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Right Triangle Trigonometry
The Pythagorean Theorem
The Pythagorean Theorem states that the following relationship exists among the lengths of
the legs, a and b, and the length of the hypotenuse, c, of any right triangle.
a2  b2  c2
Use the Pythagorean Theorem to find the value of x in each triangle.
a2  b2  c2
Pythagorean Theorem
a2  b2  c2
x2  62  92
Substitute.
x2  42  (x  2)2
x2  36  81
Take the squares.
x2  16  x2  4x  4
x2  45
x
45
x 3 5
4x  12
Simplify.
x3
Take the positive
square root and
simplify.
Find the value of x. Give your answer in simplest radical form.
1.
2.
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3.
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4.
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Reteach
The Pythagorean Theorem continued
A Pythagorean triple is a set of three
nonzero whole numbers a, b, and c
that satisfy the equation a2  b2  c2.
Pythagorean
Triples
Not Pythagorean
Triples
3, 4, 5,
5, 12, 13
2, 3, 4
6, 9, 117
You can use the following theorem to classify triangles by their angles if you know their side
lengths. Always use the length of the longest side for c.
Pythagorean Inequalities Theorem
mC  90o
mC  90o
If c2  a2  b2, then nABC is obtuse.
If c2  a2  b2, then nABC is acute.
Consider the measures 2, 5, and 6. They can be the side lengths of a triangle since
2  5  6, 2  6  5, and 5  6  2. If you substitute the values into c2  a2  b2, you
get 36  29. Since c2  a2  b2, a triangle with side lengths 2, 5, and 6 must be obtuse.
Find the missing side length. Tell whether the side lengths form a
Pythagorean triple. Explain.
5.
6.
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Tell whether the measures can be the side lengths of a triangle.
If so, classify the triangle as acute, obtuse, or right.
7. 4, 7, 9
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10. 9, 12, 15
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8. 10, 13, 16
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11. 5, 14, 20
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9. 8, 8, 11
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12. 4.5, 6, 10.2
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Applying Special Right Triangles
Theorem
Example
450-450-900 Triangle Theorem
In a 450-450-900 triangle, both legs are
congruent and the length of the hypotenuse
is 2 times the length of a leg.
In a 450-450-900triangle, if a leg
length is x, then the hypotenuse
length is x 2.
Use the 450-450-900 Triangle Theorem to find the value of x inEFG.
Every isosceles right triangle is a 450-450-900 triangle. Triangle
EFG is a 458-458-908 triangle with a hypotenuse of length 10.
10  x 2
10
2

x 2
5 2x
2
Hypotenuse is
2 times the length of a leg.
Divide both sides by
2.
Rationalize the denominator.
Find the value of x. Give your answers in simplest radical form.
1.
2.
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3.
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4.
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Applying Special Right Triangles
Theorem
Examples
300-600-900 Triangle Theorem
In a 300-600-900 triangle, the length of the
hypotenuse is 2 multiplied by the length of
the shorter leg, and the longer leg is 3
multiplied by the length of the shorter leg.
In a 300-600-900 triangle, if the shorter leg
length is x, then the hypotenuse length
is 2x and the longer leg length is x.
Use the 300-600-900 Triangle Theorem to find the values
of x and y inHJK.
12  x 3
12
3
x
4 3x
Longer leg  shorter leg multiplied by
Divide both sides by
3.
3.
Rationalize the denominator.
y  2x
Hypotenuse  2 multiplied by shorter leg.
y  2(4 3)
Substitute 4 3 for x.
y 8 3
Simplify.
Find the values of x and y. Give your answers in simplest radical
form.
5.
6.
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7.
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8.
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Trigonometric Ratios
Trigonometric Ratios
leg opposite A 4
  0.8
hypotenuse
5
leg adjacent to A 3
cos A 
  0.6
hypotenuse
5
leg opposite A
4
tan A 
  1.33
leg adjacent to A 3
sin A 
leg opposite A
hypotenuse
leg adjacent to A
You can use special right triangles to write trigonometric ratios as fractions.
sin 45  sin Q 


So sin 45 
leg opposite Q
hypotenuse
x
x 2

1
2
2
2
2
.w
2
Write each trigonometric ratio as a fraction and as a decimal
rounded to the nearest hundredth.
1. sin K
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3. cos K
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2. cos H
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4. tan H
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Use a special right triangle to write each trigonometric ratio as a
fraction.
5. cos 45
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7. sin 60
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6. tan 45
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8. tan 30
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Trigonometric Ratios continued
You can use a calculator to find the value of trigonometric ratios.
cos 38  0.7880107536 or about 0.79
You can use trigonometric ratios to find side lengths of triangles.
Find WY.
cos W 
adjacent leg
hypotenuse
cos 39 
7.5 cm
WY
Substitute the given values.
WY 
7.5
cos39
Solve for WY.
Write a trigonometric ratio that involves WY.
WY  9.65 cm
Simplify the expression.
Use your calculator to find each trigonometric ratio. Round to the
nearest hundredth.
9. sin 42
10. cos 89
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11. tan 55
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12. sin 6
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Find each length. Round to the nearest hundredth.
13. DE
14. FH
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15. JK
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16. US
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Solving Right Triangles
Use the trigonometric ratio sin A  0.8 to determine which angle of the triangle
is A.
sin 1 

leg opposite 1
hypotenuse
sin 2 
6
10

 0.6
leg opposite 2
hypotenuse
8
10
 0.8
Since sin A  sin 2, 2 is A.
If you know the sine, cosine, or tangent of an acute angle measure, then you can
use your calculator to find the measure of the angle.
Inverse Trigonometric Functions
Symbols
sin A  x  sin1 x  mA
cos B  x  cos1 x  mB
tan C  x  tan1 x  mC
Examples
sin 30 
cos 45 
1
 1
 sin1    30
2
2
 2
2
 cos1 
 45
 2 
2


tan 76  4.01  tan1 (4.01)  76
Use the given trigonometric ratio to determine which angle of the triangle
is A.
1. sin A 
1
2
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3. cos A  0.5
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2. cos A 
13
15
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4. tan A 
15
26
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Use your calculator to find each angle measure to the nearest degree.
5. sin1 (0.8)
6. cos1 (0.19)
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7. tan1 (3.4)
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 1
8. sin1  
5
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Angles of Elevation and Depression
An angle of elevation is formed by a
horizontal line and a line of sight above it.
An angle of depression is formed by a
horizontal line and a line of sight below it.
Classify each angle as an angle of elevation or an angle of depression.
1. 1
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2. 2
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Use the figure for Exercises 3 and 4. Classify each angle as an angle
of elevation or an angle of depression.
3. 3
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4. 4
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Use the figure for Exercises 5–8. Classify each angle as an angle of elevation
or an angle of depression.
5. 1
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6. 2
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7. 3
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8. 4
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
Angles of Elevation and Depression continued
You can solve problems by using angles of elevation and angles of depression.
Sarah is watching a parade from a 20-foot balcony. The angle of depression
to the parade is 47. What is the distance between Sarah and the parade?
Draw a sketch to represent the given information. Let A represent
Sarah and let B represent the parade. Let x represent the distance
between Sarah and the parade.
mB  47 by the Alternate Interior Angles Theorem. Write a sine ratio
using B.
20  leg opposite B
sin 47 
ft
 hypotenuse
x
x sin 47  20 ft
x
20
ft
sin 47
27 ft  x
Multiply both sides by x.
Divide both sides by sin 47.
Simplify the expression.
The distance between Sarah and the parade is about 27 feet.
9. When the angle of elevation to the sun
is 52, a tree casts a shadow that is
9 meters long. What is the height of
the tree? Round to the nearest tenth
of a meter.
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11. Jared is standing 12 feet from a
rock-climbing wall. When he looks up
to see his friend ascend the wall, the
angle of elevation is 56. How high up
the wall is his friend? Round to the
nearest foot.
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10. A person snorkeling sees a turtle on the
ocean floor at an angle of depression of
38. She is 14 feet above the ocean floor.
How far from the turtle is she? Round to
the nearest foot.
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12. Maria is looking out a 17-foot-high
window and sees two deer. The angle of
depression to the deer is 26. What is the
horizontal distance from Maria to the
deer? Round to the nearest foot.
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© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date __________________ Class __________________
© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry