Chapter 5 Section 5'1 Trigonometric Functions of Acute Angles

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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
5.1 Trigonometric Functions of Acute Angles

• Determine the six trigonometric ratios for a
  given acute angle of a right triangle.
• Determine the trigonometric function values of
  30, 45, and 60.
• Using a calculator, find function values for any
  acute angle, and given a function value of an
  acute angle, find the angle.
• Given the function values of an acute angle, find
  the function values of its complement.

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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Trigonometric Ratios
The figure illustrates how a right triangle is
labeled with reference to a given acute angle,
?. The lengths of the sides of the triangle
are used to define the six trigonometric
ratios sine (sin) cosecant (csc) cosine
(cos) secant (sec) tangent (tan) cotangent
(cot)
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Sine and Cosine
The sine of ? is the length of the side opposite
? divided by the length of the hypotenuse The
cosine of ? is the length of the side adjacent
to ? divided by the length of the hypotenuse.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Trigonometric Function Values of an Acute Angle ?
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example
Use the triangle shown to calculate the six
trigonometric function values of ?.
Solution
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Reciprocal Functions
Reciprocal Relationships
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Pythagorean Theorem
The Pythagorean theorem may be used to find a
missing side of a right triangle. This
procedure can be combined with the reciprocal
relationships to find the six trigonometric
function values.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example
If the find the other five
trigonometric function values of ?.
Solution Find the length of the hypotenuse.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Function Values of 30? and 60?
When the ratio of the opposite side to the
hypotenuse is ½, ? must have a measure of
30?. Using the Pythagorean theorem the missing
side is The missing angle must have a measure of
60?.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Function Values of 30? and 60?
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Function Values of 45?
The legs of this triangle must be equal, since
they are opposite congruent angles. The
hypotenuse is found by
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Function Values of 45? continued
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Summary of Function Values
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example
As a hot-air balloon began to rise, the ground
crew drove 1.2 mi to an observation station. The
initial observation from the station estimated
the angle between the ground and the line of
sight to the balloon to be 30?. Approximately how
high was the balloon at that point? (We are
assuming that the wind velocity was low and that
the balloon rose vertically for the first few
minutes.)
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example continued
Solution We begin with a drawing of the
situation. We know the measure of an acute angle
and the length of its adjacent side.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example continued
Since we want to determine the length of the
opposite side, we can use the tangent ratio, or
the cotangent ratio. The balloon is
approximately 0.7 mi, or 3696 ft, high.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Cofunctions and Complements
The trigonometric function values for pairs of
angles that are complements have a special
relationship. They are called cofunctions.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Example

• Given that sin 40? ?0.6428,
  cos 40? ? 0.7660, and tan 40? ? 0.8391, find the
  six trigonometric function values of 50?.

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Chapter 5 Section 5.2 Trigonometric Functions
of Acute Angles
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Conversion from DMS form to Decimal Degree
form
D Degree M Minute S Second
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Convert to Decimal Degree Form 54230
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Convert to DMS 72.18
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Convert to decimal degree notation. Round to two
decimal places.
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Chapter 5 Section 5.1 Trigonometric Functions
of Acute Angles
Convert to degrees, minutes, and seconds. Round
to the nearest second.
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8.2 Applications of Right Triangles
Chapter 5 Section 5.2 Applications of Right
Triangles

• Solve right triangles.
• Solve applied problems involving right triangles
  and trigonometric functions.

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Chapter 5 Section 5.2 Applications of Right
Triangles
Solving Right Triangles
To solve a right triangle means to find the
lengths of all sides and the measures of all
angles. This can be done using right triangle
trigonometry.
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Chapter 5 Section 5.2 Applications of Right
Triangles
Example
In ?ABC, find a, b, and B. Solution
B 90? ? 42? 48?
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Chapter 5 Section 5.2 Applications of Right
Triangles
Definitions

• Angle of elevation angle between the horizontal
  and a line of sight above the horizontal.

• Angle of depression angle between the
  horizontal and a line of sight below the
  horizontal.

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Chapter 5 Section 5.2 Applications of Right
Triangles
Example
To determine the height of a tree, a forester
walks 100 feet from the base of the tree. From
this point, he measures the angle of elevation to
the top of the tree to be 47?. What is the height
of the tree?
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Chapter 5 Section 5.2 Applications of Right
Triangles
Bearing
Bearing is a method of giving directions. It
involves acute angle measurements with reference
to a north-south line.
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Chapter 5 Section 5.2 Applications of Right
Triangles
Example
An airplane leaves the airport flying at a
bearing of N32?W for 200 miles and lands. How far
west of its starting point is the plane? The
airplane is approximately 106 miles west of its
starting point.
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Chapter 5 Section 5.2 Applications of Right
Triangles
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