Download Module - Trigonometric Functions of an Acute Angle

Trigonometry
Module T06
Trigonometric
Functions of an
Acute Angle
Copyright
This publication © The Northern
Alberta Institute of Technology
2002. All Rights Reserved.
LAST REVISED December, 2008
Trigonometric Functions of an Acute
Angle
Statement of Prerequisite Skills
Complete all previous TLM modules before completing this module.
Required Supporting Materials
Access to the World Wide Web.
Internet Explorer 5.5 or greater.
Macromedia Flash Player.
Rationale
Why is it important for you to learn this material?
Trigonometry is simply the study of triangles. It has been used since the time of the
Greeks. The trigonometric ratios introduced in this module are used in a wide variety of
applications today including surveying, navigation, engineering, and construction.
Learning Outcome
When you complete this module you will be able to…
Solve problems involving the six trigonometric functions.
Learning Objectives
1.
2.
3.
4.
Define and list the six trigonometric functions in terms of the sides of a right triangle.
Determine the function values of angles between 0 and 90 degrees inclusively.
Determine the angle from a given function value.
Calculate the function value of the acute angles in any right triangle, given two sides
of the triangle.
5. Determine all function values of an acute angle given one function value of the angle.
Connection Activity
Consider the following Triangle:
A
c=5
B
a=4
b
C
Your work with right triangles allows you to solve for side b. What is the ratio of side a
to side c? What if a = 8 and c = 10? What is the ratio of side a to side c now? Notice the
ratio has not changed. The proportionate increase in the sides a and c means that angle B
did not change and the ratio of the sides remained the same. This is a hint at how the
ratio of two sides of a triangle can help you determine the angle between the sides.
1
Module T06 − Trigonometric Functions of Acute Angles
OBJECTIVE ONE
When you complete this objective you will be able to…
Define and list the six trigonometric functions in terms of the sides of a right triangle.
Exploration Activity
Define and list the six trigonometric functions in terms of the sides of a right triangle.
There are six possible trigonometric functions. Following is a list of these functions and
their abbreviations.
The basic functions:
sine
-
sin
cosine
tangent
-
cos
tan
The reciprocals of the basic functions:
cosecant
-
csc
secant
cotangent
-
sec
cot
The above trigonometric functions are meaningless in themselves, you must relate them
to angles of a triangle.
These side-angle relationships are illustrated in the following right angle triangle that
identifies the sides relative to θ. These relationships hold true only for triangles with a 90º
angle.
These relationships must be memorized:
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
csc θ =
hypotenuse
1
=
sin θ
opposite
sec θ =
hypotenuse
1
=
adjacent
cos θ
adjacent
1
=
cot θ =
opposite
tan θ
Hypotenuse
Opposite θ
opposite
θ
Adjacent to θ
2
Module T06 − Trigonometric Functions of Acute Angles
NOTE:
1. The sin and csc functions are related in that the csc is the reciprocal of the sin.
2. The cos and sec functions are reciprocals of each other.
3. The tan and cot functions are also reciprocals of each other.
We can name our right triangle ABC, and identify the functions in terms of the lettered
sides.
sin A =
a
c
cos A =
b
c
tan A =
a
b
csc A =
c
a
sec A =
c
b
cot A =
b
a
B
c
A
a
b
C
NOTE: When labeling a triangle the same letter is used for the angle and the side
opposite it. Use capitals for angles and lowercase for sides.
Observe the reciprocal relationships in these fractions, i.e.
sin A =
a
1
=
c csc A
csc A =
c
a
cos A =
b
1
=
c sec A
sec A =
c
b
tan A =
a
1
=
b cot A
cot A =
b
a
3
Module T06 − Trigonometric Functions of Acute Angles
If we wished to find the 6 functions of angle B in the previous diagram we would obtain.
sin B =
opp b
=
hyp c
csc B =
hyp c
=
opp b
cos B =
adj a
=
hyp c
sec B =
hyp c
=
adj a
tan B =
opp b
=
adj a
cot B =
adj a
=
opp b
Observe again the reciprocal relationships in the above functions.
Sometimes students may get confused as to which side is the "opposite" side from an
angle. Remember, the "opposite" side is the side which is not an arm of the angle.
The hypotenuse is always opposite the right angle.
The adjacent side is always an arm of the angle.
EXAMPLE 1
For the triangle shown, determine the ratios of the six trigonometric functions of angle θ.
EXACT
DECIMAL
sin θ =
12
20
3
5
0.6000
cos θ =
16
20
4
5
0.8000
12
tan θ =
16
3
4
0.7500
20
csc θ =
12
5
3
1.6667
16
sec θ =
20
16
5
4
1.2500
20 2 = 16 2 + 12 2
cot θ =
16
12
4
3
1.3333
20
θ
Pythagorean Theorem Check
4
Module T06 − Trigonometric Functions of Acute Angles
12
csc, sec, cot could have also been found by using the reciprocals of the first 3 trig
functions as shown in the next 3 examples.
1. sin θ = 0.60000
1
1
cscθ =
=
= 1.6667
sin θ 0.6000
2. cos θ = 0.8000
1
1
=
= 1.2500
secθ =
cosθ 0.8000
3. tan θ = 0.7500
1
1
cot θ =
=
= 1.3333
tan θ 0.7500
5
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity One
1. In the figure, choose either angle P or angle R.
a)
OP
= tan ?
OR
b)
PR
= sec ?
OP
c)
OP
= cos ?
PR
d)
OP
= sin ?
PR
e)
PR
= csc ?
OP
P
O
R
2. The three sides of a right triangle are 5, 12, and 13. Let A be the acute angle opposite
the side 5 and let B be the other acute angle. Find the six functions of A and B.
Show Me
3. In the figure, if x = r, find the six functions of θ.
R
z
x
θ
X
r
Z
4. In the figure in question 3, x = 2r, find the sine, cosine, and tangent of θ.
6
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity One Answers
1. a) R, b) P, c) P, d) R, e) R
sin B = 0.9231
2. sin A = 0.3846
cos A = 0.9231
cos B = 0.3846
tan A = 0.4167
tan B = 2.4000
csc A = 2.6000
csc B =1.0833
sec A =1.0833
sec B = 2.6000
cot A = 2.4000
cot B = 0.4167
x
r
z
z
3. sin θ = ; cos θ = ; tan θ = 1; cscθ = ; secθ = ; cos θ = 1
z
z
x
r
2r
r
4. sin θ = ; cos θ = ; tan θ = 2
z
z
7
Module T06 − Trigonometric Functions of Acute Angles
OBJECTIVE TWO
When you complete this objective you will be able to…
Determine the function values of angles between 0 and 90 degrees inclusively.
Exploration Activity
Now we will use the calculator to find the ratios of the 6 trigonometric functions.
Refer to your calculator manual for this objective.
To find the SINE, COSINE, and TANGENT of any angle, you enter the desired function
and the desired angle.
EXAMPLE Take: 32°
a) Find the trigonometric ratio of sin 32º
Step 1: press the sin key
Step 2: press 32º and then the = key
Step 3: display: 0.5299
b) cos 32°
Step 1: press the cos key
Step 2: enter 32º
Step 3: display: 0.8480
c) tan 32°
Step 1: press the tan key
Step 2: enter 32º
Step 3: display: 0.6249
Ensure calculator is in degree mode when the angle is given in degrees and you
are using any of the trig functions on your calculator!!
8
Module T06 − Trigonometric Functions of Acute Angles
EXAMPLES
TRIGONOMETRIC RATIOS OF RECIPROCAL FUNCTIONS
Finding the COSECANT, SECANT, and COTANGENT of any angle is explained in the
following examples:
Find csc 79º, sec 79º and cot 79º to four decimal places:
PREFERRED METHOD
ALTERNATE METHOD
Example 1:
Find the sin 79º, then take the reciprocal.
This will be equal to the csc 79º.
csc 79º
Since csc 79º =
1
sin 79°
Enter: 1 ÷ sin 79 =
Display: 1.0187
sec 79º
1
cos 79°
Enter: 1 ÷ cos 79 =
Display: 5.2408
Step 1:
Step 2:
Step 3:
Step 4:
press the cos key.
press 79 and then the = key.
press the x −1 key.
display: 5.2408 (= sec 79º)
Find the tan 79º, then take the reciprocal.
This will be equal to the cot 79º.
Example 3:
cot 79º
Since cot 79º =
press the sin key.
press 79 and then the = key.
press the x −1 key.
display: 1.0187 (= csc 79º)
Find the cos 79º, then take the reciprocal.
This will be equal to the sec 79º.
Example 2:
Since sec 79º =
Step 1:
Step 2:
Step 3:
Step 4:
1
tan 79°
Enter: 1 ÷ tan 79 =
Display: 0.1944
Step 1:
Step 2:
Step 3:
Step 4:
press the tan key.
press 79 and then the = key.
press the x −1 key.
display: 0.1944 (= cot 79º)
9
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Two
Exercise Set A
Perform the following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
sin 64°
cos 75°
tan 74°
tan 12º
cos 15º
sin 17°
cos 28°
tan 34°
cos 63°
sin 30°
Exercise Set B
Perform the following:
1. csc 43°
2. cot 14°
3. tan 55°
4. sec 35°
5. csc 30°
6. cos 55°
7. cot 15°
8. sec 14°
Show Me
9. csc 4°
10. sin 18°
11. sin 0°
12. tan 0°
Experiential Activity Two Answers
Answers for Exercise Set A
1. 0.8988
3. 3.4874
5. 0.9659
7. 0.8829
9. 0.4540
2.
4.
6.
8.
10.
0.2588
0.2126
0.2924
0.6745
0.5000
Answers for Exercise Set B
1. 1.4663
3. 1.4281
5. 2.0000
7. 3.7321
9. 14.3356
11. 0.0000
2.
4.
6.
8.
10.
12.
4.0108
1.2208
0.5736
1.0306
0.3090
0.0000
10
Module T06 − Trigonometric Functions of Acute Angles
OBJECTIVE THREE
When you complete this objective you will be able to…
Determine the angle from a given function value.
Exploration Activity
The trigonometric ratio of 0.5976 is the ratio of the sides
opposite
for θ = 36.7º.
hypotenuse
When we are given sin θ = 0.5976 and are asked to determine θ, we know that the only
acute angle which has a sine of 0.5976 is 36.7º. Therefore, we should be able to find the
angle.
Example:
sin θ = 0.5976
θ = arc sin 0.5976
θ = 36.7º
or
sin θ = 0.5976
θ = sin−1 0.5976
θ = 36.7º
Arc sin and sin−1 are the two names used to express this operation.
It is very important you understand that sin-1 θ means the inverse operation of finding the
sine of the angle.
1
sin−1 θ is not the same as
sin θ
Do not confuse this with the reciprocal x−1 function on your calculators!!
EXAMPLE 1
Given: sin θ = 0.5976. Find θ in degrees.
SOLUTION:
The problem is to find the angle. Therefore by our definition
θ = Arc sin 0.5976
or
θ = sin−1 0.5976
This means we want to find the angle whose sine is 0.5976
11
Module T06 − Trigonometric Functions of Acute Angles
Using the calculator:
Refer to your calculator manual as the procedure varies from calculator to calculator.
Step 1: Press the 2nd F key
Step 2: press sin key
Step 3: enter 0.5976 and then press the = key
Step 4: display: 36.7º
Check: sin 36.7º = 0.5976
NOTE:
θ = Arc sin 0.5976 is read as "θ is the angle whose sine is 0.5976".
θ = Arc cos 0.3276 is read as "θ is the angle whose cosine is 0.3276 .
θ = Arc tan 4.364 is read as "θ is the angle whose tangent is 4.364"
.... and so on.
It is sometimes desirable to find the angle in radians. Again your calculator will do this
operation as long as you change it to radian mode.
EXAMPLE 2
Find the value of θ in radians for cos θ = 0.9690
θ = Arc cos 0.9690
SOLUTION: Again refer to your own calculator manual for finding an angle in radians,
given a trigonometric function. Using your calculator:
Step 1: Put your calculator in radian mode
Step 2: Press the 2nd F key
Step 3: press cos key
Step 4: enter 0.9690 and then press the = key
Step 5: display: 0.2496
Therefore θ = 0.2496
Check: cos(0.2496) = 0.9690
Using only csc, sec and cot, find the angle in degrees or radians.
12
Module T06 − Trigonometric Functions of Acute Angles
EXAMPLE 3
Given sec θ = 5.2411, find angle θ in degrees
Since the calculators do not have a secant function, it is necessary first to convert this to
the cosine function.
Since
1
secθ
1
cos θ =
5.2411
cos θ =
⎛ 1 ⎞
θ = arc cos ⎜
⎟ or
⎝ 5.2411 ⎠
⎛ 1 ⎞
θ = cos−1 ⎜
⎟
⎝ 5.2411 ⎠
θ = 79.0º
EXAMPLE 4
Given csc θ = 1.7965, find angle θ in degrees
SOLUTION:
Since
1
cscθ
1
sin θ =
1.7965
sin θ =
⎛ 1 ⎞
θ = arc sin ⎜
⎟ or
⎝ 1.7965 ⎠
⎛ 1 ⎞
θ = sin-1 ⎜
⎟
⎝ 1.7965 ⎠
θ = 33.8º
13
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Three
Exercise Set A
Solve for θ in degrees, given:
1. sin θ = 0.9135
2. cos θ = 0.8241
3. tan θ = 0.4142
4. sin θ = 0.4305
5. tan θ = 1.2059
6. cos θ = 1.2000
7. sin θ = 0.7071
8. tan θ = 0.2493
9. cos θ = −1.0000
Exercise Set B
Find θ in Radians, given:
1. sin θ = 0.5821
2. cos θ = 0.4555
3. tan θ = 3.5105
4. sin θ = 0.3778
5. tan θ =1.0761
6. cos θ = 0.0279
7. sin θ = 0.7804
8. cos θ = 0.9763
Exercise Set C
(a) Find the angle in degrees, given the following:
1. csc θ =1.970
2. sec θ = 2.000
3. cot θ =1.962
4. csc θ = 5.495
5. sec θ = 6.795
6. cot θ = 6.543
7. sec θ = 3.540
8. cot θ = 6.354
Show Me
9. csc θ = 4.945
10. sec θ = 6.514
(b) For the exercise above, find the angles in radians.
14
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Three Answers
Answers for Exercise Set A
1. 66.0º
2. 34.5º
3. 22.5º
4. 25.5º
5. 50.3º
6. none Å this is because cos θ and sin θ are limited to values between −1 and 1
7. 45.0º
8. 14.0º
9. 180º
Answers for Exercise Set B
1. 0.6213
2. 1.0979
3. 1.2933
4. 0.3874
5. 0.8220
6. 1.5429
7. 0.8953
8. 0.2181
Answers for Exercise Set C
(a.)
(b.)
1. 30.5º
1. 0.5324
2. 60.0º
2. 1.0472
3. 27.0º
3. 0.4714
4. 10.5º
4. 0.1830
5. 81.5º
5. 1.4231
6. 8.7º
6. 0.1517
7. 73.6º
7. 1.2844
8. 8.9º
8. 0.1561
9. 11.7º
9. 0.2036
10. 81.2º
10. 1.4167
15
Module T06 − Trigonometric Functions of Acute Angles
OBJECTIVE FOUR
When you complete this objective you will be able to…
Calculate the function value of the acute angles in any right triangle, given two sides of
the triangle.
Exploration Activity
If we are given any two sides of a right triangle then according to Pythagoras Theorem
we can calculate the third side. This theorem states that in a right triangle the square of
the hypotenuse equals the sum of the squares on the other two sides.
In the adjoining triangle we can find side b by using Pythagoras' formula:
c2 = a2 + b2
A
5 2 = 42 + b 2
b2 = 25 − 16
c=5
b= 9
b
so: b = 3
B
a=4
C
Once we know the three sides, we can easily find the trigonometric ratios of both angle A and
angle B
Trigonometric Ratios of ∠A
A
hypotenuse
c=5
B
sin A =
cos A =
tan A =
csc A =
sec A =
cot A =
Trigonometric Ratios of ∠B
hypotenuse
c=5
b =3
Adjacent to ∠A
a=4
Opposite ∠A
C
opp
4
= = 0.8000
hyp
5
3
= 0.6000
5
4
= 1.3333
3
5
= 1.2500
4
5
= 1.6667
3
3
= 0.7500
4
B
sin B =
cos B =
tan B =
csc B =
sec B =
cot B =
a=4
Adjacent to ∠B
A
b =3
Opposite ∠B
C
opp
3
= = 0.6000
hyp
5
4
5
3
4
5
3
5
4
4
3
= 0.8000
= 0.7500
= 1.6667
= 1.2500
= 1.3333
16
Module T06 − Trigonometric Functions of Acute Angles
EXAMPLE 1
If a = 3 and b = 4 in the right triangle
ABC to the right, find sin A and tan B.
since
c2 = a2 + b2
B
c2 = 42 + 32
c2 = 25
c
a=3
c = 25
c2 = a2 + b2
A
c=5
b=4
C
so:
3
= 0.6000 and
5
4
tan B = = 1.3333
3
sin A =
NOTE: The trigonometric ratio is equally acceptable in either fraction form or in decimal
form. See example 1 above. However when entering answers into the computer the
decimal form must be used. i.e. 3/5 would be entered as 0.6000. Enter answers with 4
decimal places.
17
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Four
1. Find the indicated ratios from the given triangle where a = 1, b = 3 , and c = 2.
Write your ratios in exact form.
a)
b)
c)
d)
sin A, sec B,
csc A, sin B,
tan A, cos B,
tan B, csc B,
B
cot A
cot B
sec A
cos A
c
A
a
C
b
2. Find the indicated ratios from the given triangle where b =2.89, and c = 3.41, and
side a is unknown. Round your ratios to 3 significant digits.
a)
b)
c)
d)
sin A,
csc A,
tan A,
tan B,
B
sec B, cot A
sin B, cot B
cos B, sec A
csc B, cos A
c
A
a
C
b
3. Determine the indicated ratios. The listed sides are those shown in the given triangle.
Write your ratios in exact form.
a)
b)
c)
d)
a = 3, b = 4. Find tan A and cos B
a = 5, c = 13. Find cos A and csc B
b = 9, c = 41. Find cot A and cos B
a = 8, c = 19. Find sin A and sec B
Round your ratios to 3 significant digits.
e) a = 2, c = 4. Find sec A and tan B
f) b = 14, c = 23. Find csc A and cos B
g) a = 132, b = 75. Find sin A and tan B
B
c
A
a
b
18
Module T06 − Trigonometric Functions of Acute Angles
C
4. Given the coordinates as shown, find the values requested:
Round trigonometric ratios to 3 significant digits. Round angles to the one decimal place.
tan A
= __________
∠A
= __________
Y
P(3.6, 4.2)
csc A = __________
Show Me
A
X
5. Given the coordinates as shown, find the values requested:
Round trigonometric ratios to 3 significant digits. Round angles to the one decimal place.
∠B
= __________
sin B
= __________
sec B
= __________
Y
P(5.8, 2.3)
B
X
19
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Four Answers
1. a) sin A =
1
, sec B = 2, cot A =
2
3
3
1
, cot B =
2
3
1
1
2
c) tan A =
, cos B = , sec A =
2
3
3
b) csc A = 2, sin B =
d) tan B =
3 , csc B =
2
3
, cos A =
3
2
2. a) sin A = 0.531, sec B = 1.88, cot A = 1.60
b) csc A = 1.88, sin B = 0.848, cot B = 0.626
c) tan A = 0.626, cos B = 0.531, sec A = 1.18
d) tan B = 1.60, csc B = 1.18, cos A = 0.848
3
3
, cos B =
4
5
12
13
b) cos A =
, csc B =
13
12
9
40
c) cot A =
, cos B =
40
41
8
19
d) sin A =
, sec B =
19
8
3. a) tan A =
e) sec A = 1.15, tan B = 1.73
f) csc A = 1.26, cos B = 0.793
g) sin A = 0.869, tan B = 0.568
4. a) tan A = 1.17, ∠A = 49.4º, csc A = 1.32
5. a) ∠B = 21.6º, sin B = 0.369, sec B = 1.08
20
Module T06 − Trigonometric Functions of Acute Angles
OBJECTIVE FIVE
When you complete this objective you will be able to…
Determine all function values of an acute angle given one function value of the angle.
Exploration Activity
EXAMPLE 1
Find the values of the trigonometric ratios of ∠A
B
12
given that sin A =
.
13
We know that sin A =
opposite
hypotenuse
c
a
Therefore:
sin A =
12
13
A
b
Since 12 is the side opposite ∠A and 13 is the
hypotenuse we can identify the following sides:
C
B
a = 12 and c = 13
Use the Pythagorean theorem to find side b:
c2 = a2 + b2
132 = 122 + b2
Hypotenuse
c = 13
a =12
Opposite ∠A
b = 25
b=5
A
C
b=5
Adjacent to ∠A
Trigonometric ratios of ∠A
12
sin A =
= 0.9231
13
csc A =
13
= 1.0833
12
cos A =
5
= 0.3846
13
tan A =
12
= 2.4000
5
sec A =
13
= 2.6000
5
cot A =
5
= 0.4167
12
21
Module T06 − Trigonometric Functions of Acute Angles
EXAMPLE 2
Given that tan A =
Since tan A =
1
, find sin A and csc A.
4
opp
1
and from the question tan A = , we can identify the following sides
adj
4
on the triangle below:
a = 1 and b = 4
B
c
a
A
b
C
Use the Pythagorean Theorem to find side c:
c2 = a2 + b2
c2 = 12 + 42
c = 17
sin A =
opp
=
hyp
1
= 0.2425
17
csc A =
hyp
=
opp
17
= 4.1231
1
22
Module T06 − Trigonometric Functions of Acute Angles
Experiential Activity Five
1. Find the values of the other five trigonometric functions of A, given in exact values.
Write your answers in exact form.
7
25
1
d) cos A =
2
24
7
7
e) cos A =
11
a) sin A =
b) cot A =
c) tan A =
3
5
f) cot A =
3
2
Experiential Activity Five Answers
24
, tan A =
25
7
b) sin A =
, cos A =
25
1. a) cos A =
c) sin A =
3
34
7
, csc A =
24
24
, tan A =
25
, cos A =
25
25
24
, sec A =
, cot A =
7
24
7
7
25
25
, csc A =
, sec A =
24
7
24
34
34
5
5
, csc A =
, sec A =
, cot A =
3
3
5
34
d) sin A =
3
1
2
, tan A = 3 , csc A =
, sec A =2, cot A =
2
3
3
e) sin A =
72
, tan A =
11
72
11
11
, csc A =
, sec A =
, cot A =
7
7
72
2
3
f) sin A =
7
, cos A =
7
, tan A =
2
3
, csc A =
7
, sec A =
2
7
72
7
3
Practical Application Activity
Complete the Trigonometric Functions of an Acute Angle assignment in TLM.
Summary
This module introduced the student to the six trigonometric functions.
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Module T06 − Trigonometric Functions of Acute Angles