2
Acute Angles
and Right
Triangle
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2.2-1
Acute Angles and
2 Right Triangles
2.1 Trigonometric Functions of Acute Angles
2.2 Trigonometric Functions of Non-Acute
Angles
2.3 Finding Trigonometric Function Values
Using a Calculator
2.4 Solving Right Triangles
2.5 Further Applications of Right Triangles
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2.2-2
2.5
Further Applications of Right
Triangles
Bearing ▪ Further Applications
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2.2-3
Bearing
There are two methods for expressing bearing
When a single angle is given, such as 164°, it is
understood that the bearing is measured in a
clockwise direction from due north.
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Bearing
The second method for expressing bearing starts
with a norht-south line and uses an acute angle to
show the direction, either east or west, from this
line.
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Example 1
SOLVING A PROBLEM INVOLVING
BEARING (FIRST METHOD)
Radar stations A and B are on an east-west line,
3.7 km apart. Station A detects a plane at C, on a
bearing on 61°. Station B simultaneously detects the
same plane, on a bearing of 331°. Find the distance
from A to C.
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Caution
A correctly labeled sketch is crucial
when solving bearing applications.
Some of the necessary information
is often not directly stated in the
problem and can be determined only
from the sketch.
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Example 2
SOLVING PROBLEM INVOLVING
BEARING (SECOND METHOD)
The bearing from A to C is S 52° E. The bearing from
A to B is N 84° E. The bearing from B to C is S 38° W.
A plane flying at 250 mph takes 2.4 hours to go from
A to B. Find the distance from A to C.
To draw the sketch, first draw
the two bearings from point A.
Choose a point B on the
bearing N 84° E from A, and
draw the bearing to C, which is
located at the intersection of the
bearing lines from A and B.
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Example 2
SOLVING PROBLEM INVOLVING
BEARING (SECOND METHOD) (cont.)
The distance from A to B is about 430 miles.
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Example 3
USING TRIGONOMETRY TO MEASURE
A DISTANCE
A method that surveyors use to determine a small
distance d between two points P and Q is called the
subtense bar method. The subtense bar with
length b is centered at Q and situated perpendicular
to the line of sight between P and Q. Angle θ is
measured, then the distance d can be determined.
(a) Find d with θ = 1°23′12″ and b = 2.0000 cm.
From the figure, we have
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Example 3
USING TRIGONOMETRY TO MEASURE
A DISTANCE (continued)
A method that surveyors use to determine a small
distance d between two points P and Q is called the
subtense bar method. The subtense bar with
length b is centered at Q and situated perpendicular
to the line of sight between P and Q. Angle θ is
measured, then the distance d can be determined.
(a) Find d with θ = 1°23′12″ and b = 2.0000 cm.
From the figure, we have
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Example 3
USING TRIGONOMETRY TO MEASURE
A DISTANCE (continued)
Convert θ to decimal degrees:
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Example 3
USING TRIGONOMETRY TO MEASURE
A DISTANCE (continued)
(b) Angle θ usually cannot be measured more
accurately than to the nearest 1″. How much
change would there be in the value of d if θ were
measured 1″ larger?
Since θ is 1″ larger, θ = 1°23′13″ ≈ 1.386944.
The difference is
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION
From a given point on the ground, the angle of
elevation to the top of a tree is 36.7°. From a second
point, 50 feet back, the angle of elevation to the top of
the tree is 22.2°. Find the height of the tree to the
nearest foot.
The figure shows two
unknowns: x and h.
Since nothing is given about the length of the hypotenuse,
of either triangle, use a ratio that does not involve the
hypotenuse, tangent.
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION (continued)
In triangle ABC:
In triangle BCD:
Each expression equals h, so the expressions must be
equal.
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION (continued)
Since h = x tan 36.7°, we can substitute.
The tree is about 45 feet tall.
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION (continued)
Graphing Calculator Solution
Superimpose coordinate axes
on the figure with D at the origin.
The coordinates of A are (50, 0).
The tangent of the angle between the x-axis and the graph
of a line with equation y = mx + b is the slope of the line.
For line DB, m = tan 22.2°.
Since b = 0, the equation of line DB is
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION (continued)
The equation of line AB is
Use the coordinates of A and
the point-slope form to find the
equation of AB:
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Example 4
SOLVING A PROBLEM INVOLVING
ANGLES OF ELEVATION (continued)
Graph y1 and y2, then find the point of intersection. The
y-coordinate gives the height, h.
The building is about 45 feet tall.
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