Download Chapter 7 Trigonometry of Right Triangles 7.1 – The Pythagorean

1 Chapter 7
Trigonometry of Right Triangles
7.1 – The Pythagorean Theorem
7.2 – The Sine Ratio
7.3 – The Cosine Ratio
7.4 – The Tangent Ratio
7.5 – Solving Right Triangles
37
Name: ________________________________
Math 10 Workplace
Marsh
2 Math 10 Workplace
Marsh
3 7.1 – The Pythagorean Theorem
A. Labelling Triangles
The corners of triangles are labelled using _____________ case letters
and the corresponding sides are labelled using __________ case letters.
Example 1: For ∆ABC label the sides.
A
C
B
The sides can also be labelled using the upper case letters.
Side a is also called side _________
Side b is also called side _________
Side c is also called side _________
Worksheet: Labelling Triangles
Math 10 Workplace
Marsh
4 Labelling Triangles Worksheet
1) Label each side of the triangles below using a sing lower case letter corresponding to
the opposite vertex (angle).
2) Label each vertex of the triangle below using a single upper case letter corresponding
to the opposite vertex.
Math 10 Workplace
Marsh
5 B. The Pythagorean Theorem
A right triangle is characterized as having one right angle.
Directly across from the right angle is the longest side of the
triangle and is called the ___________________.
The relationships between the hypotenuse and the remaining two arms of a right
triangle is stated using the ________________________________.
Example 2: For ∆PQR solve for the unknown side, q.
Example 3: For ∆XYZ solve for the unknown side, z.
Worksheet: The Pythagorean Theorem
Math 10 Workplace
Marsh
6 The Pythagorean Theorem Worksheet
1) A ladder, ℓ, is placed against the side of a house, h. The foot of the ladder is a distance
d from the base of the house. Draw a diagram and express the relationship that exists
between ℓ, h, and d.
2) Rearrange the Pythagorean Theorem to solve first for x and then for y.
x2  y2  z2
3) Find the missing side in each triangle. Round answers to nearest tenth if necessary.
a)
c)
Math 10 Workplace
b)
d)
Marsh
7 4) Calculate the values of x and y.
5) A 40 foot ladder reaches 38 feet up the side of a house. How far from the base of the
house is the foot of the ladder? Draw a diagram.
6) A field is 120 m by 180 m. How much shorter is your route if you walk diagonally
across the field rather than walking around the edge to the opposite corner? Draw a
diagram.
7) A stairway rises 6 feet 4 inches over a horizontal distance of 8 feet 6 inches. What is
the diagonal length of the stairway? Draw a diagram.
Math 10 Workplace
Marsh
8 8) A 28-metre long guy wire is attached to a point 24 m up the side of a tower. How far
from the base of the tower is the guy wire attached? Draw a diagram.
9) The construction plans for a ramp show that it rises 3.5 metres over a horizontal
distance of 10.5 metres. How long will the ramp surface be? Draw a diagram.
10) The advertised size of a TV screen is the distance between opposite corners. Sally
bought a 52-inch TV. If the height of the TV is 32 inches, how wide is it? Draw a
diagram.
11) A boat sailed due north at a rate of 12 km/h for 3 hours, then due east at a rate of 18
km/h for 2 hours. How far was it from its starting point, measuring the shortest
distance? Draw a diagram.
Answers:
3) a) 5 in b) 5 mi c) 8.6 km d) 14.1 mi 4) x = 8.0 cm , y = 6.7 cm 5) 12.5 ft
6) 83.7 m shorter 7) 10.6 ft 8) 14.4 m 9) 11.1 m
10) 41 in 11) 50.9 km
Math 10 Workplace
Marsh
9 7.2 – The Sine Ratio
A. Sine Ratio Introduction:
Another relationship exists between the ratio of the length of the side opposite a given
angle to the length of the hypotenuse. In two similar triangles this ratio will always be
the same. It is known as the __________________________.
sin  
length of side opposite 
length of hypotenuse
sin  
Example 1: Use your scientific calculator to determine the following sine ratios.
Round to four decimal places.
Note: make sure your calculator is in degree mode.
a) sin 10º
c) sin 50º
b) sin 45º
d) sin 70º
Example 2: Calculate the angle to the nearest degree.
a) sin A = 0.5164
Math 10 Workplace
b) sin B = 0.8461
Marsh
10 B. Labelling Triangles
In order to solve angles and side in right triangles, it is important to be able to label them
properly. The three sides are: ______________________, ______________________,
______________________,
The _________________ is always directly opposite the right angle and it is the longest
side.
The ______________ side is always next to the angle of interest.
The _______________ side is always directly across the angle of interest.
A
Example 3: Consider ∆ABC
θ
Label all of the sides given θ and ϕ.
ϕ
B
C
Worksheet: Introduction to Trigonometry
Introduction to Trigonometry Worksheet
1) Calculate the following sine ratios. Round to four decimal places.
a) sin 15 °
c) sin 80 °
b) sin 60°
d) sin 100 °
2) Calculate the value the angle to the nearest degree.
a) sin
0.5879
c) sin
b) sin
Math 10 Workplace
0.9994
d) sin
0.2635
0.4569
Marsh
11 3) For each triangle below, name:
(i) the hypotenuse
(ii) the side opposite the angle marked θ
(iii) the side adjacent the angle marked θ.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Answers: 1) a) 0.2588 b) 0.8660 c) 0.9848 d) 0.9848 2) a) 36° b) 88° c)15° d) 27°
Math 10 Workplace
Marsh
12 A
C. Three types of Trigonometry Questions:
1) Solving for the angle
Solve for A:
sin A 
11.2 ft
6.4 ft
opp
hyp
Steps:
B
7.3 ft
C
1) Label triangle according to the angle of interest (in this case A) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. For the angle you must use the inverse function of sine. (1 mark)
Practice:
a)
b)
Math 10 Workplace
Marsh
A
13 2) Solving for the top of the ratio
sin C 
opp
hyp
4.3 cm
c
46º
Steps:
B
C
1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. (1 mark)
Practice:
a)
b)
Math 10 Workplace
Marsh
A
14 3) Solving for the bottom of the ratio
sin C 
opp
hyp
b
6 ft
65º
Steps:
B
C
1) Label triangle according to the angle of interest (in this case C) – hyp, opp, and adj.
2) Write down required formula (1 mark).
3) Fill in known values and leave unknown value (1 mark).
4) Solve. (1 mark)
Practice:
a)
b)
Worksheet: The Sine Ratio
Math 10 Workplace
Marsh
15 The Sine Ratio Worksheet
1) Use your calculator to determine the value of each of the following sine ratios to four
decimal places.
a) sin 30°
b) sin 48°
c) sin 62°
d) sin 77°
2) Calculate the angle to the nearest degree.
a) sin D = 0.5491
b) sin H = 0.9998
3) Solve for the indicated angle in the following diagrams.
a)
b)
c)
Math 10 Workplace
Marsh
16 d)
e)
4) Find the opposite side in the following diagrams. Round answers to one decimal
place.
a)
b)
c)
d)
Math 10 Workplace
Marsh
17 5) Find the length of the hypotenuse, to one decimal place, in the following diagrams.
a)
b)
c)
d)
6) A rafter makes an angle of 28° with the horizontal. If the rafter is 15 feet long, what
is the height at the rafter’s peak? Draw a diagram.
Math 10 Workplace
Marsh
18 7) How high is a weather balloon that is tied to the ground if it is attached to a 15 metre
string and the angle between the string and the ground is 35°? Draw a diagram.
8) How long is a guy wire that is attached 4.2 metres up a pole if it makes an angle of
52° with the ground? Draw a diagram.
9) A boat is carried with the current at an angle of 43° to the shore. If the river is
approximately 15 metres wide, how far does the boat travel before reaching the
opposite shore? Draw a diagram.
Answers:
1) a) 0.5000 b) 0.7431 c) 0.8829 d) 0.9744 2) a) 33° b) 89° 3) a) 38.7° b) 39.8°
c) 43.4° d) 32.4° e) 50° 4) a) 10.8 cm b) 11.3 cm c) 2.8 mm d) 8.7 km 5) a) 199.8 mm
b) 8.6 m c)154.7 mm d) 25.6 mm 6) 7.04 ft 7) 8.6m 8) 5.3 m 9) 22m
Math 10 Workplace
Marsh
19 D. Angle of Elevation and Depression
Trigonometry is useful in many real life situations. Two common types of angles are the
angles of elevation and depression.
When you look up the _____________
_____________________ is the angle
formed between the horizontal and your line
of sight.
When you look down the ___________
____________________ is the angle
formed between the horizontal and your line
of sight.
Example: You are flying a kite overhead.
The angle of elevation is 65º. The length of
string used is 75 ft. How high is the kite?
Angle of Elevation and Depression Worksheet
1) George is in a hot air balloon that is 125 metres high. The angle of elevation from a
house below, to the balloon, is 18°. How far is George from the house?
2) The angle of elevation of a road is 4.5°. What is the length of the section of road if it
rises 16 metres?
Math 10 Workplace
Marsh
20 3) The angle of elevation of a slide that is 3.6 metres long is 32°. How high above the
ground is the top of the slide?
4) A ramp with a length of 21.2 metres has an angle of elevation of 15°. How high up
does it reach?
5) The angle of elevation from the bottom of a waterslide to the platform above is 20°. If
the waterslide is 25 metres long, how high is the platform?
6) A man walks at an angle of 68° north of east for 45 metres. How north of his starting
point is he?
Answers:
1) about 404.5 m 2) about 203.9 m
5) about 8.6 m 6) about 41.7 m
Math 10 Workplace
3) about 1.9 m
4) about 5.5 m
Marsh
21 7.3 The Cosine Ratio
cosine A 
length of side adjacent A
length of hypotenuse
cos  
adj
hyp
Example 1: Use your scientific calculator to determine the following cosine ratios.
Round to four decimal places.
Note: make sure your calculator is in degree mode.
a) cos 45°
b) cos 30°
Example 2: Calculate the angle to the nearest degree.
b) cos A = 0.9565
b) cos B = 0.0121
Examples: Using the same skills learned for the sine ratio, solve for the missing side or
angle.
1) Solving for the angle.
Solve for angle G.
Practice:
Math 10 Workplace
Marsh
22 2) Solving for the top of the ratio.
Solve for side c.
Practice:
3) Solving for the bottom of the ratio.
Solve for side z.
Practice:
Math 10 Workplace
Marsh
23 Word Problems: Draw a diagram.
1) How far from the base of a house is a 40 foot ladder if the angle of elevation is
72º?
2) The angle of a cable from a point 12.5 metres from its base is 52º. How long is
the cable?
Worksheet: Cosine Ratio
Math 10 Workplace
Marsh
24 Cosine Ratio Worksheet
1) Use your calculator to find the following pairs of ratios to four decimal places.
a) cos 23° =
sin 67° =
b) cos 83° =
sin 7° =
c) cos 45° =
sin 45° =
2) Find the measure of the indicated side or angle in each triangle (to one decimal
place).
a)
b)
c)
d)
Math 10 Workplace
Marsh
25 e)
f)
DRAW A DRAGRAM FOR ALL WORD PROBLEMS
3) How far from the base of a flagpole must a guy wire be fixed if the wire is 12
metres long and it makes an angle of 63° with the ground?
4) Reba walks 25 yards across the diagonal of a rectangular field. If the angle
between the width and the diagonal is 67°, how wide is the field?
Math 10 Workplace
Marsh
26 5) Aaron needs to string a bridge line
across the river from A to B. What
must the length of the bridge line
be, given his measurements?
6) What is the length of a rafter that makes an angle of 35° with the floor of an attic
whose centre is 9.5 metres from the edge?
7) An airplane starts descending at an angle of depression of 5°. If the horizontal
distance to its destination is 500 km, what is the actual distance the airplane will
travel before it lands?
8) A screw conveyor is sometimes used to move grains and other materials up an
incline. How far from the base of a barn must a 20-metre screw conveyor be
placed if the angle of elevation 30°?
Math 10 Workplace
Marsh
27 9) What is the slant height of a cone if the diameter is 20 cm and the angle made
with it is 65°?
10) A hot air balloon travels 1.2 km horizontally from its take-off point. The angle of
elevation from the take-off point to the balloon is 15°. How far did the balloon
travel?
11) What horizontal distance has a car travelled if the incline of the road averages
3.2° and the car’s odometer reads 8.5km?
12) The horizontal distance between two clothesline poles is 3.4m. When wet clothes
are hung in the middle of the line, it sags at an angle of depression of 6°. How
long is the clothesline?
Answers:
1) a) 0.9205 b) 0.1219 c) 0.7071 2) a) 48.2° b) 11.3 mm c) 4.2 m d) 23.0 cm e) 56.3°
f) 162.8 cm 3) 5.4 m 4) 9.8 yd 5) 14.1 m 6) 11.6m 7) 502 km 8) 17.3 m 9) 23.7 cm
10) 1.24 km 11) 8.49 km 12) 3.42 m
Math 10 Workplace
Marsh
28 7.4 The Tangent Ratio
tangent A 
length of side opposite A
length of side adjacent A
tan  
opp
adj
Example 1: Use your scientific calculator to determine the following cosine ratios.
Round to four decimal places.
Note: make sure your calculator is in degree mode.
a) tan 15°
b) tan 83°
Example 2: Calculate the angle to the nearest degree.
b) tan A = 0.5231
b) tan B = 1.651
Examples:
F
1) Solving for the angle.
Solve for angle G.
opp
tan G 
adj
4.3 ft
H
7.3 ft
G
Practice:
Math 10 Workplace
Marsh
29 X
2) Solving for the top of the ratio.
Solve for side y.
tanY 
opp
adj
y
65º
Z
6 ft
Y
Practice:
A
3) Solving for the bottom of the ratio.
Solve for side c.
tan C 
opp
adj
4.3 cm
46º
B
C
c
Practice:
Worksheet: Tangent Ratio
Math 10 Workplace
Marsh
30 The Tangent Ratio Worksheet
1) Find the value of the unknown in each of the following triangles.
a)
b)
c)
d)
e)
Math 10 Workplace
Marsh
31 f)
2) The angle of depression to a boat from the top of a 150-metre cliff is 20°. How far
is the boat from the base of the cliff?
3) When sand is piled onto a flat surface, it forms a cone. If the pile is 8m wide, and the
angle between the ground and the slope of the pile is 28°, what is the height of the
pile?
4) A 1.7 metre tall man stands 12 m from the base of a tree. He views the top of the tree
at an angle of elevation of 58°. How tall is the tree?
5) Two buildings are 18.5 metres apart. The angle of elevation from the top of one
building to the top of the other is 18°. If the taller building is 15 metres tall, how tall
is the shorter building?
Math 10 Workplace
Marsh
32 6) How far from the base of the house is the foot of a ladder if the angle of elevation is
70° and it reaches 15 feet up the side of the house?
7) About how tall is a tower if the angle of depression from its top to appoint 75 metres
from the base is 62°?
8) A rafter’s angle of elevation with the horizontal is 25°. How far from the corner could
a 6-foot man stand up straight?
9) Determine the distance, AB, across the river, given the following measurements.
Answers:
1) a) 60.9 cm b) 15 m c) 52° d) 379 mm e) 12°f) 25.5 km 2) 412 m
6) 5.5 ft 7) 141.1 cm 8) 12.9 ft 9) 373 m
Math 10 Workplace
3) 2.1 m 4) 21 m 5) 9 m
Marsh
33 7.5 Solving Right Triangles
A. SOH CAH TOA
Now that we have practiced all three of the trigonometry ratios, we need to be able to
determine the correct ratio to use. Labelling your triangle correctly is very important.
sin  
opp
adj
opp
cos  
tan  
hyp
hyp
adj
Examples: Determine the value of the unknown side or angle.
a)
b)
c)
Worksheet: Solving Lengths and Angles
Math 10 Workplace
Marsh
34 Solving Lengths and Angles Worksheet
1) Calculate the angle to the nearest degree.
a) sin D = 0.5491
c) tan G = 2.3548
b) cos F = 0.8964
d) sin H = 0.9998
2) In right triangle ΔXYZ, the ratio of the side opposite  X to the hypotenuse is
What is the approximate size of  X?
7
.
8
3) What is the approximate size of an angle in a right triangle if the ratio of the side
opposite the angle to the side adjacent to the angle is ?
4) What is the angle of depressions from the top of a 65 metre cliff to an object
48 metres from its base?
Math 10 Workplace
Marsh
35 5) At what angle to the ground must you place a support if it is 6.8 metres long and
must reach 4.2 metres up the side of a tower?
6) At what angle to the ground is an 8 metre long conveyor belt if it is fastened 5 metres
from the base of a loading ramp?
7) If a boat is 150 metres from the base of a cliff that is 90 metres high, what is the
angle of elevation from the boat to the cliff top?
8) Find the indicated angles in each of the following diagrams.
Answers:
1) a) 33° b) 26° c) 67° d) 89° 2) 61° 3) 11° 4) 54°
Math 10 Workplace
5) 38° 6) 51° 7) 31° 8) a) 31° b) 48°
Marsh
36 B. Solving Right Triangles
It is common to solve all missing angles and sides of a right triangle. You have several
tools that can be used to accomplish this:
1) ________________________________________________________
2) ________________________________________________________
3) ________________________________________________________
Examples: Solve all the missing sides and angles.
a = _________
c = _________
B = _________
b = _________
c = _________
B = _________
Worksheet: Solving Triangles
Math 10 Workplace
Marsh
37 Solving Right Triangles Worksheet
1) Solve the given triangle without using the Pythagorean Theorem.
2) The two equal angles of an isosceles triangle are each 70°. Determine the
measures of the rest of the triangle if it has a height of 16cm.
3) Solve the following triangles.
Math 10 Workplace
Marsh
38 4) What height is a pole, and how far away from it is a cable attached to the ground,
if the angle of elevation is 25° and the cable is 18 m long?
5) Find the values of a, b, c, and d.
Math 10 Workplace
Marsh
39 6) In a right triangle, one acute angle is 22° and the hypotenuse is 70 cm. Find the
lengths of the legs and the other angle measure.
Answers:
1) r = 7.3m q = 4.9m 2) 40°, 17cm, 11.6cm 3) a) t = 3.2 m R = 28° S = 62°
b) l = 6.6 cm M = 46° L = 44° 4) height = 8 m cable 16 m away 5) a = 23 cm b = 18 cm
A = 68°, c = 26 cm, a = 65 cm
Math 10 Workplace
Marsh