Download 9.6 Warmup x cos 1.

9.6 Warmup
Find the value of x. Then find the ratio of sin θ ,
cos θ , and tan θ for the triangle.
12
1.
2.
3.
12
θ
5
θ
6
Find the value of the unknown sides.
4.
5.
April 1, 2016
Geometry 9.6 Solving Right Triangles
6.
? Classify.
7, 9, 12
1
Geometry
9.6 Solving Right Triangles
9.6 Essential Question
When you know the lengths of the sides
of a right triangle, how can you find the
measures of the two acute angles?
April 1, 2016
Geometry 9.6 Solving Right Triangles
3
Goals



Use inverse trig functions to find angle
measures.
Solve right triangles.
Solve problems using right triangles.
April 1, 2016
Geometry 9.6 Solving Right Triangles
4
Solving a triangle means…


Finding the lengths of the three sides.
Finding the measure of the three
angles.
In a right
A
c
B
April 1, 2016
a
Geometry 9.6 Solving Right Triangles
b
C
triangle,
one angle
is always
90, thus
we don’t
need to
worry
about it.
5
Our Tools to Solve Triangles:





Trig equations
Pythagorean Theorem
Inverse trig functions
The sum of int. angles of a triangle
A calculator (“Best Friend Calculator Guy”) –
for speed and accuracy
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Inverse Trig Function
𝑨 𝑖𝑠 𝑎𝑛 𝑎𝑛𝑔𝑙𝑒 𝑥 𝑖𝑠 𝑎 𝑡𝑟𝑖𝑔 𝑟𝑎𝑡𝑖𝑜
Trig Function

sin 𝐴 = 𝑥 means: the sine of an angle is a
trig ratio.
Inverse Trig Function
sin−1 𝑥 = 𝐴 means: the inverse sine of a trig
ratio is an angle.
Example:
 sin 80 = 0.9848
 sin-1 .9848 = 80

April 1, 2016
Geometry 9.6 Solving Right Triangles
7
Inverse Trig Functions

If sin A = x, then sin-1x = A.



𝑜
−1
sin
ℎ
=𝐴
If cos A = x, then cos-1x = A.


sin 𝐴 =
𝑜
ℎ
𝑐𝑜𝑠 𝐴 =
𝑎
ℎ
If tan A = x,

April 1, 2016
𝑡𝑎𝑛 𝐴 =
𝑜
𝑎
𝑎
−1
 c𝑜𝑠
=𝐴
ℎ
then tan-1x = A.

−1 𝑜
tan
𝑎
=𝐴
Geometry 9.6 Solving Right Triangles
8
Example 1



sin A = 0.7660. What is A?
A = sin-1(.766)
A  50
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 2



cos A = 0.2079. What is A?
A = cos-1(.2079)
A  78
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Geometry 9.6 Solving Right Triangles
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Example 3



tan A = 0.1051. What is A?
A = tan-1(.1051)
A  6
April 1, 2016
Geometry 9.6 Solving Right Triangles
11
Example 4: Solving a triangle
First, we will find A.
A
tan A = 7/12
A = tan-1(7/12)
c
12
A  30.3
7
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
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Example 4: Solving a triangle
Now find B.
A
tan B = 12/7
12
30.3
B = tan-1(12/7)
c
A  59.7
7
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
13
Example 4: Solving a triangle
Or…
A
12
30.3
The acute angles of a
right triangle are
complementary.
c
B = 90 – 30.3 = 59.7
59.7
7
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
14
Example 4: Solving a triangle
Find side c.
A
12
30.3
Pythagorean Theorem is
best because it doesn’t
use rounded data.
c
c  12  7
2
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2
c  144  49
2
59.7
7
2
B
c 2  193
c  13.9
Geometry 9.6 Solving Right Triangles
15
Example 4: Solving a triangle
The triangle is solved.
A
12
Notice: the measures are
all approximate.
30.3
13.9
59.7
7
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
16
You try it. Solve the triangle.
First, find angle A.
tan A = 32/15
A
A = tan-1(32/15)
c
A  64.9
15
32
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
17
You try it. Solve the triangle.
Next, find angle B.
A
tan B = 15/32
B = tan-1(15/32)
64.9
c
B  25.1
15
32
April 1, 2016
B
or…
90 – 64.9 = 25.1
Geometry 9.6 Solving Right Triangles
18
You try it. Solve the triangle.
c  15  32
Now find side c.
2
A
2
2
c  225  1024
2
64.9
c
c  1249
15
2
25.1
32
B
c  1249
c  35.3
April 1, 2016
Geometry 9.6 Solving Right Triangles
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You try it. Solve the triangle.
The triangle is solved.
A
64.9
35.3
15
25.1
32
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
20
Example 5: Solve the triangle.
A
52
b
16.5
Find A first, since it’s
the complement of the
other acute angle.
A = 90 – 38 = 52
38
a
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Geometry 9.6 Solving Right Triangles
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Example 5: Solve the triangle.
A
Now use sine to find a.
52
b
16.5
38
a
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a
sin 52 
16.5
16.5sin 52  a
a  13
Geometry 9.6 Solving Right Triangles
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Example 5: Solve the triangle.
A
Now use cosine to find b.
52
b
16.5
38
13.0
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b
cos52 
16.5
16.5cos52  b
b  10.2
Geometry 9.6 Solving Right Triangles
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Example 5: Solve the triangle.
A
The triangle is solved.
52
10.2
16.5
38
13.0
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Important



You can solve a triangle in any order
you want to, as long you have the data
you need for each step.
It’s best to not use rounded data in
any calculation.
Be very careful using a calculator.
CHECK EVERYTHING TWICE!!
April 1, 2016
Geometry 9.6 Solving Right Triangles
25
Your Turn: Solve this triangle.
A
25
c
10
April 1, 2016
B
Geometry 9.6 Solving Right Triangles
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Your Turn: Solution
A
c2 = 252 + 102
c2 = 725
25
c  26.9
c26.9
tan B = 25/10
B = tan-1(25/10)
68.2
10
B
B = 68.2
A = 90 – 68.2 = 21.8
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Indirect Measure


One of the most powerful uses of trig
is to measure things that can’t be
measured directly. This is indirect
measure.
It’s a fundamental process used in
surveying, map making, astronomy
and other applications.
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 6: Using a transit.
Jim the Surveyor uses a transit to
measure distances. He knows the
distance between the tree and the fire
hydrant is 110 ft. And to move from one
to the other he swings his transit
through 7.5. How far is he from each
object?
110 ft.
Jim
7.5
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 6: Solution
110
tan 7.5 
x
110
x
tan 7.5
x  835.5
110 ft.
Jim
7.5
x
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 6: Solution
110
sin 7.5 
y
110
y
sin 7.5
y  842.7
y
Jim
110 ft.
7.5
835.5
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 6: Is this correct?
YES!
110 ft.
Jim
7.5
835.5
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Example 6: Indirect Measure
Using trig, Jim can
determine the distances to
the tree and the fire
hydrant without measuring
them directly.
110 ft.
Jim
7.5
835.5
April 1, 2016
Geometry 9.6 Solving Right Triangles
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Summary



Solving a triangle means to find all six
parts: 3 angles, 3 sides.
Use inverse trig function (sin-1, cos-1,
tan-1) to find angles.
Use the given data to calculate values,
when possible.
April 1, 2016
Geometry 9.6 Solving Right Triangles
34
Homework
April 1, 2016
Geometry 9.6 Solving Right Triangles
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