3.1 Trig Functions and Periodic Functions (AT 5.4) Name: ________________________________

Obj.: I will be able to use a unit circle to find values of sine, cosine, and tangent. I will be able to find the domain and range of sine and cosine. I will understand even and odd trigonometric functions and be able to use their periodicity.

VocabularyUnit Circle Circular Functions Even Functions Odd Functions Periodic FunctionPeriod

Notes Unit Circle

o A _________ ___________ is a circle of radius _____, with its center at the _________ of a rectangular coordinate system.

o Equation: _____________________o In a unit circle, the _____________ measure of the _____________ ___________ is

____________ to the length of the ___________________ _______. o If the angle, t, is a real number in radians, then for each t, there corresponds a point P(x , y) on the unit circle.

____________________ ______________________

____________________ ______________________

____________________ ______________________

o Because this definition expresses _____________ ____________ in terms of coordinates of a point on a unit circle, the trigonometric functions are sometimes called ______________ ______________

Domain and Range of Sine and Cosine Functions o The ____________ of a trigonometric function at the ________ number ___ is its value

at an _____________ of ___ radians.o The _____________ and _____________ of each trigonometric function can be found

from the unit circle definition: _______________ and _______________

o Sine: t can be any real number, so the _______________ is ____________ because the radius of the unit circle is 1, the ____________ is ____________

o Cosine: t can be any real number, so the _________________is ____________ because the radius of the unit circle is 1, the ____________ is ____________

3.1 Trig Functions and Periodic Functions (AT 5.4) Name: ________________________________

Even and Odd Trigonometric Functionso A function is even if _________________and odd if_______________.o The ____________ function is an _____________ functiono The __________ function is an __________ function. o On the figure to the right:

P :¿ Q :¿ The x-coordinates of P and Q are the ________, thus ______________________ The y-coordinates of P and Q are __________________ of each other, thus

¿¿ o Cosine and secant functions are even

____________________ ______________________

o Sine, cosecant, tangent, and cotangent functions are odd.

____________________ ______________________

____________________ ______________________

Periodic Functionso Certain patterns in nature repeat again and again. A behavior is periodic if the repetition

continues infinitely.o A period is the time it takes to complete one full cycle.o A function f is periodic if there exists a positive number p such that

_____________________for all t∈the domainof f. The smallest positive number p for which f is periodic is called

the period of f. o Sine and cosine functions are periodic and have period 2π:

____________________ and _____________________

o Tangent and cotangent functions are periodic and have period π:

____________________ and _____________________

o Repetitive behavior of sine, cosine, and tangent functions:

__________________ _________________ _________________

where n is the number of full cycles.

3.1 Trig Functions and Periodic Functions (AT 5.4) Name: ________________________________

Practice

A point P(x,y) is given from the unit circle corresponding to a real number t. Find the values of the six trigonometric functions at t.

1.P( 513 ,−1213 ) 2.

P(−35 , 45 )

Use the unit circle to find the value of the trigonometric function at the indicated real number t, or state that the expression is undefined.

3. sin 11π6

4. cos 13π4

Let sin t=a ,cos t=b ,∧tan t=c. Write the expression in terms of a, b, and c.5. 7sin(−t)−sin (t) 6. 8sin(−t)−sin (t)

7. −sin(−t−8π ¿)−cos (−t−2π )−tan (−t−π )¿

8. −sin(−t−4 π¿)−cos (−t−8π )−tan (−t−9π )¿

9. If f ( x )=sin x and f (a )=49 , find the value of f (a )+f (a+2π )+ f (a+4 π )+ f (a+6 π).

10. If f ( x )=sin x and f (a )=13 , find the value of f (a )+f (a+2π )+ f (a+4 π )+ f (a+6 π).

11. People who believe in biorhythms claim there are three cycles that rule our behavior − the physical, emotional, and mental. Each is a sine function of a certain period. The function for our emotional fluctuations is

E=sin π12t (Equation may not be based on actual studies.)

where t is measured in days starting at birth. Emotional fluctuations, E, are measured from − 1 to 1, with 1 representing peak emotional well-being, − 1 representing the low for emotional well-being, and 0 representing feeling neither emotionally high nor low.a. Find E corresponding to t = 18 , 24 , 30 , 36 , and 42. Describe what you observe.

3.1 Trig Functions and Periodic Functions (AT 5.4) Name: ________________________________

b. What is the period of the emotional cycle?

3.2 Graphs of Sine (AT 5.5) Name: ________________________________

Obj.: I will be able to graph sine functions and the variations. I will be able to determine amplitude, period, and phase shift for a sine graph.

VocabularySine Curve Amplitude Phase Shift

Notes The Graph of y = sin x

o The _________________________ functions can be graphed in a rectangular coordinate system by plotting points whose coordinates ____________ the ___________________.

o In this case, we graph _____________by listing some points on the graph. Graph the function on the interval _________ because the period is_______. The rest of the graph is made up of ___________________ of this portion.

Approximate_________________ in order to plot it. Mark off the __________ in terms of ______ to avoid rounding the x-values. Connect the points with a ________________ _______________

The graph of y=sin x will have the following ____________________: Domain:___________ and Range: ______________ Period: _____________ Odd function: _______________________

Graphing _____________________ of y = sin xo One complete cycle of the sine curve includes __________ __-______________, _____

___________________ point, and ______ _________________ point. o The ________ ________ on the graph can be found using the equations below to find

the x-values, and ______________ the function at each ____ to ________the ___-values.

3.2 Graphs of Sine (AT 5.5) Name: ________________________________

o Graphing _____________________________ Amplitude

Maximum ________ of the graph Equal to ________

Period ___________ Phase Shift

The _________________ shift to the sine graph

New starting point: _______ One period is _______________

o Steps for graphing: Identify the ___________, ___________________, and _________ __________. Find the __-values for the ____ key points, starting at the beginning of the cycle. Find the __-values for the ____ key points by evaluating the function at each

value of x from the second step. ______________ the five key points with a smooth curve and graph one

______________________ ____________. ______________ the graph to the _________ or ____________ as needed.

3.2 Graphs of Sine (AT 5.5) Name: ________________________________

Practice

Determine the amplitude for each function, then graph the function and y=sin x for 0≤ x≤2π .

1. y= 14sin x 2. y=1

2sin x

Determine the amplitude, period, and phase shift for each function, then graph the function for −2π ≤ x≤2π .

3. y=−2sin (4 x−π ) 4.y=3sin (x+ π4 )

5. Using the equation y = A sin Bx, if either A or B is replaced with its opposite, the graph of the resulting equation is a reflection of the graph of the original equation about the x-axis.

3.3 Graphs of Cosine (AT 5.5) Name: ________________________________

Obj.: I will be able to graph cosine functions and the variations. I will be able to determine amplitude, period, and phase shift for a cosine graph.

VocabularySinusoidal Graphs

Notes The Graph of y = cos x

o Graph ________________by listing some points on the graph. Graph the function on the interval ___________ because the period is______. The rest of the graph is made up of ___________________ of this portion.

Approximate ¿¿ in order to plot it. Mark off the x-axis in terms of ____ to avoid rounding the x-values. _______________ the points with a smooth curve.

The graph of y=cos x will have the following ________________: Domain:____________ and Range: ____________ Period: ______ _________ function: ___________________

The graph of y=cos x is the graph of y=sin x with a phase shift of ______. Tracing the cosine graph from _________ to ___________ is one

complete cycle of the ________ curve. Identity: ____________________ Because of the similarity, the graphs of sine functions and cosine

functions are called _______________ ________. Graphing ______________ of y = cos x

o Graphing __________________________ Amplitude: ____ Period: _____ Phase Shift: ____ (One period is __________________)

o ____________ for graphing are the same as for graphs of _______ functions.

3.3 Graphs of Cosine (AT 5.5) Name: ________________________________

Practice

Find the amplitude and period of the following functions.

1. y=16cos(3 x ) 2. y= 1

4cos (2 x)

Determine the amplitude for each function, then graph the function and y=cos x for 0≤ x≤2π .

3. y=17 cos x 4. y=15cos x

Determine the amplitude, period, and phase shift for each function, then graph the function for −2π ≤ x≤2π .

5. y=3cos (2πx−4 π ) 6. y=5cos (2πx−4 π )

3.4 Vertical Shifts and Periodic Models (AT 5.5) Name: ________________________________

Obj.: I will be able to translate the graph of a sine and/or cosine function vertically. I will be able to apply graphing sine and cosine functions to modeling data.

Notes Vertical Shifts of Sinusoidal Graphs

o Forms: _____________________ and ______________________o Constant ____ causes a vertical shift in the graph.

If D is ___________, the graph shifts ___________ ____ units If D is ____________, the graph shifts ______________ ______ units Oscillations are now around line _______

_______________ value of y is ________ ______________ value of y is ________

Modeling Periodic Behavioro Periodic data can be better understood by ________________. o Additionally, some graphing calculators have a _______ _______________ feature.

Minimum of ________ ____________ _______________ Gives the function in the form ________________ of the best fit for wavelike

data.

3.4 Vertical Shifts and Periodic Models (AT 5.5) Name: ________________________________

Practice

Use a vertical shift to graph one period of the function.

1. y=sin x−8 2. y=3cos 12x−4

Find an equation for the graph. Type the equation of the given graph in the form y=AsinBx or y=AcosBx.

3. 4.

5. Rounded to the nearest hour, a city averages 16 hours of daylight in June, 8 hours in December, and 12 hours in March and September. Let x represent the number of months after June and let y represent the number of hours of daylight in month x. Make a graph that displays the information from June of one year to June of the following year.

6. A clock with an hour hand that is 13 inches long is hanging on a wall. At noon, the distance between the tip of the hour hand and the ceiling is 25 inches. At 3 P.M., the distance is 38 inches; at 6 P.M., 51 inches; at 9 P.M., 38 inches; and at midnight the distance is again 25 inches. If y represents the distance between the tip of the hour hand and the ceiling x hours after noon, make a graph that displays the information for 0 ≤ x ≤ 24.

3.5 Applications (AT 5.8) Name: ________________________________

Obj.: I will be able to solve right triangles. I will be able to apply trigonometric functions to trigonometric bearings and simple harmonic motion.

VocabularySolving a right triangle

Bearing Simple Harmonic Motion

Equilibrium Position

Frequency

Notes Solve Right Triangles

o Solving a right triangle means finding the missing ______________ of its sides and the measurements of its ____________.

o In addition to the right angle, you will need at least _________ additional pieces of information to solve a right triangle.

o The sum of the two acute angles is ______.o ______, ___________, and/or _______________ can be used to find any missing sides. o Recall that if you know two sides of a right triangle but not either of the acute angles,

you can use the ______________ of a trigonometric function to solve for the ________. Trigonometry and Bearings

o In navigation and surveying problems, the word ___________ is used to specify the __________ of one point relative to another.

o The bearing from point O to point P is the acute angle, measured in __________, between ray OP and a __________-_________ line.

Each bearing has three parts: a letter (__________), the measure of an __________ angle, and a letter (________).

Start with the north/south side (___ if the north side of the north-south line, ___ if the south side of the north-south line)

Write the measure of the acute angle. If the acute angle is measured on the east side of the north-south line, write ___

last. If on the west side, write ___ last. Simple Harmonic Motion

o Trigonometric functions are used to model _______________ or _______________ motion (such as the motion of a vibrating guitar string, the swinging of a pendulum, or the bobbing of an object attached to a spring).

Consistent oscillations are called simple harmonic motion. The rest position is called the ________________ position

(horizontal line that runs through the _______ of the graph).o An object that moves on a coordinate axis is in simple harmonic

motion if its distance from the origin, d, at the t is given by either________________ or ________________

Amplitude:_______ (maximum _________________) Period:_____________________ (time for one complete cycle) Frequency: ___________________

Describes the _______________ of complete cycles per unit ________. Reciprocal of the period:¿¿

3.5 Applications (AT 5.8) Name: ________________________________

Practice

Solve the right triangle shown in the figure. 1. B=16.9 ° , b=30.5 2. B=17.5 ° ,b=36.5

Find the bearing from O to A.

3. 4.

5. An object moves in simple harmonic motion described by the equation d=−sin π3twhere t is

measured in seconds 3 and d in inches. Find the maximum displacement, the frequency, and the time required for one cycle. Find (a) the maximum displacement, (b) the frequency, and (c) the time required for one cycle.

6. A ramp is to be built beside the steps to the campus library. Find the angle of elevation of the 22 -foot ramp, to the nearest tenth of a degree, if its final height is 5 feet.

7. A jet leaves a runway whose bearing is N 34°E from the control tower. After flying 6 miles, the jet turns 90° and flies on a bearing of S 56°E for 7 miles. At that time, what is the bearing of the jet from the control tower?

8. An object is attached to a coiled spring. The object begins at its rest position at t = 0 seconds. It is then propelled downward. Write an equation for the distance of the object from its rest position

3.5 Applications (AT 5.8) Name: ________________________________

after t seconds, if the amplitude is 2 inches and the period is 4 seconds.