Download Section 2.1 Trigonometric Functions of Acute Angles

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TRIG-Fall 2011-Jordan
Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009
Chapter 2: Acute Angles and Right Triangles
Section 2.1
Trigonometric Functions of Acute Angles
Right-Triangle Based Definitions of Trigonometric Functions
For any acute angle A in standard position,
sin A =
y
opp
=
r
hyp
cos A =
x
adj
=
r
hyp
tan A =
y
opp
=
x
adj
csc A =
hyp
r
=
opp
y
sec A =
r
hyp
=
x
adj
cot A =
adj
x
=
opp
y
Example 1
Find the values of sin A, cos A, and tan A in the right triangle shown.
Cofunction Identities
For any acute angle A,
sin A = cos (90° - A)
cos A = sin (90° - A)
tan A = cot (90° - A)
csc A = sec (90° - A)
sec A = csc (90° - A)
cot A = tan (90° - A)
Example 2
Write each function in terms of its cofunction.
a) cos 38°
b) sec 78°
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Special Triangles
Example 3
Give the exact value.
a) cos 30°
b) cot 45°
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Function Values of Special Angles
Memorize at least the first two columns and be able to generate the other columns.
Example 4
Find the exact value of each part labeled with a variable.
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Section 2.2
Trigonometric Functions of Non-Acute Angles
Reference Angles
A reference angle for an angle θ is the positive acute angle made by the terminal side of
angle θ and the x-axis.
Caution: The reference angle is always found with reference to the x-axis, never the yaxis.
Example 1
Find the reference angle for each angle.
a) 218°
b) 1387°
Finding Trigonometric Function Values for any Nonquadrantal Angle θ
If θ > 360°, or if θ < 0°, then find a coterminal angle by adding or subtracting 360° as
many times as needed to get an angle greater than 0° but less than 360°.
Find the reference angle θ′.
Find the trigonometric function values for reference angle θ′.
Determine the correct signs for the values found above. This gives the values of the
trigonometric functions for angle θ.
Example 2
Find exact values of the six trigonometric functions for 210°.
Example 3
Find exact values of the six trigonometric functions for -1020°.
Finding the Angle Given the Trigonometric Function Value
Example 4
value.
Find all values of θ in the interval [0°, 360°) that has the given function
tan   3
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Section 2.3
Calculator
Finding Trigonometric Function Values Using a
Function Values Using a Calculator
When evaluating trigonometric functions of angles given in degrees, remember that the
calculator must be set in degree mode.
Remember that most calculator values of trigonometric functions are approximations.
Example 1 Use a calculator to give a decimal approximation for each value. Give as
many digits as your calculator displays.
a) sin 38° 24′
b) cot 68.4832°
Finding Angle Measures Using a Calculator
Graphing calculators have three inverse trigonometric functions.
If sin θ = x, then θ = sin -1 x for θ in the interval [0°, 90°].
The inverse trigonometric functions are not reciprocal functions.
Example 2
Find a value of θ in the interval [0°, 90°] that satisfies each statement.
a) sin θ = .8535508
b) sec θ = 2.486879
Grade Resistance
Grade resistance is a force due to gravity on an automobile as it travels uphill or
downhill on a highway.
F = W sin θ
F is the force measured in pounds, W is the weight of the automobile in pounds, and θ
is the grade in degrees. If the car is traveling downhill, then θ is negative. If the car is
traveling uphill, θ is positive.
Example 3 A 2400-lb car has a grade resistance of 288 lb. What is the angle of the
grade? Round to the nearest hundredth of a degree.
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Section 2.4
Solving Right Triangles
Significant Digits for Angles
A significant digit is a digit obtained by actual measurement.
It is helpful to write the number in scientific notation—the number of digits in the first
factor of scientific notation represents the number of significant digits.
Determine the least number of significant digits in the given numbers and round your
final answer to the same number of significant digits as this number.
Number of
Significant
Digits
2
3
4
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Angle Measure to Nearest:
Degree
Ten minutes, or nearest tenth of a degree
Minute, or nearest hundredth of a degree
Tenth of a minute, or nearest thousandth of a
degree
Example
52˚
52˚ 30′ = 52.5˚
52˚ 45′ = 52.75˚
52˚ 40.5′ = 52.675˚
Solving Triangles
To solve a triangle means to find the measures of all the angles and sides of the
triangle.
Denote the angles of a triangle by capital letters. Then use the corresponding lower
case letters to denote the respective opposite sides.
If the letters A, B, and C, are used to denote the angles of a right triangle, then it is
usually assumed that C is the right angle.
Example 1
Solve the following right triangle.
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Example 2 Solve right triangle ABC if b = 219 m and c = 647 m and C = 90˚.
(When two sides are given, give angles in degrees and minutes.)
Solving Applied Trigonometry Problems
Draw a sketch, and label it with the given information. Label the quantity to be found
with a variable.
Use the sketch to write an equation relating the given quantities to the variable.
Solve the equation, and check that your answer makes sense.
Example 3 Find the altitude of an isosceles triangle having base 184.2 cm if the angle
opposite the base is 68˚ 44′.
Angles of Elevation and Depression
When a horizontal line of sight is used as a reference line, the angle measured above
the line of sight is called an angle of elevation, while the angle measured below the line
of sight is called an angle of depression.
Example 4 An airplane is flying 10,500 feet above the level ground. The angle of
depression from the plane to the base of a tree is 13˚ 50′. How far horizontally must the
plane fly to be directly over the tree?
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Section 2.5
Further Applications of Right Triangles
Bearing
When a single angle is given as a bearing, it is understood that the bearing is measured
in a clockwise direction from the north.
bearing of 32°
bearing of 229°
The second method for expressing bearing starts with a north-south line and uses an
acute angle to show the direction, either east or west, from this line.
Example 1 A ship travels 50 km on a bearing of 27°, then travels on a bearing of 117°
for 140 km. Find the distance traveled from the starting point to the ending point.
Example 2 The bearing from Atlanta to Macon is S 27° E, and the bearing from
Macon to Augusta is N 63° E. An automobile traveling at 60 mph needs 1 ¼ hour to go
from Atlanta to Macon and 1 ¾ hour to go from Macon to Augusta. Find the distance
from Atlanta to Augusta.
Problems Involving Angles of Elevation/Depression
Example 3 Sean wants to know the height of a Ferris wheel. From a given point on
the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3°. He
then moves back 75 ft. From the second point, the angle of elevation to the top of the
Ferris wheel is 25.4°. Find the height of the Ferris wheel.