Transcript Trigonometry

Trigonometry
Basic Calculations of Angles and
Sides of Right Triangles
1
Introduction
• You can use the three trig functions (sin,
cos, and tan) to solve problems involving
right triangles.
2
Introduction
• If you have a right triangle, and you know
an acute angle and the length of one side,
you have enough info to compute the length
of either remaining side.
You could compute the
length of this side
(hypotenuse)...
7”
40°
…or this side.
3
Introduction
• If you have a right triangle, and you know
the lengths of two sides, you have enough
info to compute the size of either acute
interior angle.
55 mm
28 mm
…or this
angle.
You could compute this
angle...
4
Use trigonometry to determine the size
of an angle.
5
Determine an unknown angle
Example 1
• To start, we will determine the size of an
unknown angle when two sides of the right
triangle are known.
5.5”
A
12”
6
Determine an unknown angle
Example 1
• Let the unknown angle A be the reference
angle.
5.5”
A
12”
7
Determine an unknown angle
Example 1
• Now label the sides of the right triangle...
opposite
5.5”
hypotenuse
A
adjacent 12”
8
Determine an unknown angle
Example 1
• Note that we only know the lengths of the
opposite and adjacent sides.
opposite
5.5”
hypotenuse
A
adjacent 12”
9
Determine an unknown angle
Example 1
• So we need to pick a trig function that has
the opposite and adjacent sides in it...
opposite
5.5”
A
adjacent 12”
10
Determine an unknown angle
Example 1
• Which trig function should you pick?
opp
sin A 
hyp
opposite
adj
cos A 
hyp
5.5”
A
opp
tan A 
adj
You need to pick the
tangent function since it
is the only one that has
both opposite and
adjacent sides in it.
12”
adjacent
11
Determine an unknown angle
Example 1
• Now plug-in the numbers you have into the
tangent function...
opp
tan A 
adj
tan A  0.458333 ...
5.5
tan A 
12
Now use your
calculator to
solve. Type-in
.458333, press th
2nd function key,
then press the tan
key
A = 24.6°
opposite
5.5”
A
adjacent 12”
12
Determine an unknown angle
Example 1
• How could you determine the size of the
remaining angle?
…this one must be 65.4° degrees.
(Since 180° - 90° - 24.6° = 65.4°)
65.4°
5.5”
..and this one
was computed
to be 24.6°…
This angle is
90°…
24.6°
12”
13
Determine an unknown angle
Example 2
• Let’s try another one…
• Determine the size of angle A.
35 mm
A
31.5 mm
14
Determine an unknown angle
Example 2
• First, label the sides of the triangle...
hypotenuse
35 mm
opposite
A
31.5 mm
adjacent
15
Determine an unknown angle
Example 2
• Since you know the lengths of the adjacent
side and the hypotenuse, pick a trig function
that has both of these...
hypotenuse
35 mm
A
31.5 mm
adjacent
16
Determine an unknown angle
Example 2
• Which trig function should you pick?
opp
sin A 
hyp
adj
cos A 
hyp
hypotenuse
35 mm
A
31.5 mm
adjacent
You need to pick the
cosine function since it
is the only one that has
both the adjacent side
and hypotenuse in it.
opp
tan A 
adj
17
Determine an unknown angle
Example 2
• Now plug-in the numbers you have into the
Now use your calcula
cosine function...
solve. Type-in 0.9, pr
adj
31.5
cos A 
cos A 
hyp
35 cos A  0.9
hypotenuse
35 mm
2nd function key, then
the cos key
A  25.8o
A
31.5 mm
adjacent
18
Determine an unknown angle
Example 2
• Now that you know how big angle A is,
determine the size of the remaining angle.
35 mm
25.8°
31.5 mm
19
Determine an unknown angle
Example 2
• To determine the other angle:
• 180° - 90° - 25.8° = 64.2°
35 mm
64.2°
25.8°
31.5 mm
20
Determine an unknown angle
Example 3
• Let’s try one more.
• Determine the size of angle A.
125 mm
132 mm
A
21
Determine an unknown angle
Example 3
• Label the sides of the triangle...
opposite
125 mm
adjacent
132 mm
hypotenuse
A
22
Determine an unknown angle
Example 3
• Since you know the lengths of the opposite side
and the hypotenuse, pick a trig function that
contains them...
opposite
125 mm
132 mm
hypotenuse
A
23
Determine an unknown angle
Example 3
• Which trig function should you pick?
opp
sin A 
hyp
opposite
125 mm
132 mm
A
hypotenuse
adj
cos A 
hyp
opp
tan A 
adj
You need to pick the
sine function since it is
the only one that has
both the opposite side
and hypotenuse in it.
24
Determine an unknown angle
Example 3
• Now plug-in the numbers you have into the
sine function...
opposite
125 mm
Now use your calculator to solve. Type-in 0.947, press
the 2nd function key, then press the sin key
132 mm
hypotenuse
125
oppo
.3
sin AA  71
0.947
132
hyp
A
25
Determine an unknown angle
Example 3
• What is the size of the remaining angle?
125 mm
132 mm
71.3°
26
Determine an unknown angle
Example 3
• The angle is computed to be 18.7°.
125 mm
18.7°
132 mm
71.3°
27
Summary of Part I
• By now you should feel like you have a
pretty good chance of determining the size
of an angle when any two sides of a right
triangle are known.
• Click to see one more problem like the last
three you have done...
28
Summary of Part I
Example 4
• Determine the size of angle A.
• Solve the problem, then click to see the
answer.
25.5 ft
A
23 ft
29
Summary of Part I
Example 4
• Selecting the cos function will allow you to
determine the size of angle A.
23
adj
o
cos
 0.6.902
cos
A A25
25
.5
hyp
hypotenuse
25.5 ft
A
23 ft
adjacent
30
Use trigonometry to determine the
length of a side of a right triangle.
31
Determining the length of a side
• Recall that if you have a right triangle, and
you know an acute angle and the length of
one side, you have enough info to compute
the length of either remaining side.
You could compute the
length of this side
(hypotenuse)...
7”
40°
…or this side.
32
Determining the length of a side
Example 5
• In this problem, we will determine the
length of side x.
x
9”
26°
33
Determining the length of a side
Example 5
• As always, first label the sides of the
triangle...
opposite x
hypotenuse
9”
26°
adjacent
34
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...
opposite x
hypotenuse
9”
26°
35
Determining the length of a side
Example 5
• Which trig function should you pick?
opp
sin A 
hyp
adj
cos A 
hyp
opposite
hypotenuse
x
9”
26°
opp
tan A 
adj
You need to pick the
sine function since it is
the only one that has
both the opposite side
and hypotenuse in it.
36
Determining the length of a side
Example 5
• Now set-up the trig function:
Use basic algebra to solve this equation.
Multiply both sides of the equation by 9 to clear
the fraction.
opposite x
xx
opp
99sin
((sin
0.26
438
3
.95
xx (9)
A26
 ")
99
hyp
oo
hypotenuse
9”
26°
37
Determining the length of a side
Example 5
• Now you know the opposite side has a
length of 3.95”.
opposite 3.95”
hypotenuse
9”
26°
38
Determining the length of a side
Example 6
• Let’s try another one.
• Determine the length of side x.
x
75 mm
47°
39
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...
opposite
adjacent x
47°
75 mm
hypotenuse
40
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...
adjacent x
47°
75 mm
hypotenuse
41
Determining the length of a side
Example 6
• Which trig function should you pick?
opp
sin A 
hyp
adjacent
adj
cos A 
hyp
x
75 mm
hypotenuse
47°
You need to pick the
cosine function since it
is the only one that has
both the adjacent side
and hypotenuse in it.
opp
tan A 
adj
42
Determining the length of a side
Example 6
• Set-up your trig function...
To finish,
evaluate
47°this
(which
is 0.682) and
Use
basic algebra
tocos
solve
equation.
multiplyboth
by 75.
Multiply
sides of the equation by 75 to clear
the fraction.
adjacent x
47°
xx
oo adj
75
(
0
.
682
)

cos
cos
A
)   xx(75)
5147
.47
1 mm
75(cos
75
hyp
75
75 mm
hypotenuse
43
Determining the length of a side
Example 6
• Now you know the length of the adjacent
side is 51.1 mm.
51.1 mm
adjacent
47°
75 mm
hypotenuse
44
Determining the length of a side
Example 7
• Let’s try a little bit more challenging
problem.
• Determine the length of side x.
x
12 ft
35°
45
Determining the length of a side
Example 7
• Label the sides of the right triangle...
hypotenuse
x
opposite
12 ft
35°
adjacent
46
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.
hypotenuse
x
opposite
12 ft
35°
adjacent
47
Determining the length of a side
Example 7
• Which trig function should you pick?
opp
sin A 
hyp
adj
cos A 
hyp
hypotenuse
x
opposite
12 ft
35°
opp
tan A 
adj
You need to pick the
sine function since it is
the only one that has
both the opposite side
and hypotenuse in it.
48
Determining the length of a side
Example 7
• Now set-up your trig function.
12
opp
sin35
35
x ) 
A
20 .9 ft(ox)
x(sin
sin
x35
hyp
oo
Use algebra to solve this equation. Multiply both
Next, divide both sides by sin35° to isolate the
sides of the equation by x to clear the fraction.
unknown x.
hypotenuse
x
opposite
12 ft
35°
49
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.
35
35
35
35
35
opp
xx(tan
xx50
A
50
( x( x) )
x50
)2)
x(tan
(tan
) 35
tan
50
9.4
tan
o
tan
50xxx
1.1918
adj
o ooo
35 cm
50°
x
50
Determining the length of a side
Example 9
• Let’s try one more example.
• Determine the lengths of sides x and y.
65°
45.5 mm
y
x
51
Determining the length of a side
Example 9
• To start, you must determine which side (x
or y) you want to solve for first.
• It really doesn’t matter which one you pick.
65°
45.5 mm
y
x
52
Determining the length of a side
Example 9
• Let’s compute the length of side y first...
65°
45.5 mm
y
x
53
Determining the length of a side
Example 9
• Label the sides of the triangle...
65°
adjacent
y
hypotenuse
45.5 mm
x
opposite
54
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°
adjacent
y
hypotenuse
45.5 mm
x
opposite
55
Determining the length of a side
Example 9
• Which trig function should you pick?
opp
sin A 
hyp
adj
cos A 
hyp
adjacent
65°
hypotenuse
45.5 mm
y
x
opposite
You need to pick the
cosine function since it
is the only one that has
both the adjacent side
and hypotenuse in it.
opp
tan A 
adj
56
Determining the length of a side
Example 9
• Now set-up the trig function and solve for y.
yyyy
adj
o m
45
.
5
(
0
.
4226
)

19
.
2
m
45.5(cos
cos
(45.5)
cos65
A )
45
45.5.5
hyp
65°
adjacent
y
hypotenuse
45.5 mm
x
opposite
57
Determining the length of a side
Example 9
• Now we know side y is 19.2 mm long.
• The next question is, “How long is side x?”
65°
45.5 mm
19.2 mm
x
58
Determining the length of a side
Example 9
• You could use trig to solve for x, but why not use
the Pythagorean Theorem?
65°
45.5 mm
19.2 mm
x
59
Determining the length of a side
Example 9
• You know a leg and the hypotenuse of a right
triangle, so use this form of the theorem:
leg  hypotenuse 2  leg 2
leg  45.52  19.22
65°
45.5 mm
19.2 mm
leg  1701.61
leg  41.3 mm
x
60
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
19.2 mm
25°
41.3 mm
61
Summary
• After viewing this lesson you should be able
to:
– Compute an interior angle in a right triangle
when the lengths of two sides are known.
x
5.25”
8.75”
62
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.
60°
7.5”
x
63
Final Practice Problem
Example 10
• Determine the lengths of sides x and y and
the size of angle A.
• When you are done, click to see the answers
on the next screen.
x
A
y
15°
85 cm
64
Final Practice Problem
Example 10
• The answers are shown below...
88 cm
75°
22.8 cm
15°
85 cm
65
End of Presentation
66