Transcript 8-4 Trigonometry

8-4 Trigonometry
The student will be able to:
1. Find trigonometric ratios using right triangles.
2. Use trigonometric ratios to find angle measures in
right triangles.
Trigonometric Ratios
Trigonometric ratio – a ratio of the lengths of two sides
of a right triangle.
By AA Similarity, a right triangle with
the same acute angle measure is
similar to every other right triangle
with the same acute angle measure.
In a right triangle, if you know the measures of two sides or
if you know the measures of one side and an acute angle,
then you can find all the measures of the missing sides or
angles of the triangle.
The three most common trigonometric ratios are:
sin =
opp
hyp
cos =
adj
hyp
tan =
opp
adj
Example 1:
Find sin J, cos J, tan J, sin K, cos K, and tan K. Express each ratio
as a fraction and as a decimal to the nearest hundredth.
opp
hyp
5
=
13
sin J =
adj
hyp
12
=
13
cos J =
≈ 0.92
≈ 0.38
sin K =
opp
hyp
12
13
≈ 0.92
=
cos K =
adj
hyp
5
13
≈ 0.38
=
opp
adj
5
=
12
tan J =
≈ 0.42
opp
adj
12
=
5
tan K =
≈ 2.4
Example 2:
Use a special right triangle to express the cosine of 45° as a
fraction and as a decimal to the nearest hundredth.
1st – Draw a 45°-45°-90° right triangle,
label the side lengths with x as the length
of the legs.
2nd – How do we find cos?
adj
hyp
x
cos 45 =
x 2
1
cos 45 =
2
1 æ 2ö
cos 45 =
ç
÷
2è 2ø
1 2
2
cos 45 =
or
2
2
cos =
cos45 » 0.71
x 2
x
x
As a fraction.
As a decimal.
Example 3: Real Life Situation
The front of the vacation cottage shown is an isosceles triangle.
What is the height (x) of the cottage above its foundation? What
is the length (y) of the roof? Explain your reasoning.
Hint: If it’s an isosceles triangle, then the
base is bisected.
1st – This is a 30°-60°-90° right triangle.
Which acute angle do we know? 60°
Hint: When you are given the angle degree
your calculator must be in degree mode.
2nd
- How can you find the measure of x with
the 60° angle? Adjacent side Opposite side
32.5
3rd – Which trigonometric ratio uses the
opposite side & the adjacent side?
opp
adj
x
tan60 =
32.5
tan =
(32.5)tan 60 = x
56 ft ≈ x
4th – What is the length of the roof? 65 ft
It’s an equilateral triangle.
All sides are equal.
You Try It:
1. Express each ratio of angle L as a fraction and as a decimal to
the nearest hundredth.
opp
hyp
12
=
37
sin L =
adj
hyp
35
=
37
cos L =
≈ 0.32
≈ 0.95
opp
adj
12
=
35
tan L =
≈ 0.34
2. A fitness trainer sets the incline on a treadmill to 7°. The
walking surface is 5 feet long. Approximately how many inches
did the trainer raise the end of the treadmill from the floor. ≈ 7.3 in
opp
= sin 7
hyp
y
= sin 7
60
y = (60)sin 7
y ≈ 7.3 in
Use Inverse Trigonometric Ratios
To find an angle measure when you have two side measures:
1. Determine which trig ratio applies.
2. Use the inverse of the trig function on your calculator
(Hint: use the 2nd key).
3. The total of the angle measures may not equal 180° due
to rounding.
4. Round angles to the nearest degree and sides to the
nearest tenth.
If sin N = x, then sin-1 x = mÐN.
æ 35 ö
sin ç ÷ = mÐN
è 37 ø
-1
If cos N = x, then cos-1 x = mÐN.
æ 12 ö
cos ç ÷ = mÐN
è 37 ø
-1
If tan N = x, then tan-1 x = mÐN.
æ 35 ö
tan ç ÷ = mÐN
è 12 ø
-1
Example 4:
Find x. Round to the nearest degree if necessary.
1st – Which angle are you looking for?
ÐA
2nd – Which trig ratio applies?
opp
= sin A
hyp
3rd – What’s the inverse?
æ
ö
-1 opp
sin ç
÷ = mÐA
è hyp ø
-1 æ 12 ö
sin ç ÷ = mÐA
è 20 ø
Example 5:
Use a calculator to find the measure of ÐA to the nearest degree if
necessary.
1st – Which angle are you looking for?
ÐA
2nd – Which trig ratio applies?
opp
= tan A
adj
3rd – What’s the inverse?
æ
ö
-1 opp
tan ç
÷ = mÐA
è adj ø
-1 æ 6 ö
tan ç ÷ = mÐA
è 20 ø
Solving a Right Triangle
Remember: To solve a right triangle you must know (1) two side
lengths or (2) one side length and the measure of one acute angle.
Solve the right triangle. Round side measures to the nearest tenth
and angle measures to the nearest degree.
1st – You’re given two sides. Find
the measure of the missing side.
a2 + b2 = c 2
b2 = 144
52 + b2 = 132
b = 12
25 + b2 = 169
2nd – Choose one of the acute angles
and solve for it. Let’s start with F.
æ opp ö
sin ç
÷ = mÐF
è hyp ø
-1 æ 5 ö
sin ç ÷ = mÐF
è 13 ø
-1
12
3rd – Solve for the remaining
acute angle.
æ adj ö
cos ç
÷ = mÐG
è hyp ø
æ5ö
cos-1 ç ÷ = mÐG
è 13 ø
-1
Example 6: Solve the right triangle. Round side measures to the
nearest tenth and angle measures to the nearest degree.
1st
– You’re given one side and one angle. Find
the measure of one of the missing sides.
adj
hyp
x
cos62 =
10
4.7
8.8
cos =
(10)cos 62 = x
4.7 ≈ x
2nd – You know two sides. Find the 3rd – Find the measure of the
measure of the missing side.
missing acute angle.
æ
ö
a2 + b2 = c 2
-1 opp
b2 = 77.9
sin ç
÷ = mÐC
2
2
2
4.7 + b = 10
è hyp ø
b ≈ 8.8
-1 æ 4.7 ö
22.1 + b2 = 100
sin ç ÷ = mÐC
è 10 ø
You Try it:
1. Use a calculator to find the measure of ÐPto the nearest
degree. 47
æ
ö
-1 adj
cos ç
÷ = mÐP
è hyp ø
-1 æ 13 ö
cos ç ÷ = mÐP
è 19 ø
2. Solve the right triangle. Round side measures to the nearest
hundredth and angle measures to the nearest degree.
a2 + b2 = c 2
42 + 72 = c2
16 + 49 = c2
65 = c2
8.06 = c
æ opp ö
tan ç
÷ = mÐA
è adj ø
æ4ö
tan -1 ç ÷ = mÐA
è 7ø
-1
æ opp ö
tan ç
÷ = mÐB
è adj ø
æ 7ö
tan -1 ç ÷ = mÐB
è4ø
-1