Right Triangles and Trigonometry

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Right Triangles and Trigonometry

• Chapter 8

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Aim 8-1 How do we use the converse of the
Pythagorean Theorem?

• Pythagorean Theorem
• Theorem 8-1
• The right triangle, the sum of the squares of the
  lengths of the legs is equal to the square of the
  length of the hypotenuse.

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• Pythagorean Triple is a set of nonzero whole
  numbers a, b, and c that satisfy the equation
• Here are some examples 3, 4, 5
• 5, 12,
  13 8, 15, 17 7, 24, 25

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Pythagorean Triples

• Find the length of the hypotenuse of ?ABC. Do the
  lengths of the sides of ?ABC form a Pythagorean
  Triple?

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Pythagorean Triples

• A right triangle has a hypotenuse of length 25
  and a leg length 10. Find the length of the other
  leg. Do the lengths of the sides form a
  Pythagorean triple?

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Using Simplest Radical Form

• Find the value of x. Leave your answer in
  simplest radical form.

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Using Simplest Radical Form

• The hypotenuse of a right triangle has length 12.
  One leg has length 6. Find the length of the
  other leg. Leave your answer in simplest radical
  form.

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The Converse of the Pythagorean Theorem

• You can use the Converse of the Pythagorean
  Theorem to determine whether a triangle is a
  right triangle.
• Converse of the Pythagorean Theorem
• Theorem 8-2
• If the square of the length of one side of a
  triangle is equal to the sum of the squares of
  the lengths of the other two sides, then the
  triangle is right triangle.

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Is It a Right Triangle?

• Is this a right triangle?

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Is It a Right Triangle?

• A triangle has sides of lengths 16, 48 and 50. Is
  the triangle a right triangle?

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• You can use the squares of the lengths of the
  sides of a triangle to find whether the triangle
  is acute or obtuse.

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Theorem 8-3

• If the square of the length of the longest side
  of a triangle is greater than the sum of the
  squares of the lengths of the other two sides,
  the triangle is obtuse.
• If , the triangle is obtuse.

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Theorem 8-4

• If the square of the length of the longest side
  of a triangle is less than the sum of the squares
  of the lengths of the other two sides, the
  triangle is acute.
• If , the triangle is
  acute.

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Classifying Triangles as Acute, Obtuse or Right

• Classify the triangle whose sides are 6, 11, and
  14 as acute, obtuse or right.
• Note Substitute the greatest length for c.

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Classifying Triangles as Acute, Obtuse or Right

• A triangle has sides of lengths 7, 8, and 9.
  Classify the triangle by its angles.

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Summary Answer in complete sentences.

• Find a third whole number so that the three
  numbers form a Pythagorean triple.
• 20, 21
• Explain how you can classify a triangle as being
  obtuse or acute.

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Aim 8-2 How do we use properties of special
right triangles?

• The acute angles of an isosceles right triangle
  are both 45 angles. Another name for an
  isosceles right triangle is a 45-45-90
  triangle. If each leg has length x and the
  hypotenuse has length y, you can solve for y in
  terms of x.

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Theorem 8-5 45-45-90 Triangle Theorem

• In a 45-45-90 triangle, both legs are
  congruent and the length of the hypotenuse is
  times the length of a leg.
• Hypotenuse leg

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Finding the Length of the Hypotenuse

• Find the value of each variable.
• 1. 2.

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Finding the Length of the Hypotenuse

• Find the length of the hypotenuse of a
  45-45-90 triangle with legs of length 5 .

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Finding the Length of a Leg

• What is the value of x?
• a) 3 b) 3 c) 6 d) 6

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Finding the Length of a Leg

• Find the length of a leg 45-45-90 triangle
  with a hypotenuse of length 10.

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Using 30-60-90 Triangles

• In a 30-60-90 triangle, the length of the
  hypotenuse is twice the length of the shorter
  leg. The length of the longer leg is times
  the length of the shorter leg.
• Hypotenuse 2 shorter leg
• Longer leg shorter leg

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Using the Length of One Side

• Find the value of each variable.
• 5 d longer leg shorter leg
• Solve for d.
• f 2d hypotenuse 2 shorter leg

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Using the Length of One Side

• Find the value of each variable.

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Summary Answer in complete sentences.

• Sandra drew the triangle below. Rita said that
  the lengths couldnt be correct. With which
  student do you agree? Explain your answer.

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Aim 8-3 How do we use the tangent ratios to
determine side lengths in triangles?

• Activity Tangent RatiosWorking in your groups
  complete the following.Have your group select
  one angle measure from 10, 20, ..80. Then have
  each member of your group draw a right triangle,
  triangle ABC, where angle A has the selected
  measure. Make the triangles different
  sizes.Measure the legs of each triangle ABC to
  the nearest millimeter.Compute (the ratio leg
  opposite ltA)/(leg adjacent to ltA) and round to
  two decimal places.Compare the ratios in your
  group. Make a conjecture.

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• For each family of complementary angles, ltA and
  ltB, there is a family of similar right triangles.
  In each family, the ratio
  is constant no matter what the size of
  triangle ABC. This trigonometric ratio is called
  the tangent ratio.
• Tangent of ltA

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Writing Tangent Ratios

• Write the tangent ratio for ?T and ?U.

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Writing Tangent Ratios

• Solution

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Writing Tangent Ratios

• Write the tangent ratios for ltK and ltJ.
• How is tan K related to tan J?

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Finding Distance Using the Tangent Ratio.Your
goal in Bryce Canyon National Park is the distant
cliff. About how far is the cliff?SolutionSuppo
se you find the mlt1 86. Thedistance you walk to
lt1 is 50 ft. To find the distance to the cliff
use the tangent ratio.tan 8650(tan86) x
To find tan 86 use a calculator.The cliff is
about 715 ft away.
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Find the value of w to the nearest tenth.

• 1. 2.

w
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Using the Inverse of Tangent

• If you know leg lengths for a right triangle, you
  can find the tangent ratio for each acute angle.
  Conversely, if you know the tangent ratio for
  angle, you can use inverse of tangent, tan-1, to
  find the measure of each angle.

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Using the Inverse of Tangent

• The lengths of the sides of ? BHX are given.
• Find m?X to the nearest degree.
• Find the tangent ratio.
• m?X tan-1(0.75) Use the inverse of
  tangent.
• tan-1(0.75)36.869898
• So m?X ? 37

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Using the Inverse of Tangent

• Find the m? Y to the nearest degree.

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Summary Answer in complete sentences.

• Without using a calculator, find the angle whose
  tangent equals 1. Explain.

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Aim 8-4 How do we use sine and cosine to
determine side lengths in triangles?

• Sine of ?A
• Cosine ?A

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• The above equations can be written as
• Sin A Cos A

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Writing Sine and Cosine Ratios

• Use the triangle to write each ratio.
• Sin T Cos T
• Sin G Cos G

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Writing Sine and Cosine Ratios

• Write the sine and cosine ratios for ?X and ?Y.

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• When does sin X cos Y? Explain.

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• One way to describe the relationship of sine and
  cosine is to say that sin x cos (90 - x)for
  values of x between 0 and 90. This type of
  equation is called an identity because it is true
  for all the allowed values of the variable.

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Using the Inverse of Cosine and Sine

• Find the m? L to the nearest degree.

What other strategy could I use to find the m ?L?
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Using the Inverse of Cosine and Sine

• Find the value of x. Round you answer to the
  nearest degree.

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Summary Answer in complete sentences.

• A right triangle whose hypotenuse is 18 cm long
  contains a 65 angle. Find the lengths of its
  legs to one decimal place.

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Aim 8-5 How do we use angles of elevation and
depression to solve problems?

• Suppose a person on the ground sees a hot-air
  balloon gondola at a 38 angle above a horizontal
  line.
• This angle is the angle of elevation.

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• At the same time, a person in the hot-air balloon
  sees the person on the ground at a 38 angle
  below a horizontal line.
• This angle is the angle of depression.

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Identifying Angles of Elevation and Depression

• Describe each angle as it relates to the
  situation shown.
• ? 1
• ? 4

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Identifying Angles of Elevation and Depression

• Describe each angle as it relates to the
  situation shown.
• ? 1 is the angle of depression from the peak to
  the hiker.
• ? 4 is the angle of elevation from the hut to the
  hiker.

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Identifying Angles of Elevation and Depression

• Describe each angle as it relates to the
  situation in the diagram. ?1, ?2, ?3, ?4

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Identifying Angles of Elevation and Depression

• Describe each angle as it relates to the
  situation in the diagram. ?5, ?6, ?7, ?8

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• Surveyors use two instruments, the transit and
  the theodolite, to measure angles of elevation
  and depression. On both instruments, the
  surveyors sets the horizon line perpendicular to
  the direction of gravity. Using gravity to find
  the horizon line ensures accurate measures even
  on sloping surfaces.
• Insert.

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Real-World Connection
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Real-World Connection

• Use the tangent ratio.
• Tan 48
• X 36 (tan 48)
• X 39.98
• X is about 40. To find the height of the arch,
  add the height of the theodolite. The Delicate
  Arch is about 45 feet high.

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Real-World Connection

• You sight a rock climber on a cliff at a 32
  angle of elevation. The horizontal ground
  distance to the cliff is 1000 ft.
• Find the line-of-sight distance to the rock
  climber.

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Practice

• Find the value of x. Round to the nearest tenth
  of unit.

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Real-World Connection

• To approach runaway 17 of the Ponca City
  Municipal Airport in Oklahoma, the pilot must
  begin 3 descent starting from an altitude of
  2714 ft. The airport altitude is 1007 ft. How
  many miles from the runway is the airplane at the
  start of this approach?
• a. 3. 6 mi. b. 5.7 mi. c. 6.2 mi. d. 9.8 mi.

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Real-World Connection

• Solution
• The airplane is 2714 1007 ft above the level of
  the airport.
• Sin 3
• Divide your answer by 5280 ft to covert to miles.
  The airplane is about 6.2 mi. from the runway.
  The correct answer is C.

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Real-World Connection

• An airplane pilot sights a life raft at a 26
  angle of depression. The airplane altitude is 3
  km. What is the airplanes surface distance d
  from the raft?

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Summary Answer in complete sentences.

• Two building are 30 ft apart. The angle of
  elevation from the top of one to the top of the
  other is 19. What is their difference in height?

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Aim 8-6 How do we describe vectors?

• A vector is any quantity with magnitude(size) and
  direction.
• You can use an arrow for a vector,
• The magnitude corresponds to the distance from
  the initial point K to the terminal point M. The
  direction corresponds to the
• direction in which the arrow points.

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• You can also use an ordered pair in the
  coordinate plane for a vector. The magnitude and
  direction of the vector corresponds to the
  distance and direction of from the origin.

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Describing a Vector

• Describe as an ordered pair. Give the
  coordinates to the nearest tenth.
• Use the sine and cosine ratios to find the values
  of x and y.

X? 41.78 Y ?49.79 L is in the 4th quadrant so
y is negative.
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Describing a Vector

• Describe the vector at the right as an ordered
  pair. Give the coordinates to the nearest tenth.

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Describing a Vector Direction

• You use the compass directions north, south, east
  and west to describe the direction of a vector.

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Describing a Vector Direction

• Use compass directions to describe the direction
  of each vector.

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Describing a Vector Direction

• Use compass directions to describe the direction
  of each vector.

35 East of North
25 South of East
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Describing a Vector Direction

• Sketch a vector that has the direction 30 west
  of north.
• Give a second description for the directions of
  this vector.

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Adding Vectors

• You can also use lowercase to name a vector such
  as .
• The sum or resultant of two vectors is written
  as
• You can add vectors by adding their coordinates.
  You can also show the sum geometrically.

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Adding Vectors Property

• For

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Adding Vectors

• Vectors
• Write the sum of the two vectors as an ordered
  pair.
• Then draw

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Adding Vectors

• Solution
• Draw with its initial point at the origin. Then
  draw with its initial point at the
  terminal point of . Finally draw the
  resultant from the initial point of to
  the terminal point of

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Adding Vectors

• Write the sum of the two vectors and
• as an ordered pair.

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Summary Answer in complete sentences.

• Sketch a vector with magnitude 50 and direction
  30west of north. Describe it as an ordered pair
  with coordinates rounded to the nearest tenth.