Math Analysis Notes prepared by Mrs. Atkinson 1

Math Analysis Chapter 4 Notes: Trigonometric Functions

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

1. Use degree measure 2. Use radian measure 3. Convert between degrees and radians 4. Draw angles in standard position 5. Find coterminal angles 6. 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 the initial side to the terminal side is counter-clockwise. Degree Measure of an Angle Angles 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 = 1360

of a complete rotation around a circle.

Classifying angles by there degree measurement

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

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Practice: In 1-4, Draw the given angle in standard position. State the quadrant the terminal side is in. 1. 450 2. 2250 3. 2700 4. −600 Radian Measure of an Angle Another 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. 4π 9. 4

3π

− 10. 1

• θ = 1 radian = arc length 1 radian = 1

radiusrr

= =

Conversion between Degrees and Radians

1. To convert degrees to radians, multiply degrees by 0180π .

2. To convert radians to degrees, multiply radians by 0180

π.

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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 = 6π 600 =

3π 900 =

2π 1800 = π 2700 = 3

2π 3600 = 2π

Coterminal Angles Two 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. 223π 14. 17

6π

−

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 = 23π

•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

•

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4-2 Trigonometric Functions The word trigonometry means measurement of triangles. You need to rememorize your basic right triangles: The six trigonometric functions are: Name Abbreviation Name Abbreviation sine sin cosecant csc cosine cos secant sec tangent tan cotangent cot The six trigonometric functions are defined as:

oppositesinhypotenuse

θ = hypotenusecscopposite

θ =

adjacentcos

hypotenuseθ = hypotenusesec

adjacentθ =

oppositetanadjacent

θ = adjacentcotopposite

θ =

Steps to find the trigonometric values of angle:

1. Draw angle in standard position 2. Draw an altitude from the x-axis to the terminal side of the triangle 3. 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. 56π 21. 5

3π

450-450-900 300-600-900

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

Asn #1: p458#2-74 (evens) Worksheet on right triangle trig

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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 Circle If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then

sin t y= 1csc ty

=

cos t x= 1sec tx

=

tany

tx

= cot xty

=

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.

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

3. sin6π 4. 5cos

6π 5. tanπ 6. 7csc

6π 7. 4sec

3π

Trig Identities

Reciprocal Identities 1sin

cscθ

θ= 1cos

secθ

θ= 1tan

cotθ

θ=

1cscsin

θθ

= 1seccos

θθ

= 1cottan

θθ

=

Quotient Identities

sintancos

θθθ

= coscottan

θθθ

=

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

8. 1cos2

θ = find ( )cos θ− ? 9. ( ) 2sin3

θ− = find sinθ ? 10. sin(0.2) csc(0.2)

11. 2 2sin cos3 3π π+ 12. 2 2sec tan

4 4π π− 13. Find cosθ given 6sin

7θ =

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 2 2sin cos 1θ θ+ = 2 21 tan secθ θ+ = 2 21 cot cscθ θ+ =

Math Analysis Notes prepared by Mrs. Atkinson 8

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

Asn 2: p472#2-44(even); p484#2-34(even);p499#2-86(even)

Asn 3: p484#9-19(odd); p499#61-87(odd), 88-92(all)

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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

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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 xy sin= : Values of (x, y) on the graph xy cos=

Math Analysis Notes prepared by Mrs. Atkinson 11

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

dcbxay +−= )sin( dcbxay +−= )cos( 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 = bπ2

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

Interval length = 4

period

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. 1)2sin(3 −+= πxy

2. 332

sin2 +⎟⎠⎞

⎜⎝⎛ −−= ππxy

Math Analysis Notes prepared by Mrs. Atkinson 12

3. xy31cos

21

=

4. 32

3cos2 +⎟⎠⎞

⎜⎝⎛ +−=

πxy

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

Asn 4: p485#54-60(even) P518#18, 22, 26, 30, 44, 48, 52, 62-66(even)

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

dcbxay +−= )tan( 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 = bπ

Vertical Asymptotes: can be found by solving both: 2π−

=− cbx and 2π

=− cbx

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

Interval length = 4

period

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:

Math Analysis Notes prepared by Mrs. Atkinson 14

Practice: In 1-2, Graph the tangent function. 1. ( )π42tan3 −= xy

2. ( )xy π2tan21

−=

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Graph of Cotangent

xy cot= In 3-4, Graph the Cotangent function. 3. xy πcot2= 4. ( )ππ −−= xy 2cot

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Graph of Cosecant Function

xy csc= In 5, Graph the Cosecant Function 5. ( ) 242csc +−= ππxy Graph of Secant Function

xy sec= In 6, Graph the Secant Function 6. ( )ππ 2sec += xy Asn 5 p531#6, 8, 10, 12, 18, 20, 22, 24, 30, 32, 34, 36, 38, 40

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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: xy 1sin −= or xy arcsin=

The inverse sine function is restricted to the interval: 22ππ

≤≤− x . 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. ⎟⎟⎠

⎞⎜⎜⎝

⎛−

23sin 1 2. ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

22sin 1

The inverse cosine function has two notations are commonly used to denote the inverse sine function: xy 1cos−= or xy arccos= The inverse sine function is restricted to the interval: π≤≤ x0 . 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. Practice: In 1-2, Evaluate the given function.

1. ⎟⎟⎠

⎞⎜⎜⎝

⎛−

23cos 1 2. ⎟

⎠⎞

⎜⎝⎛−−

21cos 1

Math Analysis Notes prepared by Mrs. Atkinson 18

The inverse tangent function has two notations are commonly used to denote the inverse sine function: xy 1tan −= or xy arctan=

The inverse sine function is restricted to the interval: 22ππ

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

1. ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

33tan 1 2. ( )1tan 1 −−

Practice: In 1-6, Evaluate the given function. 1. ⎟

⎠⎞

⎜⎝⎛ −

54sincos 1 2.

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

41sinsec 1

3.

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −−

31costan 1 4. ⎟

⎠⎞

⎜⎝⎛ −

32tancos 1

Math Analysis Notes prepared by Mrs. Atkinson 19

5. ( )x1costan − 6. ⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

4sinsec

2

1

x

x

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. )32.0(sin 1 −− 2. 83cos 1−

3. ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−

75cos 1 4. )25.0(sec 1 −−

Asn 6: p547#2-72(even)

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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

C a

c

b Given B = 23.50 and c = 10

Asn 7: p557#2-12(even) P558#30-36(even) Chapter 4 Review Ws