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Math AnalysisChapter 4 Notes: Trigonometric Functions

Day #1: Section 4-1: Angles and Radian Measure; Section 4-2 Trigonometric FunctionsAfter completing section 4-1 you should be able to do the following:

1. Use degree measure2. Use radian measure3. Convert between degrees and radians4. Draw angles in standard position5. Find coterminal angles6. Find the length of a circular arc

Angles

Standard Position of an Angle:

This angle θ is a positive angle.The direction of rotation from theinitial side to the terminal side iscounter-clockwise.

Degree Measure of an AngleAngles are measured by determining the amount of rotation from the initial side to the terminal side. One way to measure angles is in degrees, symbolized by a small, raised circle 0. A complete rotation around a circle is considered 3600, therefore

10 = of a complete rotation around a circle.

Classifying angles by there degree measurement

Practice: In 1-4, Draw the given angle in standard position. State the quadrant the terminal side is in.

Math Analysis Notes Mr. Hayden 1

An angle is formed by two rays that have a common endpoint (vertex). One ray is called the initial side and the other the terminal side. If the angle is in standard position the vertex is at the origin of a rectangular coordinate systerm and the initial side is always along the positive x-axis.

• Initial Side

Terminal Side Positive and Negative Angles Positive Angle: When a ray is

rotated from the initial side counter-clockwise, the angle measure is positive.

Negative Angle: When a ray is rotated from the initial side clockwise, the angle measure is negative.

•This angle θ is a negative angle. The direction of rotation from the initial side to the terminal side is clockwise.

• • • •

00 < θ < 900 θ = 900 900 < θ < 1800 θ = 1800

1. 450 2. 2250 3. 2700 4. −600

Radian Measure of an AngleAnother way to measure angles is in radians. One radian is the measure of the central angle of a circle that intercepts an arc length equal in length to the radius of the circle:

A radian is the ratio of the arc length (S) intercepted by two radii. A radian is a unit-less angle measurement.

Practice: In 5-7, Convert each angle in degrees to radians. Do not use a calculator.

5. 600 6. 2700 7. −3000

Practice: in 8-10, Convert each angle in radians to degrees. Do not use a calculator.

8. 9. 10. 1


•θ = 1 radian =

Conversion between Degrees and Radians

1. To convert degrees to radians, multiply degrees by .

2. To convert radians to degrees, multiply radians by .

The following are very important equivalent forms of radian and degree measures. The sooner you realize they are the same the better you will do in the next three chapters on Trigonometry.

300 = 600 = 900 = 1800 = π 2700 = 3600 = 2π

Coterminal AnglesTwo angles when drawn in standard form are said to be coterminal angles if they have the same terminal side.

To find Coterminal Angles: In Degree Measure: add multiples of 3600 or subtract multiples 3600 to the angle given In Radian Measure: add multiples of 2π or subtract multiples of 2π to the angle given

Practice: In 11-14, Find a positive angle less than 3600 or 2π that is coterminal with the given angle.

11. 4000 12. −1350 13. 14.

The Length of a Circular Arc

Practice: In 15-17, Find the length of the arc on a circle of radius r intercepted by a central angle θ.

15. r = 10 inches, θ = 450 16. r = 5 feet, θ = 900 17. r = 6 yards, θ =


•The angles 2250, 5850 and −1350 are said to be conterminal because they all have the same terminal side when drawn in standard position.

Let r be the radius of a circle and θ the nonnegative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is:

S = rθ

•

4.2

The word trigonometry means measurement of triangles. You need to rememorize your basic right triangles:

The six trigonometric functions are:

Name Abbreviation Name Abbreviationsine sin cosecant csccosine cos secant sectangent tan cotangent cot

The six trigonometric functions are defined as:

Steps to find the trigonometric values of angle:1. Draw angle in standard position2. Draw an altitude from the x-axis to the terminal side of the triangle3. Determine type of triangle created and use the above definitions to find the trigonometric value.

Practice: In 18-21, Find the six trigonometric values of the given angle.

18. 1350 19. −600 20. 21.


450-450-900 300-600-900

Use SOH-CAH-TOA to remember the 1st 3 trig functions.

Worksheet on Evaluating Trigonometric FunctionsMath Analysis(Part of HW #1)

Please do work on separate sheet of paper.

In 1-2, Draw the angle in standard position and find the six trigonometric value of the given angle.

1. θ = 2100 2.

In 3-10, Draw the angle in standard position and then evaluate the trigonometric function.3. sec1350 4. tan2400 5. sin(–1500) 6. csc(–4200)

7. 8. 9. 10.


Day #2: Section 4-2: Trigonometric Function; Section 4-3 Right Triangle Trigonometry; Sections 4-4 Trigonometric Functions of Any Angle

Unit Circle

A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is x2 + y2 = 1.

Definitions of the Trigonometric Functions in Terms of a Unit CircleIf t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then

Practice: In 1-2, Given a point P(x,y) is shown on the unit circle corresponding to a real number t. Find the values of the six trigonometric functions at t.

1. 2.


The Unit Circle:

Practice: In 3-7, use the unit circle to evaluate the trigonometric function.

3. 4. 5. 6. 7.

Trig Identities


Reciprocal Identities

Quotient Identities

Practice: In 8-13, Use the trigonometric identities to evaluate or simplify.

8. find ? 9. find ? 10.

11. 12. 13. Find given

4-3 Right Triangle Trigonometry

Practice: In 14-15, find the six trigonometric functions of the given right triangle.

14. 15.

Practice: In 16-19, Draw the angle in standard position then evaluate the trigonometric function

16. cos300 17. tan(−600) 18. csc2250 19. cot2100


Pythagorean Identities

Practice: In 20-21, Find the length of the missing side of the triangle.

20. 21.

Practice: In 22, a point is given on the terminal side of an angle θ is given. Find the six trig functions of θ.

22. (−4, 3)

Day #3: Review of Sections 4-1 to 4-4 and Chapter 4 Quiz


Day 4: Section 4-3 Applications of right triangles, Section 4-5 Graph of Sine and Cosine Functions

After completing today notes you should be able to do the following Solve problems involving angle of elevation Solve problems involving angle of depression Graph sine equations Graph cosine equations

4-3: Applications

Many applications of right triangle trigonometry involve the angle made with an imaginary horizontal line. An angle formed by a horizontal line and the line of sight to an object above the horizontal line is called the angle of elevation.

The angle formed by a horizontal line and the line of sight to an object that is below the horizontal line is called the angle of depression.

Practice: In 1-2, Solve each problem.

1. A flagpole is 14 meters tall casts a shadow 10 meters long. Find the angle of elevation of the sun to the nearest degree.


Observer located here

Horizontal Line

Line of sight above observer

Angle of Elevation

Observer located here

Horizontal Line

Line of sight below observer

Angle of Depression

2. On a cliff 250 feet above the sea an observer sights a ship in the water. If the angle of depression is measured to be 150, how far is the ship from the cliff?

4-5: Graphs of Sine and Cosine Functions.

Values of (x, y) on the graph :

Values of (x, y) on the graph


The sine or cosine graph can vary according to the different values of a, b, c, and d.

where a = amplitude: vertical distance from starting point to the next point x-value ¼ period.

b – helps find period: Trigonometric functions are periodic, which means that the graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the period. To find the period of a sine or a cosine function use:

period =

Interval length: How far horizontally to go to find next exact value of the graph. To find interval length use:

Interval length =

c ─ helps find phase shift. The phase shift is a horizontal shift that states the starting value of the sine graph. To find the phase shift set bx – c = 0 and solve for x. The value of x is the phase shift.

d = vertical shift:

Practice: In 1-4 Graph the Sine or Cosine function. You must graph two full periods for full credit.

1.

2.


3.

4.

Practice: In 5-6, find both the sine and cosine equation for each graph. (pg 518 #61 and 63)

5. 6.


Day 5: Section 4-6 Graph of other Trigonometric Functions

After completing today notes you should be able to do the following Graph tangent function Graph cotangent function Graph cosecant function Graph secant function

Values of (x, y) on the graph :

The tangent graph can vary according to the different values of a, b, c, and d.

where a = vertical distance from starting point to the next point.

b – helps find period: Trigonometric functions are periodic, which means that its graph has a repeating pattern that continues indefinitely. The shortest repeating portion is called a cycle. The horizontal length of each cycle is called the period. To find the period of a tangent function use:

period =

Vertical Asymptotes: can be found by solving both: and

Interval length: How far horizontally to go to find next exact value of the graph. To find interval length use:

Interval length =

c ─ helps find phase shift. The phase shift is a horizontal shift that states the starting value of the sine graph. To find the phase shift set bx – c = 0 and solve for x. The value of x is the phase shift.

d = vertical shift:


Practice: In 1-2, Graph the tangent function.

1.

2.


Graph of Cotangent

In 3-4, Graph the Cotangent function.

3.

4.

Graph of Cosecant Function


In 5, Graph the Cosecant Function

5.

Graph of Secant Function

In 6, Graph the Secant Function

6.

Day 6: Section 4-7 Inverse Trigonometric Functions


After completing today notes you should be able to do the following Understand and use the inverse sine function Understand and use the inverse cosine function Understand and use the inverse tangent function Use a calculator to evaluate inverse trigonometric functions

Remember that in order for a function to have an inverse it must past the horizontal line test.

The inverse sine function has two notations are commonly used to denote the inverse sine function:

or

The inverse sine function is restricted to the interval: . Which means the inverse sine function is only

defined in the 1st and 4th quadrants.

When you are finding the value of the inverse sine function you are finding an angle value.

Practice: In 1-2, Evaluate the given function.

1. 2.

The inverse cosine function has two notations are commonly used to denote the inverse sine function:

orThe inverse sine function is restricted to the interval: . Which means the inverse sine function is only defined in the 1st and 2nd quadrants.

When you are finding the value of the inverse cosine function you are finding an angle value.


1. 2.

The inverse tangent function has two notations are commonly used to denote the inverse sine function:


or

The inverse sine function is restricted to the interval: . Which means the inverse sine function is only

defined in the 1st and 4th quadrants.

When you are finding the value of the inverse tangent function you are finding an angle value.


1. 2.


1. 2.

3. 4.


5. 6.

Using a calculator to evaluate inverse trigonometric functions.

You need to verify what quadrant the angle lands in. Your calculator will only give you reference angles.

Practice: In 1-4, use a calculator to find the value of each expression rounded to two decimal places.

1. 2.

3. 4.


Day 7: Section 4-8 Solving Right Triangles

After completing today notes you should be able to do the following Solve Right Triangles

A right triangle has 6 parts, 3 angles and 3 sides. To solve a right triangle means to find the length of all 3 sides and find the measure of all 3 angles.

Practice: 1. Solve the right triangle.

Practice: 2. Find x.


A B

Ca

c

bGiven B = 23.50 and c = 10

Chapter 4 Review Worksheet

Please do all work on a separate piece of paper. Show all work.

In 1-2, Evaluate the six trigonometric functions of the given angle θ.

1. 2.

In 3-6, Draw an angle with the given measure in standard position.

3. 1000 4. −450 5. 6.

In 7-10, Find one positive angle and one negative angle that is coterminal with the given angle.

7. 2250 8. 600 9. 10.

In 11-18, Rewrite each degree measure in radians and each radian measure in degrees.

11. 1350 12. 400 13. 2600 14. 2150

15. 16. 17. 18.

In 19-21, Find the arc length with the given radius r and centeral angle θ.

19. r = 4 in, θ = 600 20. r = 5 m, θ = 2400 21. r = 12 cm, θ =

In 22-23, Solve ΔABC using the diagram at the right and the given measurements.

22. B = 400, a = 14

23. A = 350, a = 12


θ9

15

θ10

12

A

BC a

bc

In 24-25, Use the given point on the terminal side of an angle θ in standard position. Evaluate the six trigonometric functions of θ.

24. (−4, 5) 25. (−5, −8)

In 26-33, Draw the angle in standard position then evaluate the function without using a calculator.

26. tan 1350 27. sin(−600) 28. cos 2100 29. sec(−3150)

30. 31. 32. 33.

In 34-37, Use a calculator to evaluate the function. Round the result to four decimal places.

34. sin 180 35. sec 4 36. cot (−6.7) 37. csc2420

In 38-39, Solve each problem. Round measurements of lengths to the nearest tenth. Draw a picture, set up an equation, and then solve.

38. Nancy shines a light from a window of Rocky Rococco’s beachside mansion on a cliff 250 feet above the water level. Nick Danger, 10 feet above the water level, is on a ship off –shore find the angle of elevation of the light is 50. Find the slant distance from the ship to Rococco’s mansion.

39. An airplane is directly above a beacon that is 10,000 feet from an airport control tower. The angle of depression from the plane to the base of the control tower is 60. How high above the beacon is the plane?

In 40-45, Determine the amplitude, period, interval lengths, phase shift, and vertical shift. Then graph the function. Label the axes. Graph 2 full periods.

40. 41.

42. 43.

44. 45.

46. 47.

48.


math analysis - canyon crest academy library media …teachers.sduhsd.net/chayden/chapter 4 note...

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