9.6 - Solving Right Triangles (Part 1)

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9.6 - Solving Right Triangles (Part 1)

• Geometry
• Mr. Valdez
• Spring 2012

2
Objectives

• Solve a right triangle.
• Use right triangles to solve real-life problems,
  such as finding the glide angle and altitude of a
  space shuttle.

3
Solving a Right Triangle

• Every right triangle has one right angle, two
  acute angles, one hypotenuse and two legs. To
  solve a right triangle, means to determine the
  measures of all six (6) parts. You can solve a
  right triangle if the following one of the two
  situations exist
• Two side lengths
• One side length and one acute angle measure

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Note

• As you learned in Lesson 9.5, you can use the
  side lengths of a right triangle to find
  trigonometric ratios for the acute angles of the
  triangle. As you will see in this lesson, once
  you know the sine, cosine, or tangent of an acute
  angle, you can use a calculator to find the
  measure of the angle.

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Write This Down!

• In general, for an acute angle A
• If sin A x, then sin-1 x m?A
• If cos A y, then cos-1 y m?A
• If tan A z, then tan-1 z m?A

The expression sin-1 x is read as the inverse
sine of x.

• On your calculator, this means you will be
  punching the 2nd function button (usually in
  yellow) prior to doing the calculation. This is
  to find the degree of the angle.

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Angle (degrees)
Ratio

• sin A x
• sin-1 x m?A

Ratio
Angle (degrees)
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Ex. 1

• Solve the right triangle. Round the decimals to
  the nearest tenth.

HINT Start by using the Pythagorean Theorem.
You have side a and side b. You dont have the
hypotenuse which is side cdirectly across from
the right angle.
Find all 3 sides and all 3 angles!
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Ex. 1
(hypotenuse)2 (leg)2 (leg)2
Pythagorean Theorem
Substitute values
c2 32 22
Simplify
c2 9 4
Simplify
c2 13
Find the positive square root
c v13
Use a calculator to approximate
c 3.6
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Ex. 1 continued

• Then use a calculator to find the measure of ?B
• 2nd function
• Tangent button
• 2
• Divided by symbol
• 3

m ?B 33.7
Make sure you are in Degree Mode!!!
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Ex. 1 continued

• Because the sum of the measures of a triangle is
  180, we can write
• m?A 180 - 90 - 33.7 56.3
• ?The side lengths of the triangle are 2, 3 and
  v13, or about 3.6. The triangle has one right
  angle and two acute angles whose measure are
  about 33.7 and 56.3.

This is a complete answer!
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9.6 - Solving Right Triangles (Part 2)

• Geometry
• Mr. Valdez
• Spring 2012

12
Part 1 Recap

• Solving a right triangle consists of
• Finding all three sides
• Finding all three angles
• You can find the measure of an angle given a
  triangles sides by using inverse trigonometric
  functions

Ratio
Angle (degrees)
sin A x sin-1 x m?A
Ratio
Angle (degrees)
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Ex. 1 Solving a Right Triangle (h)

• Solve the right triangle. Round decimals to the
  nearest tenth.

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You are looking for opposite and hypotenuse which
is the sin ratio.
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.4226) h
5.5 h
Use your calculator to approximate.
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Ex. 1 Solving a Right Triangle (g)

• Solve the right triangle. Round decimals to the
  nearest tenth.

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You are looking for adjacent and hypotenuse which
is the cosine ratio.
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
13(0.9063) g
11.8 g
Use your calculator to approximate.
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Ex. 2 Solving a Right Triangle

• Solve the right triangle. Round decimals to the
  nearest tenth.

Sides AB BC CA
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8.9
8
 
48.2
41.8
90
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Ex. 3 Solving a Right Triangle

• Solve the right triangle. Round decimals to the
  nearest tenth.

Sides LM MN NL
6.6
12.4
14
 
62
90
28
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Using Right Triangles in Real Life

• Space Shuttle During its approach to Earth, the
  space shuttles glide angle changes.
• A. When the shuttles altitude is about 15.7
  miles, its horizontal distance to the runway is
  about 59 miles. What is its glide angle? Round
  your answer to the nearest tenth.

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Solution
Glide ? x
15.7 miles

• You know opposite and adjacent sides. If you
  take the opposite and divide it by the adjacent
  sides, then take the inverse tangent of the
  ratio, this will yield you the slide angle.

59 miles
opp.
tan x
Use correct ratio
adj.
15.7
Substitute values
tan x
59
Key in calculator 2nd function, tan-1 (15.7/59)
14.9
? When the space shuttles altitude is about
15.7 miles, the glide angle is about 14.9.
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Part B
Glide ? 19

• When the space shuttle is 5 miles from the
  runway, its glide angle is about 19. Find the
  shuttles altitude at this point in its descent.
  Round your answer to the nearest tenth.

h
5 miles
opp.
tan 19
Use correct ratio
adj.
h
Substitute values
tan 19
5
h
Isolate h by multiplying by 5.
5 tan 19
5
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? The shuttles altitude is about 1.7 miles.
1.7 h
Approximate using calculator
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Reminders

• Homework due tomorrow!
• pgs. 570-571 18-30
• Make sure you include all parts of the answer
  (all three side lengths and all three angle
  measures)!
• Bring calculators this week!
• Chapter 9 Test on Friday!