm up
1.
How long, to the nearest foot, is the cable that supports the pole?
2.
How high, to the nearest foot, is the cliff?
3.
A bird rises 20 meters vertically over a horizontal distance of 80 meters. What is the
angle of elevation?
4.
The length of a water ski jump is 720 cm and the angle of elevation is 35 . Find the
height of the ski jump, to the nearest cm.
5.
How far below sea level, to the nearest meter, will a porpoise be if it swims 250 meters
at a 12 angle of depression?
15
Angle of Elevation and Angle of Depression
Object
Observer
Angle of
Elevation
Observer
Angle of
Depression
Object
Example 1
At a point on the ground 40 feet from the foot of a tree, the angle of elevation of the top of
the tree is 42 . Find the height of the tree, to the nearest tenth of a foot.
Example 2
Find, to the nearest degree, the angle of elevation of the sun when a vertical post 15 feet high
casts a shadow which is 20 feet long.
16
Example 3
An observer in a balloon, which is 2000 feet above an airport, finds that the angle of
depression of a steamer ship out at sea is 21 degrees. Find, to the nearest hundred feet, the
distance between the observer in the balloon and the steamer ship at sea.
Example 4
From the top of a lighthouse 160 feet high, the angle of depression of a boat out at sea is 24 .
Find, to the nearest foot, the distance from the boat to the foot of the lighthouse. (The foot
of the lighthouse is at sea level.)
Example 5
An airplane which had taken off from an airport traveled a ground distance (horizontal) of
3,660 feet. What is the angle of elevation from the point of take-off to the point when the
plane has traveled 4,150 feet through the air? Round to the nearest degree.
17
Practice / Homework
1.
A tree casts a 60 foot shadow. The angle of elevation is 30º. This is the angle at which
you look up to the top of the tree from the ground. What is the height of the tree?
2.
An observer is 120 feet from the base of a television tower which is 150 feet tall. Find,
to the nearest degree, the angle of elevation of the top of the tower from the point
where the observer is standing.
3.
From the top of a vertical cliff which is 40 meters high, the angle of depression of an
object that is level with the base of the cliff is 34º. How far is the object from the
base of the cliff, to the nearest meter?
4.
From the top of a cliff which is 450 feet above sea level, the angle of depression of a
boat out at sea is 24 degrees. Find, to the nearest foot, the distance from the top of
the cliff to the boat.
5.
The angle of elevation of the top of a flagpole from a point on the ground 30 meters
from the base of the flagpole is 18 degrees. What is the height of the flagpole, to the
nearest meter?
18
6.
7.
8.
An airplane is flying at an altitude of 1000 meters. From the plane, the angle of
depression to the base of a tree on the ground is measured as 15°. What is the distance
from the plane to the base of the tree, rounded to the nearest tenth of a meter?
From a 200 feet high cliff a boat is
noticed floundering at sea! The boat
is approximately 300 yards from
the base of the cliff. What is the
angle of depression, to the nearest
degree, of the line of sight to the
boat?
At 10:00am , a person observes a hot air balloon climbing vertically in the air from a
point 300 meters away from the launch pad for the balloon. The angle of elevation to
the top of the balloon at this time is 25o. At 10:02am, the person observes that the
angle of elevation to the balloon is now 60o. What is the change in altitude, to the
nearest meter, for the balloon over the 2 minutes between the first and second
observations?
60
25
300
19
‘Solving’ a Right Triangle
When asked to solve a right triangle, you need to find all of the missing angle measures and
lengths of the sides of the triangle.
Example 1 Using the diagram below, solve the right triangle. Round angle measures to the
nearest degree and segment lengths to the nearest tenth.
A
42
B
12
C
m A  ____
a  ____
m B  ____
b  ____
m C  ____
c  ____
Example 2 Using the diagram below, solve the right triangle. Round angle measures to the
nearest degree and segment lengths to the nearest tenth.
A
40
65
B
C
m A  ____
a  ____
m B  ____
b  ____
m C  ____
c  ____
Example 3 Using the diagram below, solve the right triangle. Round angle measures to the
nearest degree and segment lengths to the nearest tenth.
A
C
25
m A  ____
a  ____
m B  ____
b  ____
m C  ____
c  ____
15
B
20
Example 4 Solve the right triangle, if m C  90 , m B  48 , and AB  24 . Round angle
measures to the nearest degree and segment lengths to the nearest tenth.
Example 5 Solve the right triangle, if m C  90 , AC  15 , and BA  35 . Round angle
measures to the nearest degree and segment lengths to the nearest tenth.
Example 6 Solve the right triangle, if m C  90 , AC  23 , and BC  28 . Round angle
measures to the nearest degree and segment lengths to the nearest tenth.
21
Homework
Solve each of the following right triangles. Round angle measures to the nearest degree and
segment lengths to the nearest tenth. Angle C is the right angle of the triangle.
1.
BC = 10 and AC = 20.
2.
AB = 35 and BC = 12.
3.
AC = 15 and BC = 6.
4.
AB = 22 and AC = 10.
22
5. BC = 6, and measure angle B is 23 .
6. AC = 12 and measure angle A is 42 .
7. AB = 32 and measure angle B is 36 .
8. AC = 13 and measure angle B is 47 .
23
REVIEW
Use a calculator to find the measure of angle A to the nearest degree. Angle A is in quadrant I.
1. sin A  
4
9
4. sin A   0.5677
2. tan A  
8
13
5. tan A   1.5608
3. cos A  
9
11
6. cos A   0.2435
----------------------------------------------------------------------------------------------------In a right triangle ABC , m C  90 . Round angle measures to the nearest degree and
segment lengths to the nearest tenth. Don’t forget to draw the triangle!!!!
7.
A  43 , AB  13 , find BC.
8. AC  36, AB  57, find m A.
9.
A  13 , BC  23, find AB .
10. BC  19.4, AC  12.3, solve ABC .
11.
A  55 , AC  25, solve ABC .
24
12.
A boy visiting New York City views the Empire State building from a point on the
ground, A, which is 940 feet from foot, C, of the building. The angle of elevation of the
top, B, of the building as seen by the boy is 53 degrees. Find the height of the building
to the nearest foot.
13.
From an airplane which is flying at an altitude of 3000 feet, the angle of depression of
an airport ground signal is 27 degrees. Find to the nearest hundred feet the distance
between the airplane and the airport signal.
14.
A 30-foot steel girder is leaning against a wall. The foot of the girder is 20 feet from
the wall. Find to the nearest degree the angle which the girder makes with the ground.
25
15.
An airplane A is 1000 feet above the ground and directly over a church C. The angle of
elevation of the plane as seen by a boy at a point A on the ground some distance from
the church is 22 degrees.
a. How far, to the nearest foot, is the boy is from the church?
b. How far, to the nearest foot, is the boy from the plane?
16.
The base of a rectangle measures 20 feet and the altitude measures 14 feet. Find to
the nearest degree, the measure of the angle that the diagonal makes with the base.
26