Section 5.2 Trigonometric Functions 5-1 Chapter 5 Trigonometric Functions 5.1 Angles ■ Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles Key Terms: vertex of an angle, initial side, terminal side, positive angle, negative angle, quadrantal angle Basic Terminology A counterclockwise rotation generates a ____________ measure, and a clockwise rotation generates a ____________ measure. Degree Measure EXAMPLE 1 Finding the Complement and the Supplement of an Angle For an angle measuring 40°, find the measure of (a) its complement and (b) its supplement. EXAMPLE 2 Finding Measures of Complementary and Supplementary Angles Find the measure of each marked angle in the figure.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Section 5.2 Trigonometric Functions 5-1
Chapter 5 Trigonometric Functions
5.1 Angles
■ Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles
Key Terms: vertex of an angle, initial side, terminal side, positive angle, negative angle, quadrantal angle
Basic Terminology
A counterclockwise rotation generates a ____________ measure, and a clockwise rotation generates a
____________ measure.
Degree Measure
EXAMPLE 1 Finding the Complement and the Supplement of an Angle
For an angle measuring 40°, find the measure of (a) its complement and (b) its supplement.
EXAMPLE 2 Finding Measures of Complementary and Supplementary Angles
Find the measure of each marked angle in the figure.
5-2 Chapter 5 Trigonometric Functions
Standard Position
An angle is in ____________ _____________ if its vertex is at the origin and its initial side lies on the
positive x-axis. The figure shows ranges of angle measures for each quadrant when o o0 360 .
Quadrantal Angles
Angles in standard position whose terminal sides lie on the __________ or __________, such as angles
with measures o o o90 , 180 , 270 ,and so on, are quadrantal angles.
Coterminal Angles
Angles with measures o60 and o420 have the same initial side and the same terminal side, but different
amounts of rotation. Such angles are ____________ ____________. Their measures differ by a multiple
of ________.
EXAMPLE 5 Finding Measures of Coterminal Angles
Find the angles of least positive measure that are coterminal with each angle.
(a) o908 (b) o75 (c) o800
EXAMPLE 6 Analyzing the Revolutions of a CD Player
CD Players always spin at the same speed. Suppose a player makes 480 revolutions per min. Through
how many degrees will a point on the edge of a CD move in 2 sec?
EXAMPLE 5 Finding Trigonometric Function Values of a Quadrant III Angle
Find the values of the six trigonometric
functions for 210°.
Finding Trigonometric Function Values for Any Nonquadrantal Angle
Step 1 If 360 , or if 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°.
Step 2 Find the reference angle .
Step 3 Find the trigonometric function values for reference angle .
Step 4 Determine the correct signs for the values found in Step 3. (Use the table of signs in Section 5.2,
if necessary.) This gives the values of the trigonometric functions for angle .
EXAMPLE 6 Finding Trigonometric Function Values Using Reference Angles
Find the exact value of each expression.
(a) cos 240 (b) tan675
Finding Function Values Using a Calculator
When evaluating trigonometric functions of angles given in degrees, remember that the calculator
must be set in ____________ mode.
EXAMPLE 7 Finding Function Values With a Calculator Find the value of each expression rounded to the nearest thousandth.
(a) sin 49° (b) sec97.977
(c) 1
cot51.4283 (d) sin 246
__________
__________
__________
x
y
r
5-10 Chapter 5 Trigonometric Functions
Finding Angle Measures
If x is a number in the appropriate range, then 1 1sin , cos ,x x
or 1tan ,x
gives the measure of an
angle whose sine, cosine, or tangent, respectively, is x.
EXAMPLE 8 Using Inverse Trigonometric Functions to Find Angles
Use a calculator to find an angle in the interval 0 , 90 that satisfies each condition.
(a) sin 0.96770915 (b) sec 1.0545829
EXAMPLE 9 Finding Angle Measures Given an Interval and a Function Value
Find all values of , if is in the interval 0 , 360 and 2
cos .2
Section 5.4 Solving Right Triangles 5-11
5.4 Solving Right Triangles
■ Significant Digits ■ Solving Triangles ■ Angles of Elevation or Depression ■ Bearing
■ Further Applications
Key Terms: exact number, significant digits, angle of elevation, angle of depression, bearing
EXAMPLE 1 Solving a Right Triangle Given an Angle and a Side
Solve right triangle ABC, if A = 34° 30′ = 34.5° and c = 12.7 in.
To maintain accuracy, always use given information as much as possible, and do not round in
intermediate steps.
EXAMPLE 2 Solving a Right Triangle Given Two Sides
Solve right triangle ABC, if A = 29.43 cm and c = 53.58 cm.
Angles of Elevation or Depression
In applications of right triangles, the angle of _____________________ from point X to point Y (above X)
is the acute angle formed by ray XY and a horizontal ray with endpoint at X. See the figure part (a). The
angle of _____________________from point X to point Y (below X) is the acute angle formed by ray XY
and a horizontal ray with endpoint X. See the figure part (b).
(a) (b)
Both the angle of elevation and the angle of depression are measured between the line of sight and
a_______________________ ______________.
5-12 Chapter 5 Trigonometric Functions
Solving an Applied Trigonometry Problem
Step 1 ____________ ____________ ____________, and label it with the given information. Label
the quantity to be found with a variable.
Step 2 Use the sketch to write an ___________________ relating the given quantities to the variable.
Step 3 _____________________ ________ ________________, and check that your answer makes
sense.
EXAMPLE 3 Finding a Length Given the Angle of Elevation
Pat Porterfield knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to the
top of the flagpole is 26° 45′ = 26.75°. If her eyes are 5.30 ft above the ground, find the height of the
flagpole.
Step 1
Step 2
Step 3
EXAMPLE 4 Finding an Angle of Depression
From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the angle of
depression from the top of the cliff to the base of the lighthouse.
Section 5.4 Solving Right Triangles 5-13
Bearing
Method 1 When a single angle is given, such as 164°, it is understood that the bearing is measured in a
clockwise direction from due north. Sample bearings using Method 1 are shown below.
EXAMPLE 5 Solving a Problem Involving Bearing (Method 1) 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
of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to
C.
Method 2 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. Sample bearings using Method
2 are shown below.
5-14 Chapter 5 Trigonometric Functions
EXAMPLE 6 Solving a Problem Involving Bearing (Method 2)
A ship leaves port and sails on a bearing of N 47° E for 3.5
hr. It then turns and sails on a bearing of S 43° E for 4.0 hr.
If the ship’s rate of speed is 22 knots (nautical miles per
hour), find the distance that the ship is from port.
Further Applications
EXAMPLE 7 Using Trigonometry to Measure a Distance
The subtense bar method is a method that surveyors use to determine a small distance d between two
points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of
sight between P and Q. Angle is measured, and then the distance d can be determined.
(a) Find d when 1 23 12 ≈ 1.3867° and b = 2.0000 cm.
(b) How much change would there be in the value of d if measured 1″ ≈ 0.0003° larger?
EXAMPLE 8 Solving a Problem Involving Angles of Elevation
Francisco needs to know the height of a tree. From a given point
on the ground, he finds that the angle of elevation to the top of
the tree is 36.7°. He then moves back 50 ft. From the second
point, the angle of elevation to the top of the tree is 22.2°. See
the figure. Find the height of the tree to the nearest foot.