5.1: angles and radian measure date: pre-calculus
TRANSCRIPT
5.1: Angles and Radian Measure Date: ________
Pre-Calculus
*Use Section 5.1 (beginning on pg. 482) to complete the following
Trigonometry: βmeasurement of trianglesβ
An angle is formed by two rays that have a common endpoint. One ray is called __________ side and the other the
_______________ side.
Draw and label the diagram:
An angle is in standard position if:
Its vertex is at the origin of a rectangular coordinate system
Its initial side lies along the positive x-axis
Draw and label both diagrams:
Positive angles are generated by a counterclockwise rotation. Negative angles are generated by a clockwise rotation. If the terminal side of an angle lies in a specific quadrant (I, II, III, IV), we say that the angle lies in that quadrant. If the terminal side of an angle lies on the x-axis or the y-axis, the angle is called a quadrantal angle. Measuring Angles Using Degrees
Think about a clock. One full rotation is 60 minutes and this is a rotation of 360Β°.
Fractional parts of degrees are measured in minutes and seconds
One minute, written 1β is _____ degree: 1β = _______.
One second, written 1ββ is _________ degree: 1ββ = ____________.
For example, 31Β°47β²12β²β² = (31 +47
60+
12
3600) Β° β 31.787Β°.
Many calculators have keys for changing an angle from degree-minute-second notation (π·Β°πβ²πβ²β²) to a decimal form
and vice versa.
Measuring Angles Using Radians:
Use the circle at the right:
1. Create an angle with the vertex at the center of the circle (a central angle)
2. Label the sides of the angle, r, for the radius
If the central angle intercepts an arc along the circle measuring r units, the measure of such an angle is 1 radian
3. Label the arc created by the central angle r
4. Label the central angle 1 radian
*1 radian = when arc and radius are equal
The radian measure of any central angle is the length of the intercepted arc divided by the circleβs radius.
Construct an angle π½ with measure 2 radians.
Radian Measure: Consider an arc of length s on a circle of radius r. The measure of the central angle, π (βthetaβ), that intercepts the arc is
π =π
π radians.
Relationship between Degrees and Radians: Compare the number of degrees and the number of radians in one complete rotation.
π =π
π=
2ππ
π= 2π
So, 360Β° = 2π radians. Thus, 180Β° = π radians.
Conversion between Degrees and Radians:
Using the basic relationship that 180Β° = π radians,
1. To convert degrees to radians, multiply degrees by _________________________
2. To convert radian to degrees, multiply radians by ___________________________
Angles that are fractions of a complete rotation are usually expressed in radian measure as factional multiples of π
rather than as decimal approximations. For example π =π
2.
s
r
r
π
Ex. 2: Convert each angle in degrees to radians:
a) 60Β° b) 270Β° c) β300Β°
Ex. 3: Convert each angle in radians to degrees:
a) π
4 radians b) β
4π
3 radians c) 6 radians
Drawing Angles in Standard Position:
*Goal: Learn to THINK in RADIANS
Ex. 4: Draw and label each angle in standard position.
a) π = βπ
4 b) πΌ =
3π
4 c) π½ = β
7π
4 d) πΎ =
13π
4
Terminal Side Radian Measure of Angle Degree Measure of Angle
1
12 revolution
1
8 revolution
1
6 revolution
1
4 revolution
1
3 revolution
1
2 revolution
2
3 revolution
3
4 revolution
7
8 revolution
1 revolution
Coterminal Angles- two angles with the same ________________ and _____________________ sides.
Every angle has infinitely many coterminal angles.
Increasing or decreasing the degree measure of an angle in standard position by an integer multiple of 360Β° results
in a coterminal angle. Thus, an angle of πΒ° is coterminal with angles of πΒ° Β± 360Β°π, where k is an integer.
Increasing or degreasing the radian measure of an angle by an integer multiple of 2π results in a coterminal angle.
Thus, an angle of π radians is coterminal with angles of π Β± 2ππ, where k is an integer.
Ex. 5: Find a positive angle less than 360Β° that is coterminal with each of the following:
a) a 400Β° angle b) a β135Β° angle
Ex. 6: Find a positive angle less than 2π that is coterminal with each of the following:
a) a 13π
5 angle b) a β
π
15 angle
Ex. 7: Find a positive angle less than 360Β° or 2π that is coterminal with each of the following:
a) an 855Β° angle b) a 17π
3 angle c) a β
25π
6 angle
5.1 Continued: Angles and Radian Measure Date: ________ Pre-Calculus The Length of a Circular Arc: Ex 1: The minute hand of a clock is 6 inches long and moves from 12 to 4 oβclock. How far does the tip of the minute hand move? Express your answer in terms of π and then round to two decimal places. Ex 2: The figure shows a highway sign that warns of a railway crossing. The lines that form the cross pass through the circleβs center and intersect at right angles. If the radius of the circle is 24 inches, find the length of each of the four arcs formed by the cross. Express your answer in terms of π and then round to two decimal places. Ex 3: A circle has a radius of 6 inches. Find the length of the arc intercepted by a central angle of 45Β°. Express arc
length in terms of π. Then round your answer to two decimal places.
Linear and Angular Speed: If a point is in motion on a circle of radius r through an angle of π radians in time t, then its LINEAR SPEED is: and its ANGULAR SPEED is: Note: The linear speed is the product of the radius and the angular speed, where w is the angular speed in radians per unit of time.
Ex 4: The angular speed of a point on Earth is π
12 radian per hour. The Equator lies on a circle of radius
approximately 4000 miles. Find the linear velocity, in miles per hour, of a point on the Equator.
Ex 5: A Ferris wheel has a radius of 25 feet. The wheel is rotating at two revolutions per minute. Find the linear
speed, in feet per minute, of a seat on this Ferris wheel.
Ex 6: A water wheel has a radius of 12 feet. The wheel is rotating at 20 revolutions per minute. Find the linear
speed, in feet per minute, of the water.
Ex. 9: Long before iPods that hold thousands of songs and play them with superb audio quality, individual songs
were delivered on 75-rpm and 45-rpm circular records. A 45-rpm record has an angular speed of 45 revolutions per
minute. Find the linear speed, in inches per minute, at the point where the needle is 1.5 inches from the recordβs
center.
Homework: MathXL 5.1 Homework
5.2: Right Triangle Trigonometry Date: ________ Pre-Calculus Fundamental Identities: Trigonometry Identities: Quotient Identities: Pythagorean Theorem: Divide both sides by π2 What do you observe? Pythagorean Theorem Identities:
Ex 1: Given that π πππ =1
2 and π is an acute angle, find the value of πππ π using a trig identity.
Consider the relationship between the π ππ30Β° = _______________, πππ 60Β°= _________________.
So, π πππ = cos( )
Any pair of trigonometric functions f and g for which π(π) = π(90Β° β π) are called __________________.
Cofunction Identities: If π is measured in radians, replace the 90Β° with π
2.
The value of a trigonometric function of π is equal to the cofunction of the complement of π. Cofunctions of
complementary angles are equal.
Ex 2: Find a cofunction with the same value as the given expression:
a) π ππ46Β° b) πππ‘π
12
Using a calculator to evaluate trigonometric functions:
Ex 3: Using a calculator to find the value to four decimal places:
a) π ππ72.8Β° b) ππ π1.5
Applications:
The angle formed by a horizontal line and the line of sight to an object that is above the horizontal line is called the
_____________ of _________________________.
The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the
_____________ of _________________________.
Ex. 4: A flagpole that is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation to the sun to the
nearest degree.
Ex. 5: Find the exact value of each expression. Do not use a calculator.
a) π‘ππ
π
4
2β
1
π πππ
6
b) 1 + π ππ240Β° + π ππ250Β° c) πππ 12Β°π ππ78Β° + πππ 78Β°π ππ12Β°
Ex. 6: Express the exact value of the function as a single fraction. Do not use a calculator.
If π(π) = 2π πππ β π πππ
2, find π (
π
3).
Ex. 7: At a certain time of day, the angle of elevation of the sun is 40Β°. To the nearest foot, find the height of a tree
whose shadow is 35 feet long.
Ex. 8: The Washington Monument is 555 feet high. If you are standing one quarter of a mile from the base of the
monument and looking to the top, find the angle of elevation to the nearest degree.
Homework: Math XL: 5.2 Homework
5.3: Trigonometric Functions of Any Angle Date: ________
Pre-Calculus
Trigonometric Functions of Any Angle: The point P = (x,y) is a point r units away from the origin on the terminal side
of π. A right triangle is formed by drawing a line segment from P = (x,y) _____________________ to the x-axis.
Note: y is the length of the side opposite π and x is the length of the side adjacent to π.
Definitions of Trigonometric Functions of Any Angle: Let π be any angle in standard position and let P = (x,y) be a
point on the terminal side of π. If r = _____________________ is the _______________ from (0,0) to (x,y), the six
trigonometric functions of π are defined by the following ratios:
Since P = (x,y) is any point on the terminal side of π other than the origin, π = ____________________ cannot be zero.
(Think of βrβ as the radiusβ¦.distance from the origin to P)
Ex 1: Let P = (1,-3) be a point on the terminal side of π. Find each of the six trigonometric functions of π. (Start by
drawing the angle in standard position.)
Trigonometric Functions of Quadrantal Angles:
Ex. 2: Find the values of the six trigonometric functions, if they exist, for each of the four quadrantal angles.
a) π = 0Β° = 0 b) π = 90Β° =π
2 c) π = 180Β° = π d) π = 270Β° =
3π
2
The Signs of the Trigonometric Functions: If π is not a quadrantal angle, the sign of a trigonometric function
depends on the quadrant in which π lies.
In all four quadrants, r, is positive (radius).
Since π πππ =π¦
π, the sign of this ratio depends on y. Similarly the sign of the cosine ratio depends on x. Tangent is
the ratio of sine and cosine so it is dependent on the signs of ______ x and y.
*Think alphabetical β¦ (x,y) β¦ (cosine,sine)
Now, look at all 4 quadrants:
Your bookβs phrase: A Smart Trig Class (see pg. 517)
Ex 3: If π πππ < 0 and πππ π < 0, name the quadrant in which angle π lies.
Ex. 4: Given π‘πππ = β1 and πππ π < 0, find π πππ and π πππ.
Reference Angles: Let π be a nonacute angle in standard position that lies in a quadrant. Its _______________
___________ is the ______________ ____________ angle πβ² (β__________ __________β) formed by the
terminal side of π and the x-axis.
a) π is in Quadrant II b) π is in Quadrant III c) π is in Quadrant IV
Ex. 5: Find the reference angle, πβ², for each of the following angles:
a) π = 210Β° b) π =7π
4 c) π = β240Β° d) π = 3.6
Finding Reference Angles for Angles Greater than 360Β° (2π) or less than β360Β° (β2π):
1) Find a positive angle πΌ less than 360Β° or 2π that is ________________ with the given angle.
2) Draw πΌ is standard position.
3) Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the
terminal side of πΌ and the x-axis is the _________________ _____________.
Ex. 6: Find the reference angle for each of the following angles:
a) π = 665Β° b) π =15π
4 c) π = β
11π
3
Evaluating Trigonometric Functions Using Reference Angles:
The value of a trigonometric function of any angle π is found as follows:
1. Find the associated reference angle, πβ², and the function value for πβ².
2. Use the quadrant in which π lies to prefix the appropriate sign to the function value in step 1.
Ex. 7: Use reference angles to find the exact value of the following trigonometric functions:
1. π ππ300Β° b) π‘ππ5π
4 c) π ππ (β
π
6)
Ex. 8: Use reference angles to find the exact value of each of the following trigonometric functions:
a) πππ 17π
6 b) sin (β
22π
3)
Homework: pg. 525 #3-84 (multiples of 3)---Use page 524 as a reference.
5.4: Trigonometric Functions of Real Numbers; Periodic Functions Date: ________
Pre-Calculus
Unit Circle:
The equation of this unit circle is: ___________________________.
Recall: π = ππ, so in the unit circle, where the central angle measures t radians, π = ________.
So, in a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc.
π₯ = ___________ , π¦ = ___________
Definitions of the Trigonometric Functions in Terms of a Unit Circle
If t is a real number and P=(x,y) is a point on the unit circle that corresponds to t, then
π πππ‘ = _______ ππ ππ‘ = _________ , π¦ β 0
πππ π‘ = _______ π πππ‘ = _________ , π₯ β 0
π‘πππ‘ = ________ , π₯ β 0 πππ‘π‘ = _________ , π¦ β 0
Note: π₯2 + π¦2 = 1 is equivalent to the Pythagorean identity πππ 2π₯ + π ππ2π₯ = 1.
Ex. 1: Find the values of the trig functions at t if π = (β3
2,1
2).
Ex. 2: Use a unit circle to find the values of the trigonometric functions at π‘ = π.
Domain and Range of Sine and Cosine Functions
The domain of the sine function and the cosine function is (ββ,β), the set of all real numbers. The range of these
functions is [β1,1], the set of all real numbers from -1 to 1, inclusive.
Even and Odd Trigonometric Functions
Even: π(βπ‘) = ___________ Odd: π(βπ‘) = _____________
If P(cost,sint); Q((cos(-t),sin(-t)) The x-coordinates of P and Q are the same. Thus, cos(βπ‘) = ____________ , so cosine is an _________ function.
The y-coordinates of P and Q are opposites of each other. Thus sin(βπ‘) = ______________ , so sine is an ________
function.
EVEN FUNCTIONS: ODD FUNCTIONS:
Ex. 3: Find the exact value of each trigonometric function:
a) cos (β60Β°) b) tan (βπ
6)
A function f is periodic if there exists a positive number p such that π(π‘ + π) = π(π‘) for all t in the domain of f. The
smallest positive number p for which f is periodic is called the period of f.
Periodic Properties of the Sine and Cosine Functions:
sin(π‘ + 2π) = π πππ‘ and cos(π‘ + 2π) = πππ π‘
The sine and cosine functions are periodic functions and have a period of ______. (Similarly, cosecant and secant
have a period of 2π.)
Ex. 4: Find the exact value of each trigonometric function:
a) πππ 405Β° b) π ππ7π
3
Periodic Properties of the Tangent and Cotangent Functions
tan(π‘ + π) = π‘πππ‘ and cot(π‘ + π) = πππ‘π
The tangent and cotangent functions are periodic functions and have period __________
Repetitive Behavior of the Sine, Cosine, and Tangent Functions:
For any integer n and real number t,
sin(π‘ + 2ππ) = _____________
cos(π‘ + 2ππ) = __________________
tan(π‘ + ππ) = ____________
Homework: pg. 532 #1 β 18, 20 β 36(e), 44 β 46
5.5: Graphs of Sine and Cosine Date: ________ Pre-Calculus Intro to Sine and Cosine Functions:
π¦ = π πππ₯
Amplitude: __________
Period: ___________
Scaling: __________
π¦ = πππ π₯
Amplitude: __________
Period: ____________
Scaling: _________
Graphing sine and cosine functions in general:
π¦ = π΄π ππ(π΅π₯ β πΆ) + π· and π¦ = π΄πππ (π΅π₯ β πΆ) + π·
1. |A| = amplitude (If A<0, the graph is reflected over the x-axis.)
2. Period = 2π
π΅
3. Horizontal shift (called the _________ ________): The starting point changes from π₯ = 0 to π₯ =πΆ
π΅
(Find this by setting π΅π₯ β πΆ = 0.) If πΆ
π΅> 0, the shift is to the right. If
πΆ
π΅< 0, the shift is to the left. The
number πΆ
π΅ is called the phase shift.
4. Vertical shift: The constant D causes vertical shifts in the graph. If D is positive, the shift is D units
__________. If D is negative, the shift is |D| units _____________. These vertical shifts result in graphs that
oscillate about the horizontal line ________ rather than the x-axis. Thus, the maximum y is D + |A| and
the minimum y is D - |A|. (These values give the _________ of the function.)
Examples: Graph each function. Draw two cycles. Label the max/min points and intercepts (5 key points per cycle).
Ex 1: π¦ = 3cos (π₯ + π)
Amplitude: _______
Period: _______
Scale by: ________
Vertical shift: __________
Phase shift: ____________
Ex 2: π¦ = 2 β sin (2π
3π₯)
Amplitude: _______
Period: _______
Scale by: ________
Vertical shift: __________
Phase shift: ____________
Ex 3: π¦ =2
3πππ (
π₯
2β
π
4)
Amplitude: _______
Period: _______
Scale by: ________
Vertical shift: __________
Phase shift: ____________
Ex 4: π¦ = 3 cos(π₯ + π) β 3
Amplitude: _______
Period: _______
Scale by: ________
Vertical shift: __________
Phase shift: ____________
Homework: pg. 554 #13, 20, 23, 26, 41, 44, 47, 50, 62, 64, 66, 84 β 87
5.6: Graphs of Other Trig Functions (Tangent and Cotangent) Date: ________ Pre-Calculus
π¦ = tan π₯
Period: ______
VA: ___________
Domain: ____________
Range: ____________
x-intercepts:
Symmetry:
Graphing Variations of π¦ = tan π₯ in the form π¦ = π΄ tan(π΅π₯ β πΆ)
1. Period = _______
2. Find two consecutive asymptotes by finding an interval containing one period: _______________________
The asymptotes occur at:
a.
b.
3. Identify an x-intercept, midway between the consecutive asymptotes
4. Amplitude = ______
Ex 1: π¦ = tan (π β π₯)
Amplitude: ___________
Period: ___________
Asymptotes:
π¦ = πππ‘π₯
Period: ______
VA: ___________
Domain: ____________
Range: ____________
x-intercepts:
Symmetry:
Graphing Variations of π¦ = cot π₯ in the form π¦ = π΄ cot(π΅π₯ β πΆ)
1. Period = _______
2. Find two consecutive asymptotes by finding an interval containing one period: _______________________
The asymptotes occur at:
a.
b.
3. Identify an x-intercept, midway between the consecutive asymptotes
4. Amplitude = ______
Ex 2: π¦ = cot (π₯
2)
Amplitude: ___________
Period: ___________
Asymptotes:
Ex 3: π¦ = β2cot (π₯ +π
2)
Amplitude: ___________
Period: ___________
Phase Shift: ____________
Asymptotes:
Ex 4: π¦ =1
2cot 2π₯
Amplitude: ___________
Period: ___________
Asymptotes:
Ex 5: π¦ = β2 tan1
2π₯
Amplitude: ___________
Period: ___________
Asymptotes:
Homework: pg. 567 #1 β 4, 6 β 12(e), 13 β 16, 18 β 24(e), 45, 46, 59
5.6 Continued: Graphs of Other Trigonometric Functions Date: ________ Pre-Calculus
π¦ = sec π₯ π¦ = ππ ππ₯
Reciprocal function:
Domain:
Range:
Period:
Vertical Asymptotes:
Symmetry:
π¦ = sec π₯ π¦ = csc π₯
Ex 1: π¦ = 3 cscπ₯
2
Amplitude: ________
Period: _______
Scale by: _______
Ex 2: π¦ = β1
2sec π₯
Amplitude: ________
Period: _______
Scale by: _______
Ex 3: π¦ = cscπ₯
3
Amplitude: ________
Period: _______
Scale by: _______
Ex 4: π¦ = sec(ππ₯) β 1
Amplitude: ________
Period: _______
Scale by: _______
Ex 5: π¦ = 2 csc (π₯ +π
4)
Amplitude: ________
Period: _______
Scale by: _______
Phase shift: ________________
Homework: pg. 568 #26 β 44(e), 47, 48, 61, 93 β 96
5.7: Inverse Trigonometric Functions Date: ________ Pre-Calculus Recall: In order for a function to have an inverse function, the graph of the function must pass the
_________________ ________ _________. On their entire domain, the trig functions do ______ pass this test.
INVERSE SINE
Restrict π¦ = π πππ₯ so that _______________________.
Then it will have an inverse function.
The inverse function for π πππ₯ is _____________________ or _________________________.
(For inverse functions, the domain/range are switched.)
Sine is an __________ function - symmetry with the _____________________.
Note: π¦ = π ππβ1π₯ β 1
π πππ₯.
Graphically inverses are reflections over the line ________________. (true for exp/log functions)
π π πππ
Ex 1: Find the exact value of each of the following:
a) π ππβ1 (β3
2) = ______________ b) π ππβ1 (β
β2
2) = __________________
INVERSE COSINE
Restrict π¦ = πππ π₯ so that _______________________.
Then it will have an inverse function.
The inverse function for πππ π₯ is _____________________ or _________________________.
(For inverse functions, the domain/range are switched.)
Cosine is an __________ function - symmetry with the _____________________.
π πππ π
Ex 2: Find the exact value of each of the following:
a) πππ β1 (β1
2) = ____________ b) πππ β1 (
β2
2) = ________________
INVERSE TANGENT
Restrict π¦ = π‘πππ₯ so that _______________________.
Then it will have an inverse function.
The inverse function for π‘πππ₯ is ____________________ or _________________________.
(For inverse functions, the domain/range are switched.)
Cosine is an __________ function - symmetry with the _____________________.
π π‘πππ
Ex. 3: Find the exact value of each of the following:
a) π‘ππβ1(β1) = ___________ b) π‘ππβ1 (β3
3) = ____________
Ex. 4: Use a calculator to find the value of each expression to four decimal places.
a) πππ β1 (1
3) = _______________ b) π‘ππβ1(35.85) =________________ c) π ππβ12 =__________
COMPOSITION OF FUNCTIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
Recall: π(πβ1(π₯)) = π₯ for ____________ and πβ1(π(π₯)) = π₯ for _______________.
INVERSE PROPERTIES
sin(π ππβ1π₯) = π₯ for ______________________
π ππβ1(π πππ₯) = π₯ for _______________________
cos(πππ β1π₯) = π₯ for ________________________
πππ β1(πππ π₯) = π₯ for ________________________
tan(π‘ππβ1π₯) = π₯ for _________________________
π‘ππβ1(π‘πππ₯) = π₯ for _________________________
Ex. 5: Find the exact value (if possible) for each of the following expressions:
a) cos(πππ β10.7) = ______________________
b) π ππβ1(π πππ) = ______________________
c) πππ [πππ β1(β1.2)] = ____________________
EVALUATING A COMPOSITE TRIG EXPRESSION (Draw a triangle!)
Ex. 6: Find the exact value of sin (π‘ππβ1 (3
4)).
Ex. 7: Find the exact value of πππ [π ππβ1 (β1
2)].
SIMPLIFYING EXPRESSIONS INVOLVING TRIG FUNCTIONS
Ex. 8: If π₯ > 0, write π ππ(π‘ππβ1π₯) as an algebraic expression in π₯.
Homework due Tuesday 12/13: Graphing Trig Functions Review (handout) & pg. 584 #1 β 30 (no calc for #1 β 18)
Quiz #8 (5.5 β 5.6) β NON CALCULATOR! Wednesday 12/14
Homework due Wednesday 12/14 (Section 5.7): pg. 584 #32 β 72(e), 95, 96, 116 β 121, 124 β 126 -- Make sure you
know the sin, cos, tan values in the tables!
5.8: Applications of Trigonometric Functions Date: ________ Pre-Calculus Objectives:
Solve a right triangle.
Solve problems involving bearings.
Solving Right Trianglesβfinding the missing lengths of the sides and angles of a triangle.
Generic diagram of a right triangle
Ex. 1: Let π΄ = 62.7Β° and π = 8.4. Solve the right triangle, rounding lengths to two decimal places.
Ex. 2: from a point on level ground 80 feet from the base of the Eiffel Tower, the angle of elevation is 85.4Β°.
Approximate the height of the Eiffel Tower to the nearest foot.
Ex. 3: A guy wire (support wire) is 13.8 yards long and is attached from the ground to a pole 6.7 yards above the
ground. Find the angle, to the nearest tenth of a degree that the wire makes with the ground.
Using Two Right Triangles to Solve a Problem
Ex. 4: You are standing on level ground 800 feet from Mt. Rushmore, looking at the sculpture of Abraham Lincolnβs
face. The angle of elevation to the bottom of the sculpture is 32Β° and the angle of elevation to the top is 35Β°. Find
the height of the sculpture of Lincolnβs face to the nearest tenth of a foot.
Trigonometry and Bearings
The bearing from a point O to a point P is the acute angle, measured in degrees between ππββββ β and a north-south line.
Ex. 5: Find the bearing from O to P in each diagram.
Ex. 6: A boat leaves the entrance to a harbor and travels 150 miles on a bearing of π 53Β°πΈ. How many miles north
and how many miles east form the harbor has the boat traveled?
Ex. 7: You leave your house and run 2 miles due west followed by 1.5 miles due north. At that time, what is your
bearing from your house?
Ex. 8: You leave the entrance to a system of hiking trails and hike 2.3 miles on a bearing of π 31Β° π. Then the trail
turns 90Β° and you hike 3.5 miles on a bearing of π 59Β° π. At that time:
a) How far are you, to the nearest tenth of a mile, from the entrance to the trail system?
b) What is your bearing, to the nearest tenth of a degree, from the entrance to the trail system?
Homework: pg. 595 #6, 10, 14, 16, 42 β 58(e) (We will continue section 5.8 tomorrow.)
5.8 Continued: Simple Harmonic Motion Date: ________ Pre-Calculus Simple Harmonic MotionβConsider a ball attached to a spring hung from the ceiling. You pull the ball down and
then release it. If we neglect the effects of friction and air resistance, the ball will continue bobbing up and down on
the end of the spring. These up-and down oscillations are called simple harmonic motion. The position of the ball
before you pull the ball down is d=0. This rest position is called the equilibrium position.
An object that moves on a coordinate axis is in simple harmonic motion if its distance from the origin, d, at time t, is
given by either
π = acos(ππ‘) ππ π = asin (ππ‘)
Use π = acos(ππ‘) if ____________________________________________________________
Use π = asin (ππ‘) if _____________________________________________________________
The motion has _______________ |π|, the maximum displacement of the object from its rest position.
The __________ of the motion is 2π
π, where π > 0.
The period gives the time it takes for the motion to go through one _______________ ___________.
Ex. 10: A ball on a spring is pulled 6 inches below its rest position and then released. The period for the motion is 4
seconds. Write the equation for the ballβs simple harmonic motion.
Frequency of an Object in simple Harmonic Motion
An object in simple harmonic motion given by
π = acos(ππ‘) ππ π = asin (ππ‘)
has frequency f, given by
π =π
2π, π > 0.
Equivalently, π =1
ππππππ.
Ex. 11: An object moves in simple harmonic motion described by π = 12πππ π
4π‘, where t is measured in seconds
and d in centimeters. Find:
a) the maximum displacement:
b) the frequency:
c) the time required for one cycle:
Ex. 12: An object moves in simple harmonic motion described by π = β1
2π ππ (
π
4π‘ β
π
2), where t is measured in
seconds and d in centimeters. Find:
a) the maximum displacement:
b) the frequency:
c) the time required for one cycle:
d) the phase shift of the motion:
Homework: pg. 596 #20, 24, 26, 37, 39, 60, 62, 70, 72 β 77; pg. 601 #18 β 20, pg. 603 #106 β 120(e)