LAST REVISED December, 2008
Copyright
This publication © The Northern
Alberta Institute of Technology
2002. All Rights Reserved.
Trigonometry
Module T06
Trigonometric
Functions of an
Acute Angle
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LAST REVISED December, 2008
Copyright This publication © The Northern Alberta Institute of Technology
- All Rights Reserved.
Trigonometry
Module T
Trigonometric
Functions of an
Acute Angle
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 - cos tangent - tan
The reciprocals of the basic functions:
cosecant - csc secant - sec cotangent - 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 = hypotenuse
opposite
Opposite θ
Adjacent to θ
Hypotenuse
opposite
cos θ = hypotenuse
adjacent
tan θ = adjacent
opposite
csc θ = opposite
hypotenuse (^) = sinθ
1
sec θ = adjacent
hypotenuse (^) = cosθ
1
cot θ = opposite
adjacent (^) = tanθ
1
NOTE:
- The sin and csc functions are related in that the csc is the reciprocal of the sin.
- The cos and sec functions are reciprocals of each other.
- 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 = c
a
A C
B
a
c
b
cos A = c
b
tan A = b
a
csc A = a
c
sec A = b
c
cot A = a
b
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.
1 sin csc
a A c A
= = csc A =
c a
cos 1 sec
A b c A
= = sec A =
c b
tan cot
a A b A
= = cot A =
b a
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.
- sin θ = 0. 1 1 csc 1. sin 0.
θ θ
- cos θ = 0.
secθ =
cos θ 0.
- tan θ = 0. 1 1 cot 1. tan 0.
Experiential Activity One
- In the figure, choose either angle P or angle R.
a) OR
OP = tan? P
O
R
b) OP
PR (^) = sec?
c) PR
OP (^) = cos?
d) PR
OP (^) = sin?
e) OP
PR (^) = csc?
- 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
- In the figure, if x = r , find the six functions of θ.
R
z
x
r
θ
X Z
- In the figure in question 3, x = 2 r , find the sine, cosine, and tangent of θ.
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.
b) cos 32° Step 1: press the cos key
Step 2: enter 32º
Step 3: display: 0.
c) tan 32° Step 1: press the tan key
Step 2: enter 32º
Step 3: display: 0.
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!!
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:
csc 79º
Find the sin 79º, then take the reciprocal. This will be equal to the csc 79º.
Step 1: press the sin key. Step 2: press 79 and then the = key. Step 3: press the x −^1 key. Step 4: display: 1.0187 (= csc 79º)
Since csc 79º =
sin 79°
Enter: 1 ÷ sin 79 = Display: 1.
Example 2:
sec 79º
Find the cos 79º, then take the reciprocal. This will be equal to the sec 79º.
Step 1: press the cos key. Step 2: press 79 and then the = key. Step 3: press the x −^1 key. Step 4: display: 5.2408 (= sec 79º)
Since sec 79º =
cos 79°
Enter: 1 ÷ cos 79 = Display: 5.
Example 3:
cot 79º
Find the tan 79º, then take the reciprocal. This will be equal to the cot 79º.
Step 1: press the tan key. Step 2: press 79 and then the = key. Step 3: press the x −^1 key. Step 4: display: 0.1944 (= cot 79º)
Since cot 79º =
tan 79°
Enter: 1 ÷ tan 79 = Display: 0.
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 hypotenuse
opposite (^) for θ = 36.7º.
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. θ = arc sin 0. θ = 36.7º
or
sin θ = 0. θ = sin−^1 0. 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.
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. or θ = sin−^1 0.
This means we want to find the angle whose sine is 0.
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.
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..
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.
θ = Arc cos 0.
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.
Therefore θ = 0.
Check: cos(0.2496) = 0.
Using only csc, sec and cot, find the angle in degrees or radians.
Experiential Activity Three
Exercise Set A Solve for θ in degrees, given:
- sin θ = 0.
- cos θ = 0.
- tan θ = 0.
- sin θ = 0.
- tan θ = 1.
- cos θ = 1.
- sin θ = 0.
- tan θ = 0.
- cos θ = −1.
Exercise Set B Find θ in Radians, given:
- sin θ = 0.
- cos θ = 0.
- tan θ = 3.
- sin θ = 0.
- tan θ =1.
- cos θ = 0.
- sin θ = 0.
- cos θ = 0.
Exercise Set C (a) Find the angle in degrees, given the following:
- csc θ =1.
- sec θ = 2.
- cot θ =1.
- csc θ = 5.
- sec θ = 6.
- cot θ = 6.
- sec θ = 3.
- cot θ = 6.
- csc θ = 4.945 Show Me
- sec θ = 6. (b) For the exercise above, find the angles in radians.
Experiential Activity Three Answers
Answers for Exercise Set A
- 66.0º
- 34.5º
- 22.5º
- 2 5.5º
- 5 0.3º
- none Å this is because cos θ and sin θ are limited to values between −1 and 1
- 45.0º
- 14.0º
- 180º
Answers for Exercise Set B
Answers for Exercise Set C (a.) (b.)
- 30.5º 1. 0.
- 60.0º 2. 1.
- 27.0º 3. 0.
- 10.5º 4. 0.
- 81.5º 5. 1.
- 8.7º 6. 0.
- 73.6º 7. 1.
- 8.9º 8. 0.
- 11.7º 9. 0.
- 81.2º 10. 1.
EXAMPLE 1
If a = 3 and b = 4 in the right triangle ABC to the right, find sin A and tan B.
since c^2 = a 2 + b^2 c^2 = 4 2 + 3 2 c^2 = 25
c = 25 c^2 = a^2 + b^2 c = 5
so:
A C
a = 3
c
B
b = 4
sin A = 5
3 = 0.6000 and
tan B = 3
4 = 1.
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.
Experiential Activity Four
- Find the indicated ratios from the given triangle where a = 1, b = 3 , and c = 2. Write your ratios in exact form.
a) sin A , sec B , cot A b) csc A , sin B , cot B c) tan^ A , cos^ B , sec^ A d) tan B , csc B , cos A
A C
B
a
c
b
- 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) sin^ A , sec^ B , cot^ A b) csc A , sin B , cot B c) tan A , cos B , sec A d) tan B , csc B , cos A
A C
B
a
c
b
- Determine the indicated ratios. The listed sides are those shown in the given triangle.
Write your ratios in exact form.
a) a = 3, b = 4. Find tan A and cos B b) a = 5, c = 13. Find cos A and csc B c) b = 9, c = 41. Find cot A and cos B d) 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