Gettin’ Triggy wit it H4Zo

6. 1 A n g l e s a n d T h e i r M e a s u r e

An angle is formed by joining the endpoints of two half-lines called rays. The side you measure from is called the initial side. Initial Side The side you measure to is called the terminal side. Terminal Side This is a counterclockwise rotation. This is a clockwise rotation. Angles measured counterclockwise are given a positive sign and angles measured clockwise are given a negative sign. Positive Angle Negative Angle

It’s Greek To Me! It is customary to use small letters in the Greek alphabet to symbolize angle measurement.     alpha betagamma theta

We can use a coordinate system with angles by putting the initial side along the positive x-axis with the vertex at the origin.  positive Initial Side T e r m i n a l S i d e  negative We say the angle lies in whatever quadrant the terminal side lies in. Quadrant I angle Quadrant II angle Quadrant IV angle If the terminal side is along an axis it is called a quadrantal angle.

We will be using two different units of measure when talking about angles: Degrees and Radians Let’s talk about degrees first. You are probably already somewhat familiar with degrees. If we start with the initial side and go all of the way around in a counterclockwise direction we have 360 degrees You are probably already familiar with a right angle that measures 1/4 of the way around or 90°  = 90° If we went 1/4 of the way in a clockwise direction the angle would measure -90°  = - 90°  = 360°

 = 45° What is the measure of this angle? You could measure in the positive direction  = - 360° + 45° You could measure in the positive direction and go around another rotation which would be another 360 °  = 360° + 45° = 405° You could measure in the negative direction There are many ways to express the given angle. Whichever way you express it, it is still a Quadrant I angle since the terminal side is in Quadrant I.  = - 315°

If the angle is not exactly to the next degree it can be expressed as a decimal (most common in math) or in degrees, minutes and seconds (common in surveying and some navigation). 1 degree = 60 minutes1 minute = 60 seconds  = 25°48‘29" degrees minutes seconds To convert to decimal form use conversion fractions. These are fractions where the numerator = denominator but two different units. Put unit on top you want to convert to and put unit on bottom you want to get rid of. Let's convert the seconds to minutes 30"= 0.5'

1 degree = 60 minutes1 minute = 60 seconds  = 25°48'30" Now let's use another conversion fraction to get rid of minutes. 48.5'=.808° = 25°48.5'= °

initial side terminal side radius of circle is r r r arc length is also r r This angle measures 1 radian Given a circle of radius r with the vertex of an angle as the center of the circle, if the arc length formed by intercepting the circle with the sides of the angle is the same length as the radius r, the angle measures one radian. Another way to measure angles is using what is called radians.

Arc length s of a circle is found with the following formula: arc lengthradiusmeasure of angle IMPORTANT: ANGLE MEASURE MUST BE IN RADIANS TO USE FORMULA! Find the arc length if we have a circle with a radius of 3 meters and central angle of 0.52 radian. 3  = 0.52 arc length to find is in black s = r  = 1.56 m What if we have the measure of the angle in degrees? We can't use the formula until we convert to radians, but how? s = r 

We need a conversion from degrees to radians. We could use a conversion fraction if we knew how many degrees equaled how many radians. Let's start with the arc length formula s = r  If we look at one revolution around the circle, the arc length would be the circumference. Recall that circumference of a circle is 2  r 2  r = r  cancel the r's This tells us that the radian measure all the way around is 2 . All the way around in degrees is 360°. 2  =  2  radians = 360°

Convert 30° to radians using a conversion fraction. 30° The fraction can be reduced by 2. This would be a simpler conversion fraction. 180°  radians = 180° Can leave with  or use  button on your calculator for decimal. Convert  /3 radians to degrees using a conversion fraction. = radians  0.52 = 60°

Area of a Sector of a Circle The formula for the area of a sector of a circle (shown in red here) is derived in your textbook. It is:  r Again  must be in RADIANS so if it is in degrees you must convert to radians to use the formula. Find the area of the sector if the radius is 3 feet and  = 50° = radians

A Sense of Angle Sizes See if you can guess the size of these angles first in degrees and then in radians. You will be working so much with these angles, you should know them in both degrees and radians.

Angles, Arc length, Conversions Angle measured in standard position. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative. Coterminal Angles: Angles that have the same terminal side. 60°, 420°, and –300° are all coterminal. Degrees to radians: Multiply angle by radians Radians to degrees: Multiply angle by Arc length = central angle x radius, or Note: The central angle must be in radian measure. Note: 1 revolution = 360° = 2π radians.

6.2 TRIGONOMETRY

Remember SOHCAHTOA Sine is Opposite divided by Hypotenuse Cosine is Adjacent divided by Hypotenuse Tangent is Opposite divided by Adjacent SOHCAHTOA!!!!!!

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Right Triangle Trig Definitions sin(A) = sine of A = opposite / hypotenuse = a/c cos(A) = cosine of A = adjacent / hypotenuse = b/c tan(A) = tangent of A = opposite / adjacent = a/b csc(A) = cosecant of A = hypotenuse / opposite = c/a sec(A) = secant of A = hypotenuse / adjacent = c/b cot(A) = cotangent of A = adjacent / opposite = b/a A a b c B C

Special Right Triangles 30° 45° 60°45°

6.2 Assignment (day 1) pp (1-23 odd, 27, 29) 21, 23

Basic Trigonometric Identities Quotient identities: Reciprocal Identities: Pythagorean Identities: Even/Odd identities: Even functionsOdd functions

ASTC All Students Take Calculus. Quad II Quad I Quad III Quad IV cos(A)>0 sin(A)>0 tan(A)>0 sec(A)>0 csc(A)>0 cot(A)>0 cos(A)<0 sin(A)>0 tan(A)<0 sec(A)<0 csc(A)>0 cot(A)<0 cos(A)<0 sin(A)<0 tan(A)>0 sec(A)<0 csc(A)<0 cot(A)>0 cos(A)>0 sin(A)<0 tan(A)<0 sec(A)>0 csc(A)<0 cot(A)<0

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6.3 Trigonometric Functions of Real Numbers

Exact Values Using Points on the Circle Example. Let t be a real number and P = the point on the unit circle that corresponds to t. Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = 0 = 0 ± Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = = 90 ± Answer:

Exact Values for Quadrantal Angles Example. Find the values of the trigonometric functions of µ Problem: µ = ¼ = 180 ± Answer:

Circles of Radius r Example. Problem: Find the exact values of each of the trigonometric functions of an angle θ if ({12, 5}) is a point on its terminal side. Answer:

Even-Odd Properties Theorem. [Even-Odd Properties] sin( θ ) = sin( θ ) cos( θ ) = cos( θ ) tan( θ ) = tan( θ ) csc( θ ) = csc( θ ) sec( θ ) = sec( θ ) cot( θ ) = cot( θ ) Cosine and secant are even functions The other functions are odd functions

The Sine Function The sine of a real number t is the y–coordinate (height) of the point P in the following diagram, where |t| is the length of the arc. x y P sin t 1 unit 1 1 –1 |t||t|

Properties of Sine and Cosine Functions 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.

The Sine Function Highlight those sections of the circle where sin(t) >0 sin(t) > 0

The Sine Function

Sine Function Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x

The Cosine Function The cosine of a real number t is the x– coordinate (length) of the point P in the following diagram, where |t| is the length of the arc. x y P cos t 1 unit 1 1 –1 |t||t|

The Cosine Function cos(t) > 0 Highlight those sections where cos(t) > 0

The Cosine Function

Cosine Function Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x

Graphing the Tangent Function Periodicity: Only need to graph on interval [0, ¼ ] Plot points and graph

Graphing the Cotangent Function Periodicity: Only need to graph on interval [0, ¼ ]

Graphing the Cosecant and Secant Functions Use reciprocal identities Graph of y = csc x

Graphing the Cosecant and Secant Functions Use reciprocal identities Graph of y = sec x

6.4 Values of the Trigonometric Functions

6.4 Reference Angles A reference angle for an angle , written , is the positive acute angle made by the terminal side of angle  and the x-axis. ExampleFind the reference angle for each angle. (a)218 º (b) Solution (a)  = 218 º – 180 º = 38 º (b)

6.4Special Angles as Reference Angles ExampleFind the values of the trigonometric functions for 210 º. SolutionThe reference angle for 210 º is 210 º – 180 º = 30 º. Choose point P on the terminal side so that the distance from the origin to P is 2. A 30 º - 60 º right triangle is formed.

6.4Finding Trigonometric Function Values Using Reference Angles ExampleFind the exact value of each expression. (a)cos(–240 º )(b) tan 675 º Solution (a)–240 º is coterminal with 120 º. The reference angle is 180 º – 120 º = 60 º. Since –240 º lies in quadrant II, the cos(–240 º ) is negative. Similarly, tan 675 º = tan 315 º = –tan 45 º = –1.

Reference Angles Quad I Quad II Quad III Quad IV θ’ = θθ’ = 180° – θ θ’ = θ – 180°θ’ = 360° – θ θ’ = π – θ θ’ = 2π – θ θ’ = θ – π

6.4Finding Trigonometric Function Values with a Calculator ExampleApproximate the value of each expression. (a)cos 49 º 12(b) csc º SolutionSet the calculator in degree mode.

6.4Finding Angle Measure ExampleUsing Inverse Trigonometric Functions to Find Angles (a)Use a calculator to find an angle  in degrees that satisfies sin   (b)Use a calculator to find an angle  in radians that satisfies tan  .25. Solution (a)With the calculator in degree mode, we find that an angle having a sine value of is 75.4º. Write this as sin  75.4º. (b)With the calculator in radian mode, we find tan 

6.4Finding Angle Measure ExampleFind all values of , if  is in the interval [0º, 360º) and SolutionSince cosine is negative,  must lie in either quadrant II or III. Since So the reference angle  = 45º. The quadrant II angle  = 180º – 45º = 135º, and the quadrant III angle  = 180º + 45º = 225º.

6.4 Review Answers

6.4 Review

6.5 Graphs of Trigonometric Functions

6.5 Trigonometric Graphs Objective: We will re-define the basic trig. functions and their graphs and go more in depth with vertical stretches, horizontal stretches, periods, amplitudes, and shifts up and down.

Sine graphs y = sin(x) y = sin(3x) y = 3sin(x) y = sin(x – 3) y = sin(x) + 3 y = 3sin(3x-9)+3 y = sin(x) y = sin(x/3)

Graphs of cosine y = cos(x) y = cos(3x) y = cos(x – 3) y = 3cos(x) y = cos(x) + 3 y = 3cos(3x – 9) + 3 y = cos(x) y = cos(x/3)

6.5 (Sine & Cosine) Review: y = sin xy = cos x a y = asin(bx+c) +dy = acos(bx+c) +d *y = 3 sin x *y = 2 cos x *y = ½ sin x *y = -2cos x 1 Π2Π2Π Π2Π2Π a varies height/ vertical stretch of graph amplitude = |a| a varies height/ vertical stretch of graph amplitude = |a|

Amplitude The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2sin x y = sin x

y x Example: y = 3 cos x Example: Sketch the graph of y = 3 cos x on the interval [– , 4  ]. Partition the interval [0, 2  ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)

y x y = cos (–x) Graph y = f(-x) Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). Use the identity sin (–x) = – sin x The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. Example 2: Sketch the graph of y = cos (–x). Use the identity cos (–x) = cos x The graph of y = cos (–x) is identical to the graph of y = cos x. y x y = sin x y = sin (–x) y = cos (–x)

y x Period of a Function period: 2 period: The period of a function is the x interval needed for the function to complete one cycle. For b  0, the period of y = a sin bx is. For b  0, the period of y = a cos bx is also. If 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4

y=asin(bx+c) +d y=acos(bx+c) +d b varies width/horizontal stretch Period = 2 Π b

y = 3 sin 2x amp: |a| = per: 2 Π = b ** this means there is 1 sine wave on the interval [0, Π] ** Sketch the graph.

Y = 2sin x amp: |a| = per: 2 Π = b Graph:

y x 0 20 –2 0y = –2 sin 3x 0 x Example: y = 2 sin(-3x) Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 amplitude: |a| = |–2| = 2 Calculate the five key points. (0, 0) (, 0) (, 2) (, -2) (, 0) Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: =

d shifts up & down y= 2sinx + 3 y = cosx - 2

y = asin(bx+c); phase shift = -c/b; 0≤bx+c≤2Π y = 3 sin(2x+Π) amp: |a| = 2 per: 2 Π = b p.s. = 0 ≤ 2x+Π ≤ 2Π 2

Express in the form: y = asin(bx+c) for a,b,c,> 0 a = b = p.s. =

Example Determine the amplitude, period, and phase shift of y = 2sin(3x-  ) Solution: Amplitude = |A| = 2 period = 2  /B = 2  /3 phase shift = -C/B =  /3

Example cont. y = 2sin(3x-  )

Amplitude Period: 2π/b Phase Shift: -c/b Vertical Shift

6.6 More Trig. Graphs Objective: Students will look at more trig. graphs, including the tangent and reciprocal functions, and the variations on these reciprocal functions and their graphs.

6.6 Variations on reciprocal functions:  Amplitude has no meaning (tan/cot)  “a” still varies vertical stretch  “b” still affects period tanθ, cot θ = csc θ, sec θ =  “c” still affects phase shift  “d” still varies vertical shifts

6.6 y = tan xy = a tan(bx+c) +d amplitude: period: phase shift: Successive vertical asymptotes: <bx+c< -2Π - ΠΠ 2 Π

6.6 y = tan (x + ) period: p.s.

6.6 y = cot x y = csc x y = sec x

y = csc (2x + Π)Recall: csc θ = = amp: per: p.s.

6.7 Applied Problems Objective: Students will use trig. functions in applying real-world application problems.

6.7 Ex: (#25 from assignment) SOH-CAH-TOA sin 60 = x = 433 ft + 4 = ft. x ft. 500 ft. 60°

Ch. 6 Test Review