Trig Part I .doc

Trigonometry I

Math 30-1

Trigonometry Part I

Specific Outcome: Students will demonstrate an understanding of angles in standard position, expressed in degrees and radians

Achievement Indicators:

 Sketch in standard position, an angle (positive or negative) when the measure is given in degrees.

 Describe the relationship among different systems of angle measurement, with emphasis on radians and

degrees.

 Sketch, in standard position, an angle in with a measure of 1 radian.

 Sketch in standard position, an angle with a measure expressed in the form radians, where .

 Express the measure of an angle in radians (exact value or decimal approximation), given its measure in

degrees.

 Express the measure of an angle in degrees, given its measure in radians (exact value or decimal

approximation)

 Determine the measures, in degrees or radians, of all angles in a given domain that are coterminal with a

given angle in standard position.

 Determine the general form of the measures, in degrees or radians, of all angles that are coterminal with a

given angle in standard position.

 Explain the relationship between the radian measure of an angle in standard position and the length of the

arc cut on a circle of radius r, and solve problems based upon that relationship.

Specific Outcome: Students will develop and apply the equation of the unit circle

Achievement Indicators:

 Derive the equation of the unit circle from the Pythagorean theorem.

 Describe the six trigonometric ratios, using a point that is the intersection of the terminal arm of

an angle and the unit circle.

Trigonometry I

Specific Outcome: Students will solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.

Achievement Indicators:

 Determine, with technology, the approximate value of a trigonometric ratio for any angle with a measure

expressed in either degrees or radians.

 Determine, using a unit circle or reference triangle, the exact value of a trigonometric ratio of angles

expressed in degrees that are multiples of , or for angles expressed in radians that are

multiples of and explain the strategy.

 Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified

domain, given the value of a trigonometric ratio.

 Explain how to determine the exact value of the six trigonometric ratios, given the coordinates of a point

on the terminal arm of an angle in standard position.

 Determine the measures of the angles in a specified domain in degrees or radians, given a point on the

terminal arm of an angle in standard position.

 Determine the exact values of the other trigonometric ratios, given the value of one trigonometric ratio in

a specified domain.

 Sketch in a diagram to represent a problem that involves trigonometric ratios

 Solve a problem, using trigonometric ratios.

Specific Outcome: Students will graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems.

Achievement Indicators:

 Sketch, with or without technology, the graph of .

 Determine the characteristics (amplitude, asymptotes, domain, period, range, and zeros) of the graph of

 Determine how varying the value of a affects the graphs of .

Trigonometry I

 Determine how varying the value of b affects the graphs of .

 Sketch, without technology, graphs of the form using

transformations, and explain strategies.

 Determine the characteristics (amplitude, asymptotes, domain, period, phase shift, range and zeros) of the

graph of a trigonometric function of the form .

 Determine the values of a, b, c, and d for functions of the form

that correspond to a given graph, and write the equation of the function.

 Determine a trigonometric function that models a situation to solve a problem.

 Explain how the characteristics of the graph of a trigonometric function relate to the conditions in a

problem situation.

 Solve a problem by analyzing the graph of a trigonometric function.

Lesson 1: Standard Position Angles and Trigonometric Ratios

Define: A rotation angle, in standard position, is an angle that starts at the positive x-axis and rotates an arm in a positive or negative direction.

Example: θ = 520° Example: θ = -140°

 If , then the angle rotates in a ______________________________ direction.

 If , then the angle rotates in a ______________________________ direction.

Define: Coterminalangles are two or more rotation angles which share a common terminal arm (they end at the same spot).

Example: θ = 520° Example: θ = -200°

3

I

II

III

IV

I

II

III

IV

I

Trigonometry I

 The difference between any two coterminal angles will always be a multiple of ___________°

Define: The principal angle is the smallest positive standard position angle that is coterminal with a set of angles.

Example: θ = 520° Example: θ = -200°

Define: A reference angle is the positive acute angle formed between an angle's terminal arm and the closest x-axis.

 A set of coterminal angles will have the same reference angle. Try:

a) Draw in standard position.

b) State the quadrant and the reference angle.

c) Find two other angles that are coterminal to .

d) State the principal angle.

Recall: There are three primary trigonometric ratios.

In addition to the three primary trig ratios there are also three reciprocal ratios which are the reciprocal of the ratios above

For rotation angles these ratios are expressed in terms of the coordinates of a point (x, y) on the terminal and the distance (r) from that point to the origin.

Problem: Given that the point P (-3, 2) lies on the terminal arm of a rotation angle, , state the quadrant and determine the exact values of the primary trig ratios.

In General: Only the trig ratios shown below have positive values in those quadrants, otherwise they are negative. 5 P(x, y) x y r 0 0 360 2     90 2  

180

Quadrant I Quadrant II

Quadrant III Quadrant IV

C

A

S

Problem: Given that Determine the exact values of cos θ, csc θ and cot θ.

Try: Given that Determine tan θ, sec θ and sin θ to the nearest one

hundredth.

Assignment: Page 474 #1-5, 8, MC #1

Lesson 2: Radian Measure and Arc Length

A radian is another unit of measurement for angles (instead of degrees), just as centimeters and inches are simply different units for measuring length.

Problem: Determine the size of a 360o _{angle in radians}

Degrees

Radians

Converting between Degrees and Radians

Example: Convert from degrees to radians (give your answer as an exact value in terms of ).

Example: Convert from radians to degrees.

Problem: Convert the following. Express as an exact value or as a decimal correct to the nearest hundredth.

a) b) 4 rad = _______ º c) d)

Example

Complete the following chart

sketch in standard position

reference angle

one coterminal angle

general statement for all coterminal angles

Arc Length

The arc length is the portion of the circumference contained by the arms (the radii) of a rotation angle.

Problem: Determine the length of an arc which subtends a central angle of in a circle whose radius is 12.3 cm, correct to the nearest tenth.

Assignment: Page 485 #2,3,4 Page 494 #4,5,7-11, MC #1,2

Lesson 3: Unit Circle

Investigation: The definition of a circle is the set of points that are equidistant from a central point. Given a point P(x, y) on the circle and the centre at the origin, determine the equation of the circle in terms of radius, r.

Recall Special Triangles: Triangles containing the most common reference angles.

9 2

3

30º

P(x, y)

Unit Circle

If triangles with a hypotenuse of one (unit triangles) were centered at the origin then the vertices would form a circle with a radius of one. This circle is called the unit circle (r = 1 unit). The side lengths would give the x and y coordinates of the points on the circle.

If r = 1 then the following is true:

 sin θ = ________

 cos θ = ________

 tan θ = ________

where θ is the rotation angle at the origin.

45º

2

2

2 30º

1

Example: If is the point at the intersection of the terminal arm of an angle and the unit circle, determine the exact coordinates of each of the following:

Problem: Determine one positive and one negative measure for if

Problem: Determine the exact value for the six trigonometric ratios when the unit circle passes through the

point and state the angle.

Problem: Determine the exact value for the following trig functions

a) sin 225˚ b) cos 120˚ c) cot

d) cos e) tan 600˚ f) sin

Lesson 4: Solving Equations involving Sine or Cosine

Recall: For each trig ratio there are 2 quadrants and therefore 2 angles which will share that ratio. For all angles with the same reference angle the ratio will be similar with exception to the sign of the ratio.

Recall: To determine an angle, given its ratio, the inverse of the given trig function must be used.

Recall: Remember to be in the appropriate mode depending on the domain.

Solving Trig Equations with angles on the unit circle.

Problem: Solve the following equations on the domain of .

a) b)

Try: Solve the following equations on the domain of .

Problem: Solve the trig equation on the domain of

Solving Trig Equations with Other Angles.

Problem: Determine the solutions to equations on the domain .

a) b)

Try: Solve the following equations on the domain of . Express answers to the nearest hundredth.

a) b)

Lesson 5: Exploring Sine and Cosine Curves

Materials Required:

 Uncooked spaghetti

 Paper (about 1m in length)

 Circle printout

 1 meter stick

 Piece of string

 1 black marker

Trigonometry 1

Get Ready:

 Watch the following video for instructions http://www.youtube.com/watch? feature=player_profilepage&v=AUOgD_Hq70A

 Place the paper on the floor and create an axis as shown below.

 Place the string around the circumference of the circle.

 Mark your string at every 15o _{all the way around the circle.}

 Lay the string down on your x-axis.

 Wherever there is a mark on the string, make a small mark on the x-axis. (That means at every 15o_{, you}

will now have a mark on your x-axis).

Get to Work:

 Place a piece of spaghetti on the circle to represent the radius for the 15o _mark.

 Question: When we’re looking at coordinates, which represents sine? Cosine?

 The next steps depend on whether or not you are creating the sine graph or the cosine graph. Only do the steps that match YOUR graph.

Sine Graph Cosine Graph

 Break a piece of spaghetti to the length of the vertical leg of this triangle, from the 15o _{mark on the circle to the x-axis. This}

spaghetti piece will represent the y-value where x=15o_.

 Tape the spaghetti to your graph, placing it so that it is touching but above the x axis at 15o _{(since you placed the spaghetti piece}

in your circle above the x axis).

 Continue creating triangles and transferring the vertical lengths for all marks on the unit circle.

 When the spaghetti piece ends up under the

x-axis in your circle, you need to place it under the x-axis on your paper.

 Break a piece of spaghetti to the length of the horizontal leg of this triangle.

 Tape the spaghetti to your graph, placing it so that it is touching but above the x axis at 15o _{(since y is positive in this position)}

 Continue creating triangles and transferring the horizontal lengths for all marks on the unit circle.

 REMEMBER: Although the spaghetti pieces are horizontal on your circle, you need to place the perpendicular to the x -axis on your paper!

 REMEMBER: positive x values on your circle are represented by placing the spaghetti above the x-axis. Negative x

values on your circle are represented by placing the spaghetti below the x-axis.

 Every time you tape a piece of spaghetti to your paper, mark a dot at the top of the spaghetti.

Trigonometry 1

Big

Questions

1. If you change the radius of the circle (length of spaghetti), what will happen to your graph?

2. Select an angle between 0 o _{and 360}o_{. Based on the graph your created (either sine or cosine),} would you have placed the spaghetti above or below the x-axis. Why?

3. Using the same angle as before, find another angle that would require you to place the spaghetti in exactly the same way – same length, same direction. What do you notice?

4. Using the same angle as before, how many angles can you find that would be exactly the same length but on the opposite side of the x-axis? Name them. What do you notice?

5. At what angle does the graph repeat itself? (This is called the period of the graph).

6. What is the approximate height of the triangle at 30o_{? (We call this Sin30}o _{or Cos30}o _depending on which graph you created)

7. Select several other angles that we added to our graph and determine the Sin or Cos of those angles. Make sure at least two are 180o _apart.

8. Select one angle (other than a multiple of 15o_{) and determine its Sin or Cos.}

9. Why does sin300 _{equal sin150}o_?

10. What is the most important thing you would tell someone about this concept?

11. Complete the table comparing and

Trigonometry 1

Lesson 6: Graphing Trigonometric Functions

Trigonometric functions are often referred to as circular or periodic functions. This is because the values of the functions can be related to the x and y coordinates of a circle and the coordinates will repeat after a rotation of 360º or 2π radians.

Define:

Period: the horizontal length that it takes for a periodic function to repeat itself.

Amplitude: the distance from the median line to the minimum or maximum value of a periodic function.

Median Line:the average of the maximum and minimum values.

Phase Shift: the horizontal translation away from the y axis

Recall: Label the period, the amplitude, the median line and the maximum and minimum values on the graphs below. State the following: domain, range and x and y intercepts.

a) y = sin b) y = cos x

Trigonometry 1

Investigation

Investigating parameters a, b, c, and d for .

Graph the following functions with your graphing calculator

Describe transformation to the previous graph beginning with

amplitude

Period (radians) range

Summary of the Effect of the Parameters a, b, c and d for

Trigonometry 1

Problem: A trigonometric function is defined by the equation . Analyze the

function and sketch one complete period of its graph.

Problem: A sine function has a period of , an amplitude of 3, a maximum of 5 and a phase shift

Trigonometry 1

Try: For the functions stated below, describe the transformations which would occur to the graphs of or to produce the transformed functions. Determine the amplitude, period, phase shift, median line, domain, and range and sketch the graph.

a) b)

Try: The graph of undergoes the following transformations: a vertical stretch by a factor of 3 about the x-axis, a horizontal stretch by a factor of 2 about the y-axis, a reflection in the x -axis, a vertical displacement 2 units down and a phase shift of to the left. Determine the

equation of the function in the form , a < 0.

Trigonometry 1

Lesson 7: Writing Equations for Trigonometric Functions

Summary of the Effect of the Parameters a, b, c and d for or

.

Summary of the Effect of the Parameters a, b, c and d for Amplitude =

Period = (for degrees) Period = (for radians)

Horizontal phase shift = c Vertical displacement = d to right if c > 0 up if d > 0, down if d < 0

to left if c < 0

Amplitude – not applicable

a value represents:

 a vertical expansion or

 a vertical compression

Period = (degree measure)

Period = (radian measure)

Horizontal phase shift = c

 right if c > 0, left if c < 0 Vertical displacement = d

 up if d > 0

Trigonometry 1

Problem: The graph of a sinusoidal function is shown below.

a) Determine the amplitude, median line, period and phase shift of this function.

b) Write the equation of this function in the form:

i)

ii)

c) Write the equation of this function in the form:

i)

ii)

Try: The graph shown below has a range of . Determine the equation of the graph if it is a:

a) cosine function, a > 0

Trigonometry 1

Assignment: Page 534 #6,7, MC#1,2 and Page 548 #3-5, MC#1,2

Lesson 8: Applications of Trigonometric Functions

Problem: The depth of water in a harbor, d metres, t hours after midnight, can be approximated using

the function below .

a) Display the graph of this function on a grid. Choose a suitable window.

b) Determine the depth of water, correct to the nearest tenth of a metre, at 2:00 AM, and 2:00 PM.

Trigonometry 1

Problem: A ferris wheel has radius of 5 metres. Passengers get on at a point 1 m above the ground, and make 1 full revolution every 24 seconds.

a) Draw a graph representing the height (h) of a passenger in metres as a function of time (t) in seconds. Show three complete cycles.

b) Determine the equation of this graph in the form:

i) ii)

c) Determine the distance of the passenger above the ground 10 seconds after passing the lowest point of the ride, correct to the nearest tenth of a metre.

d) How long does it take for the passenger to first reach a height of 9 metres, correct to the nearest tenth of a second?