Download - Trigonometry
2
Introduction
• You can use the three trig functions (sin, cos, and tan) to solve problems involving right triangles.
Using a Calculator
You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.
Sample keystroke sequencesSample calculator
displayRounded
approximation
3.4874
0.2756
0.96130.96126169574 or 74sin Entersin
74 or 74cos cos Enter
74 or 74tan tan Enter
0.275637355
3.487414444
5
9”
26°
x
Determining the length of a side Example 5
• In this problem, we will determine the length of side x.
6
9”
26°
xopposite
hypotenuse
adjacent
Determining the length of a side Example 5
• As always, first label the sides of the triangle...
7
9”
26°
xopposite
hypotenuse
Determining the length of a side Example 5
• Since you know the length of the hypotenuse and want to know the length of the opposite side, you should pick a trig function that contains both of them...
8
hyp
oppA sin
hyp
adjA cos
adj
oppA tan
You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.
Determining the length of a side Example 5
• Which trig function should you pick?
9”
26°
x
oppo
site
hypotenuse
9
9”
26°
xopposite
hypotenuse
hyp
oppA sin
926sin
xo x)438.0(9
Determining the length of a side Example 5
• Now set-up the trig function:
x"95.3 )9(9
)26(sin9xo
Use basic algebra to solve this equation. Multiply both sides of the equation by 9 to clear the fraction.
10
9”
26°
3.95”opposite
hypotenuse
Determining the length of a side Example 5
• Now you know the opposite side has a length of 3.95”.
11
75 mm
47°
x
Determining the length of a side Example 6
• Let’s try another one.
• Determine the length of side x.
12
75 mm
47°
x
hypotenuse
opposite
adjacent
Determining the length of a side Example 6
• Since the known angle (47°) will serve as your reference angle, you can label the sides of the triangle...
13
75 mm
47°
x
hypotenuse
adjacent
Determining the length of a side Example 6
• You know the length of the hypotenuse and want to know the length of the adjacent side, so pick a trig function which contains both of them...
14
hyp
oppA sin
hyp
adjA cos
adj
oppA tan
You need to pick the cosine function since it is the only one that has both the adjacent side and hypotenuse in it.
Determining the length of a side Example 6
• Which trig function should you pick?
75 mm
47°
x
hypotenuse
adja
cent
15
75 mm
47°
x
hypotenuse
adjacent
hyp
adjA cos
7547cos
xo
Use basic algebra to solve this equation. Multiply both sides of the equation by 75 to clear the fraction. )75(
75)47(cos75
xo xmm 1.51 x)682.0(75
To finish, evaluate cos 47° (which is 0.682) and multiply by 75.
Determining the length of a side Example 6
• Set-up your trig function...
16
75 mm
47°
51.1 mm
hypotenuseadjacent
Determining the length of a side Example 6
• Now you know the length of the adjacent side is 51.1 mm.
17
12 ft
35°
x
Determining the length of a side Example 7
• Let’s try a little bit more challenging problem.
• Determine the length of side x.
18
12 ft
35°
x oppositehypotenuse
adjacent
Determining the length of a side Example 7
• Label the sides of the right triangle...
19
12 ft
35°
x oppositehypotenuse
adjacent
Determining the length of a side Example 7
• Which trig function will you pick? You know the length of the side opposite and want to know the length of the hypotenuse.
20
hyp
oppA sin
hyp
adjA cos
adj
oppA tan
You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it.
Determining the length of a side Example 7
• Which trig function should you pick?
35°
xhypotenuse
12 ft
opposite
21
12 ft
35°
x oppositehypotenuse
hyp
oppA sin
xo 12
35sin ft 9.20x
Use algebra to solve this equation. Multiply both sides of the equation by x to clear the fraction.
)(12
)35(sin xx
x o ox
35sin
12
Next, divide both sides by sin35° to isolate the unknown x.
Determining the length of a side Example 7
• Now set-up your trig function.
22
Determining the length of a side Example 8
• The reason the last problem was a little bit more difficult was the fact that you had an unknown in the denominator of the fraction.
• Keep clicking to see a similar trig function solved.
adj
oppA tan
xo 35
50tan ox
50tan
35 9.42x )(
35)50(tan x
xx o )(
35)50(tan x
xx o 35)50(tan ox
1918.1
35x
50°
35 cm
x
23
Determining the length of a side Example 9
• Your turn.
• Determine the lengths of sides x and y.
65°
45.5 mm
x
y
27
Determining the length of a side Example 9
• Since you want to know the length of side y (adjacent) and you know the length of the hypotenuse, which trig function should you select?
65°
45.5 mm
x
y
hypotenuse
opposite
adjacent
33
Determining the length of a side Example 9
• Both sides have been determined, one by trig, the other using the Pythagorean Theorem.
• Also the size of the other acute interior angle is indicated...
65°
45.5 mm
41.3 mm
19.2 mm
25°
35
Summary
• After viewing this lesson you should be able to:– Compute the length of any side of a right
triangle as long as you know the length of one side and an acute interior angle.
7.5”x60°