NOTES Unit 5 Trigonometry Honors Common Core Math 2 23tisciamath.weebly.com/uploads/2/5/8/7/25873786/...Inverse operations of sine, cosine and tangent i. Sine ii. Cosine iii. Tangent

NOTES Unit 5 Trigonometry Honors Common Core Math 2 23

Spaghetti Lab UNIT CIRCLE

Materials needed (per group): Double check that you have ALL these BEFORE we leave class!!

Butcher paper (about 8 feet long) Uncooked spaghetti (5-6 pieces) 2 or 3 Pencils

Masking tape (4-6 pieces) Protractor Compass

Meter stick 2 ft of twine that does not unravel 2 Colored markers

1. Study the above diagram. You will be creating this diagram along with the function graph that will be in the

space under the header “Function Graph”. This is a discovery activity and you are discovering what the

function graph looks like.

2. Tape the large piece of paper to the top of the lockers. Break a spaghetti noodle in half. Use your compass

to make a circle, at the far left of your paper, which has a radius equal to half of a spaghetti noodle. DO NOT

draw the circle freehand because it needs to be an accurate circle not a “sort of” circle. This circle is referred

to as a “unit circle”, meaning the radius is “one unit of measurement”, which in this case is a half a piece of

spaghetti.

3. Mark the 0° location as shown in the diagram.

4. Place your protractor on the center of the circle and make a mark along the edge of the protractor at the 15°

mark.

5. Draw a line that connects the center of the circle to this mark and extends to touch the circle in quadrant I

and III. This is illustrated in the diagram for the 15° mark.

6. Repeat steps 4 and 5 for every 15° increment in quadrant I. You should draw a diameter, a full line through

the circle, for 15° increment each mark.

7. Continue marking angles every 15° until you get to 180°. Similar to what you did for the marks in quadrant

I, connect the marks you made in quadrant II with the center of the circle while extending the line through

the IV quadrant Your degree marks in quadrant II will create diameters going from quadrant II into

quadrant IV. OR Extend your marks in quadrant II all the way through to the other side of the circle in

quadrant IV.

8. Label all the marks made along the circle with 15° increments. A complete circle is 360°, which will be the

same location as 0°.

9. You will need to transfer all of the 15° marks from the circle to the string, which will then be used to create

the x-axis for the function graph. To make this transfer, one person should hold the end of the string at the

0° location while a second person holds the string along the path of the curve and a third person marks the

locations on the string using a marker.

I

III IV

II


10. Now use the meter stick to make the x-axis and y-axis of the function graph. Then, to create the x-axis labels

for the function graph, hold the end of the string that was at 0° at the origin of the function graph and extend

the string along the x-axis. Make tick marks on the x-axis at the same locations as the marks on the string.

Label these tick marks.

11. What component from the unit circle do the x-values on the function graph represent?

x-values = ________________________

12. Use the length of your spaghetti radius to mark one unit above and below the origin on the y-axis of the

function graph. Label these marks 1 and –1, respectively.

13. Draw a right triangle in the unit circle where the hypotenuse is the radius of the circle to the 15° mark and

the legs lie along and perpendicular to the x-axis. (Reference the diagram below.)

14. Break a piece of spaghetti to the length of the vertical leg of this triangle, which is from the 15° mark on the

circle to the x-axis (Step 1 in diagram below). This piece of spaghetti represents the y-value for the point on

the function graph where x = 15°. Place the spaghetti piece appropriately on the function graph and make a

dot at the top of it (Step 2 below). Note: Since this point is above the x-axis in the unit circle, the

corresponding point on the function graph should also be above the x-axis.


15. Continue constructing triangles and transferring lengths for all marks in quadrants I and II on the unit circle.

16. Now, continue in a similar manner for the marks in quadrant III and IV but think carefully about what is

important to note about these points on the unit circle. Since the points are below the x-axis on the unit circle

this means the y-value corresponding to the “height” of the triangle is a negative value. As a result, should

the corresponding point on the function graph be above or below the x-axis?_____________________

17. After you have constructed all the triangles, transferred the lengths of the vertical legs to the function graph,

and added the dots, using a color marker draw a smooth curve to connect the dots.

18. The vertical leg of a triangle in the unit circle, which is the y-value on the function graph, represents what

function of the related angle measure?

y-values = ________________________

19. You have completed the graph of . Label the graph

Graph of 20. Using the same circle and graph, use a piece of spaghetti for the length of the horizontal leg of each

triangle, starting again with the 15° mark on the circle to the y-axis. After you have transferred the lengths of

the vertical legs to the function graph and added the dots, using a different color marker draw a smooth curve

to connect the dots.

The horizontal leg of a triangle in the unit circle, which is the x-value on the function graph, represents what

function of the related angle measure?

x-values = ________________________

21. Now you have completed the graph of g(x) = cos x. Label the graph g(x) = cos(x)

Stop: Raise your hand when you get this far so your teacher can check your work.

OK _________

After getting checked, keep going….lab questions on next page


Lab Questions

1. The period of a sin or cos function is defined as the length of one cycle.

What is the period of the sine curve? ____________ The cosine curve? __________

2. The amplitude of a sin or a cos functions is defined as the vertical distance from the midline,

or in this case the x axis. What is the amplitude for the sin and cos functions?____________

3. What are the zeroes of the sin function? _________________________

The cos function? _____________________

4. What are the x values of the maxima on the sine function? ________________

The cosine function? ______________

4. What are the x values of the minima on the sine function? ________________

The cosine function? ______________

5. Imagine this function as it continues in both directions. Explain how you can predict the value of the sine and

cosine of 390°.

6. In what ways are the sine and cosine graphs similar? Be sure to include a discussion of intercepts, amplitude,

maxima, minima, and period.

7. In what ways are the sine and cosine graphs different? Again, be sure to include a discussion of intercepts,

amplitude, maxima, minima, and period.


Day 9: Evaluating Trig Functions

Warm-Up:

1. A water skier must be at least a horizontal distance of 50 feet from the boat in order to safely avoid undertow from the propeller. If the angle of elevation is 35° from the skier to the pole how long is the rope?

2. A 21-foot tree needs trimming. Safety guidelines say the angle made by the ladder and the ground should be 70°. How long should the ladder be to reach the top of the tree?

Day 9 (Part 1): Exploring Sine, Cosine, and Tangent Angle Restrictions

Using your calculator, complete the chart. Round to the nearest thousandth.

Angle sin(angle) cos(angle) tan(angle)

0

30

60

90

120

150

180

210

240

270

300

330

360

1. What do you notice about the sine column? Describe the pattern.

2. What do you notice about the cosine column? Describe the pattern.

3. What do you notice about the tangent column? Describe the pattern.


Remember!!

a. Angles are measured in ____________ or____________

b. We have to check our mode to make sure the calculator knows what measure we

are using!

i. In this class, we will always use _________, but you should know that

radians exist!

Make sure Degree is highlighted!

Day 9 (Part 2): Solving Trig Equations

1. Use the inverse trig functions on your calculator to solve the following equations:

a. sin (x) = 0.3

b. sin (x + 2) = 1.5

c. 3 sin(x) = 2

Solving Sine, Cosine and Tangent Equations

1. We can solve equations involving ___________, _____________ and

_________________ just like any other equation!

2. Inverse operations of sine, cosine and tangent

i. Sine

ii. Cosine

iii. Tangent

3. Solve the equations and express your answer to the nearest tenth degree:

1. sin (x) = 0.6 2. cos (x) = 1.5 3. tan (x) = -6.7

4. cos (x) = -0.87 5. 3sin (x) = 1.5 6. 4sin(x) = 1.2


Practice

Solve the following equations and express your answer to the nearest tenth degree:

1) sin (x) = 0.8

2) cos (x) = -0.78 3) tan (x) = -9.5 4) sin (x) = 0.366

5) sin (x) = -0.768

6) 3tan (x) = -12.8

7) 3sin (x) + 4 = 1.57 8) 4cos (x) – 6 = -5.2

Day 9 (Part 3): Calculator Trig functions

An exploration Use your graphing calculator to answer the following questions.

1. Use your calculator to find the following trig ratios. Round your answers to the nearest thousandth.

Sin (20) = Cos (40) = Tan (70) =

Sin (83) = Cos (75) = Tan (25) =

2. Find the sine, cosine, and tangent of a right triangle with a hypotenuse of 1 and angle of elevation of 45°.

a) What is the sine of 45°, rounded to the nearest thousandth? _______________

b) What is the cosine of 45°, rounded to the nearest thousandth? _______________

c) What is the tangent of 45°, rounded to the nearest thousandth? _______________ d) What is special about the sine and cosine of 45°?

e) What is special about the tangent of 45°?

3. Use your calculator to find the following sine and cosine ratios.

Cos (20) = Sin (70) =

Cos (30) = Sin 60 =

Cos (60) = Sin (30) =

Cos (75) = Sin (15) =

What do you notice about sine and cosine when the angles add to 90°?

4. Use your calculator to find the following:

Tan(40) = sin 40

cos 40 Tan(50) =

sin 50

cos 50

What conclusion can you draw about the relationship between the tangent function and sine and cosine?


Day 10: Graphs of Sine, Cosine and Tangent

Warm-up: Solve the trig equations:

1.) 1 + cos (x) = 0 2.) 2sin(x)cos(x) + cos(x) = 0

3.) Find the area of the triangle if b = 11, a = 8, and Angle C = 37.

4.) Solve the triangle in problem #3.

Complete the table below: Make sure your calculator is in degree mode!!

Degree sin(x) Point (Degree, sin(x))

0 0 (0,0)

30

60

90

120

150

180

210

240

270

300

330

360

Using the points above (degree, sin(x)), sketch a graph of y = sin(x).


Complete the table below:

Degree Cos(x) Point (Degree, Cos(x))

0 1 (0,1)

30

60

90

120

150

180

210

240

270

300

330

360

Using the points above (degree, cosx), sketch a graph of y = cos(x).


Complete the table below:

Degree Tan(x) Point (Degree, Tan(x))

0 0 (0,0)

30

60

90

120

150

180

210

240

270

300

330

360

4. Using the points above (degree, tanx), sketch a graph of y = tanx.

What happens to tangent at 90o and 270o? Why is this happening?


Day 10 Notes: The Graphs of Sine, Cosine, and Tangent


Day 11 Warm Up

Warm-up:

1. Graph the following and state the vertex and axis of symmetry:

1. y = 3x2

2. y = x2 +5

3. y = 3(x-4)2 -7

2. Solve the triangle if Angle A = 60, c = 8, b = 10

3. Solve the trigonometric equation: 2tan(x)sin(x) = 2tan(x)

Day 11 Notes: Amplitudes and Midlines of Trig Functions

How are y = sin(x), y = 2sin(x), and y = ½ sin(x) alike? How are they different?

I. Amplitude

a. A graph in the form ________________ or _________________ has an

amplitude of ______________.

b. The amplitude of a standard ____________ or _____________ graph is

______.


c. The amplitude of a sine or cosine graph can be found using the following formula:

d. Find the amplitude for each of the following:

1. y = 3sinx

2. y = -4cos5x

3. y = (1/3)sinx +5

II. Midline

a. The midline is the line that ________________________________

b. The midline is halfway between the __________ and ___________

c. The midline can be found using the following formula:

d. When there is no vertical shift, the midline is always ____________.

III. Period

a. A period is the length of one ________________.

b. y=sin(x) has a period of ___________.

c. y=cos(x) has a period of ___________.

d. y=tan(x) has a period of ___________.

e. When f(x) = Asin(Bx) the formula for period is:


4. y = 0.5 sin (x) Amplitude: ________ Midline: ________ Period: ________

5. y = 5 sin (x) + 1 Amplitude: ________ Midline: ________ Period: ________

6. y = -2 sin (3x) Amplitude: ________ Midline: ________ Period: ________

7. y = cos (2x) + 1 Amplitude: ________ Midline: ________ Period: ________


Day 12 Warm up

Warm-up: 1. Find the amplitude, period and midline. Then, graph each Trig Function with 1 cycle in the negative direction and

1 cycle in the positive direction. a. y = -4 sin(3x) b. y = cos (2x) + 1

2. Solve the problem. Round answer(s) to the nearest degree

a. 2sin(x)cos(x) = - 2 sin(x) b. -2cos(5x) = 3

3. The graph shown displays the level of water at a boat dock, which varies due to the tides. Determine the amplitude,

midline, and period of the graph.

Day 12 Notes: Interpreting Graphs of Trig Functions

I. Amplitude and Midline

a. The amplitude can be found by using the following formula:

b. The midline can be found using the following formula:

# of Hours after Midnight

Depth of

Water (ft)


c. Find the amplitude and midline for each of the following graphs:

1. 2.

3. 4.

Day 13 Warm Up

1. Identify the amplitude, period, and midline of the following trig function. Hint: it may help to trace out one cycle.

State the amplitude, period, and midline of each of

the following:

1. y = (1/2)sin (x)

2. y = -5cos (3x)

3. y = sin(x +5) - 6

4. y = 2cos (x) + 3


Day 13 Notes: Graphing Practice, Writing Equations of Trig Functions

Graphing Practice: Graph the following functions over two periods, one in the positive direction and one in the negative directions. Label the axes appropriately.

1. y = -2 sin (3x) Amplitude: ________ Midline: ________ Period: ________

2. y = cos (2x) - 1 Amplitude: ________ Midline: ________ Period: ________

3. y = 3 sin (1/2x) Amplitude: ________ Midline: ________ Period: ________


4. y = -2 cos (4x) + 1 Amplitude: ________ Midline: ________ Period: ________

Notes: Writing an equation given a trig graph

To write an equation of a trigonometric function when given a graph, first determine

______________________, _________________, and _________________ of the graph.

**HINT: tracing one cycle of the graph can help determine these values AND decide if sine or cosine is better.

Then use those values and the formulas to calculate a, b, and d of the standard equation y = a sin(bx) + d or

y = a cos(bx) + d.

Formulas we must know

Write the equation for the following trigonometric functions.

1) A radio transmitter sends a

radio wave from the top of a 50-

foot tower. The wave is

represented by the

accompanying graph.

2) The accompanying graph

represents a portion of a sound

wave.

3)

Amplitude =

Period =

Midline =

45° 90° 135° 180°


You Try! Write the equation for the following trigonometric functions.

4)

5)

6)

Day 14 Warm Up – Review Day

1. Graph one period in the positive and negative direction for y = -2cos(3x) – 1.

2. Solve the triangle given b = 16, a = 10, and angle A = 30.

3. The pilot of an airplane finds the angle of depression to an airport to be 16 degrees. If the altitude

of the plane is 6000 meters, find the horizontal distance to the airport.

The figure at the left shows that the depth of water at a boat dock

varies with the tides. The depth is 6 feet at low tide and 12 feet at high

tide. On a certain day, low tide occurs at 6 AM and high tide occurs at

12 Noon.

The figure above shows that the

depth of water at a boat dock

varies with the tides. On a certain

day, low tide is at 6 AM and high

tide is at 12 Noon.

Documents

NOTES Unit 5 Trigonometry Honors Common Core Math 2 23tisciamath.weebly.com/uploads/2/5/8/7/25873786/...Inverse operations of sine, cosine and tangent i. Sine ii. Cosine iii. Tangent