Section 6.1
Trigonometric Functions of Acute
Angles
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives
Determine the six trigonometric ratios for a given acute angle of a right triangle.
Determine the trigonometric function values of 30º, 45º, and 60º.
Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle.
Given the function values of an acute angle, find the function values of its complement.
Right Triangles and Acute AnglesAn acute angle is an angle with measure greater than 0º and less than 90º.
Greek letters such as (alpha), (beta), (gamma), (theta), and (phi) are often used to denote an angle.
Side opposite
Side adjacent to
Hypotenuse
We label the sides with respect to angles. The hypotenuse is opposite the right angle. There is the side opposite and the side adjacent to .
Trigonometric Ratios
The lengths of the sides of a right triangle are used to define the six trigonometric ratios:
sine (sin)
cosine (cos)
tangent (tan)
Side opposite
Side adjacent to
Hypotenuse
cosecant (csc)
secant (sec)
cotangent (cot)
Trigonometric Function Values of an Acute Angle Let be an acute angle of a right triangle. Then the six trigonometric functions of are as follows:
sin side opposite
hypotenuse
cos side adjacent to
hypotenuse
tan side opposite
side adjacent to
csc hypotenuse
side opposite
sec hypotenuse
side adjacent to
cot side adjacent to side opposite
Example
In the triangle shown, find the six trigonometric function values of (a) and (b) .
a) sin opp
hyp
12
13
13
5
12
cos adj
hyp
5
13
tan opp
adj
12
5
csc hyp
opp
13
12
sec hyp
adj
13
5
cot adj
opp
5
12
Example
In the triangle shown, find the six trigonometric function values of (a) and (b) .
a) sin opp
hyp
5
13
13
5
12
cos adj
hyp
12
13
tan opp
adj
5
12
csc hyp
opp
13
5
sec hyp
adj
13
12
cot adj
opp
12
5
Reciprocal Functions
Note that there is a reciprocal relationship between pairs of the trigonometric functions.
csc 1
sin1
seccos
cot 1
tan
Example
Given that sin 4
5, cos
3
5, and tan
4
3,
Solution:
csc 1
sin
14
5
5
4
sec 1
cos
13
5
find csc , sec , and cot .
5
3
cot 1
tan
14
3
3
4
Example
If and is an acute angle, find the other five trigonometric function values of .
sin 6
7
Solution:
6
7
opp
hyp
Use the definition of the sine function that the ratio
and draw a right triangle.
7
a
6
Use the Pythagorean equation to find a. a2 b2 c2
a2 62 72
a2 36 49
a2 49 36 13
a 13
Example (cont)
Use the lengths of the three sides to find the other five ratios.
sin 6
7
cos 13
7
tan 6
13
6 13
13
csc 7
6
sec 7
13
7 13
13
cot 13
6
Function Values of 45º
A right triangle with one 45º, must have a second 45º, making it an isosceles triangle, with legs the same length. Consider one with legs of length 1.
sin 45ºopp
hyp
1
2
2
20.7071
45º
45º
1
1
2 cos 45ºadj
hyp
1
2
2
20.7071
tan 45ºopp
adj
1
11
Function Values of 30º
A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it.
sin 30º1
20.5,
30º
60º
2
1
3cos 30º
3
20.8660,
tan 30º1
3
3
30.5774
Function Values of 60º
A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it.
sin60º3
20.8660,
cos60º1
20.5,
3tan 60º 3 1.7321
1
30º
60º
2
1
3
Example
As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.)
Solution:
Draw the situation, label the acute angle and length of the adjacent side.
Example (cont)
tan 30ºopp
adjh
1.2
1.2 tan 30ºh
1.23
3h
0.7 h
The balloon is approximately 0.7 mi, or 3696 ft, high.
Function Values of Any Acute Angle
Angles are measured either in degrees, minutes, and seconds: 1º = 60´, 1´ = 60´´; referred to as the DºM´S´´ form
61 degrees, 27 minutes, 42 seconds 61º 27 42
or are measured in decimal degree form, expressing the fraction parts of degrees in decimal form
61º 27 42 61.451
Examples
Find the trigonometric function value, rounded to four decimal places, of each of the following:
0.5703899297
Solution:
Check that the calculator is in degree mode.
a) tan29.7º
a) tan29.7º 0.5704
b) sec 48º c) sin84º1 0 3 9
1
cos 48ºb) sec 48º 1.49447655
c) sin84º1 0 3 9
1.49445
0.9948409474 0.9948
Example
A window-washing crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position?
Solution:
Draw the situation, label the hypotenuse and length of the side adjacent to .
Example (cont)
cos adj
hyp
6.5 ft
25 ft
0.26
74.92993786º
Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground.
Use a calculator to find the acute angle whose cosine is 0.26:
Cofunction Identities
Two angles are complementary whenever the sum of their measures is 90º. Here are some relationships.
sin cos 90º
90º – cos sin 90º tan cot 90º cot tan 90º sec csc 90º csc sec 90º
Example
Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan 18º ≈ 0.3249, find the six trigonometric function values of 72º.
csc18º1
sin18º3.2361
Solution:
sec18º1
cos18º1.0515
cot18º1
tan18º3.0777
sin 72ºcos18º0.9511
cos 72ºsin18º0.3090
tan 72ºcot18º3.0777
cot 72ºtan18º0.3249
sec 72ºcsc18º3.2361
csc 72ºsec18º1.0515