Download Trigonometry Lesson #1

Trigonometry Lesson #8.3
Why learn trig??
Part A: Introduction to Trigonometry
 For all right angle triangles there are consistent relationships between sides
and acute angles of the triangle.
 Trigonometric ratios can be written to solve for missing variables.
Given triangle ABC, where C = 90,
Opposite
side to  B
Hypotenuse
B
Adjacent
side to  B
Primary Trigonometric Ratios:
(Always in relation to the angle given or the one that is trying to be solved)
Name
Abbreviation
Ratio
sine  A
sin A
side opposite  A
hypotenuse
cosine  A
cos A
side adjacent  A
hypotenuse
tangent  A
tan A
side opposite  A
side adjacent  A
**Remember: SOH CAH TOA**
Example #1
1. Given Triangle DEF, state the primary trigonometric ratios for  D and  E.
D
5
3
F
sin D 
sin E 
cos D 
cos  E 
tan  D 
tan  E 
E
4
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Example #2
Solve for the missing side and then state the primary trig ratios for the
two acute angles.
Q
17.4
7.6
R
q
P
Example #3
Given the following information
 Draw a triangle and label the given information
 Calculate the length of the missing side
 State the primary trig ratios for both acute angles
ABC, B  90, b  7, c  5
2
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Lesson #8.3 Part A Assignment
1. Given  FGH, state the following ratios
(a) sin F=
(d)
sin E=
(b) cos F=
(e)
cos E=
E
28
15
(c) tan F=
(f)
tan E=
G
20
F
2. Calculate the missing side, to the nearest tenth, and then give the primary
trig ratios for  B.
(a)
sin B=
c
A
B
cosB=
5.3
13.5
C
tanB=
(b)
A
14.5
B
b
11.8
C
3
3. For each of the following triangles
 Draw a triangle and label the given information
 Calculate the length of the missing side
 State the primary trig ratios for both acute angles
(a) PQR,  P = 90º, side q = 14.3 cm, side r = 11 cm.
(b)  STV, S = 90º, side s = 18.9m, side t = 9.8m
4. Calculate the cosine value for  B, rounded to the nearest hundredth.
A
31.4
13.8
B
25.1
C
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5. Calculate the sine value of  N, rounded to the nearest hundredth.
N
10.6
5.4
L
M
6.3
6. The tangent value for  R is
T
127.9
84.0
V
153.0
R
Lesson #8.3 Part A Answer Key
1 (a)
(d)
2. (a)
(b)
3. (a)
sin F = 15/28
sin E = 20/28
c = 14.5
b = 8.4
p = 18.0 cm
(b) v=16.2 m
4. 0.80
(b) cos F = 20/28
(e) cos E = 15/28
sin B = 5.3/14.5
sin B = 8.4/14.5
sin R = 11/18
sin Q = 14.3/18
sin V = 16.2/18.9
sin T = 9.8/18.9
5. 0.59
(c) tan F = 15/20
(f) tan E = 20/15
cos B = 13.5/14.5
tan B = 5.3/13.5
cos B = 11.8/14.5
tan B = 8.4/11.8
cos R = 14.3/18
tan R = 11/14.3
cos Q = 11/18
tan Q = 14.3/11
cos V = 9.8/18.9
tan V = 16.2/9.8
cos T = 16.2/18.9
tan T = 9.8/16.2
6. 0.66
5
Part B: Calculator Trigonometry
You can now use your calculator to generate the angle associated with the
trigonometric ratio or generate a ratio given any angle.
Example #1
Calculate to 4 decimal places the following trig ratios.
(a) sin 68º
Steps used:
(b) cos 34º
(c) tan 78º
(d) cos 87º
Note: Make sure your calculator is in degree mode.
**All calculators are not the same. It is your responsibility to know how to use
the trig functions on your own calculator.**
Example #2
Given the ratio, generate the angle, to the nearest degree, associated with it.
(a) sin D = 0.8090
Steps used:
(b) cos A = 0.4384
(c) tan F = 0.2126
(d) sin P = 0.5592
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Lesson #8.3 Part B Assignment
1. Find the value of each trigonometric ratio, to four decimal places.
(a) tan 26º
(b)
sin 35º
(c) cos 23º
(d)
tan 63º
(e) sin 73º
(f)
cos 9º
(g) sin 50º
(h)
cos 71º
(i) tan 82º
1. Find the angle, to the nearest degree, that is associated with the following
trig ratios.
(a) sin A = 0.2588
(b) cos B = 0.9063
(c) tan C = 1.7321
(d) cos D = 0.1045
(e) tan E = 0.0875
(f) sin F = 0.9272
Lesson #8.3 Part B Answer Key
1. (a) 0.4877 (b) 0.5736
(f) 0.9877 (g) 0.7660
2. (a) 15º
(b) 25º
(c) 0.9205
(h) 0.3256
(c) 60º
(d) 1.9626
(i) 7.1154
(d) 84º
(e) 0.9563
(e) 5º
(f) 68º
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Trigonometry Lesson #8.4
PUTTING IT ALL TOGETHER:
In the right triangle below, the lengths of two sides are given. Calculate the
measure of the two acute angles and the missing side.
First, use the Pythagorean Theorem to solve for the missing side
X
Y
16.5 cm
11.0 cm
z=
Z
As a fraction:
As a decimal
The angle is:
sin X= _____
sin X=
X=
cos X= _____
cos X=
X=
tan X= _____
tan X=
X=
sin Z= _____
sin Z=
Z=
cos Z= _____
cos Z=
Z=
tan Z= _____
tan Z=
Z=
**Note: the two angles (42 o and 48o) will always add up to 90 o, and all three
angles (42 o + 48o + 90o) will always add up to 180 o.
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Part A: Solving Triangles
 When you are asked to solve a triangle, you must solve for all unknown
sides and angles
Steps to Solving a Triangle
1. Make a list of or label all known information.
2. Use primary trigonometric ratios (SOH CAH TOA) and/or Pythagoras’
Theorem to find all missing angles and side lengths.
3. Clearly label answers.
EXERCISE: In the right triangle below, you will be given the length of one side and the
size of one angle (other than the 90o). Write the trig ratio you would use to
find the two missing sides.
L
N
M
a. M = 26o, n = 15. Find sides l and m
1. Decide which side you want to
figure out first?
2. Based on the “names” for each side,
which Trig Ratio will you use?
3. Substitute and solve for the
missing side.
Do you need to use SOH CAH TOA
to find the 3rd side, or can you use
Pythagoras instead?
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b. M = 18o, l = 9. Find sides m and n
c. L = 72o, m = 12.5. Find sides l and n
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Example #1
Given triangle ABC solve for the missing side and the two acute angles.
24.6 cm
A
B
15.6 cm
C
Example #2
Given triangle DEF, solve for the missing angle and two unknown sides.
D
70.5º
E
58.3 mm
F
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Lesson #8.4 Part A Assignment
1. Given the following triangles, solve for all missing sides and/or angles.
Round all side measurements to the nearest tenth and all degrees
measurements to the nearest degree.
R
(a)
53
12 cm
S
T
X
(b)
8.3cm
Y
6.5 cm
Z
A
(c)
16
C
5.5 cm
B
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2. For each of the following:
 Draw a sketch of the triangle
 Record the given information on to the triangle
 Solve for any missing sides and/or angles
(a) Triangle ABC,  A = 35,  B = 90, and b = 6 cm
(b) In triangle PQR,  P = 46, R = 90, and r = 12 m
(c) In triangle KLM,  K = 90, m = 7.9cm and k = 23.7 cm
(d) In triangle RST,  S = 90, t = 108m and r = 142m
Lesson #8.4 Part A Answer Key
1. (a) r =15.9 cm; t =19.9 cm; S=37
(c) b = 20.0 cm; c = 19.2 cm; C=74
2. (a) a =3.4 cm; c =4.9 cm; C=55
(c) l =22.3 cm; L=71; M=19
(b) y = 10.5 cm; X=38; Z=52
(b) p=8.6 m; q=8.3 m; Q=44
(d) s = 178.4 m; R=53;T=37
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Part B: Solving Word Problems with Trigonometry
Ratios
Steps for Solving Trigonometry Word Problems
1.
2.
3.
4.
Draw a diagram (if one is not provided).
Make a list or label all known sides and angles.
Determine what the question is asking you to solve.
Use primary trigonometric ratios (SOH CAH TOA) to set up an equation,
and then solve.
5. Clearly label answer to problem.
Example #1
A 56 m wire, supporting a TV tower 45 m tall, joins the top of the tower to
an anchor point on the ground. What is the angle, , to the nearest degree,
that the wire makes with the ground?
56 m
45 m

Example #2
A 3.6 m tall cactus stands in the middle of the desert. If the suns rays hit the
ground at an angle of 36, how long of a shadow, to the nearest hundredth of
a metre, will the cactus have?
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Lesson #8.4 Part B Assignment
1. Calculate the missing side for each of the following triangles. (Round your
answers to the nearest tenth.)
(a)
(b)
x
72 cm
12 cm
a
32
48
(c)
(d)
12 cm
d
36 cm
51
68
f
2. Solve for the missing angle in each of the following triangles. (Round your
answers to the nearest degree)
(a)
(b)
.

45 mm
5.2 cm

6.7cm
21 mm
15
(c)
(d)

7mm
1.6 cm
3.2 cm
33mm

3. A tree is 3.9 m high and casts a shadow along the ground. The sun’s rays
make an angle of 52 with the ground. How long is the shadow of the tree,
rounded to the nearest tenth?
4. A ladder rests against a wall and forms an angle of 68 with the ground. If
the ladder is 7 m long and Jordan slips off the ladder, how far, to the nearest
tenth of a meter, will she fall?
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5. A ladder leans against a wall and forms an angle of 35 with the wall. If the
ladder is 2.5 m from the base of the wall, find:
(a) length of the ladder
(b) the height the ladder reaches up the wall
6. A roof rafter, PQ, is 7.0m long. The rise, PR, is 3.3 m. Find  Q, rounded
to the nearest degree.
P
rafter
rise
S
R
Q
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7. A wire 3 m long, supports a small tree and forms an angle of 47 with the
ground. How tall is the tree? (Record your answer to the nearest tenth of a
metre)
8. A tree, 12 m high, casts a shadow 19 m long. Calculate the angle, to the
nearest degree, that the suns rays make with the ground.
Lesson #8.4 Part B Answer Key
1. (a) 45.0 cm
2. (a) 52
3. 3.0 m
6. 28
(b) 8.9 cm
(b) 62
4. 6.5 m
7. 2.2 m
(c) 38.8 cm
(c) 63
5. (a) 4.4 m
8. 32
(d) 7.6 cm
(d) 12
(b) 3.6 m
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