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Ch4.1A – Radian and Degree Measure r
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• Ch4.1A Radian and Degree Measure r
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• Ch4.1A Radian and Degree Measure s ~3.14 arcs = half circle r = 1 radian One radian the measure of the angle when the arc length = the radius 1 revolution (360) = 2 radians (~6.28 arc lengths) revolution ( ) = radians revolution ( ) = radians 1/3 revolution ( ) = radians 1/8 revolution ( ) = radians
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• Ch4.1A Radian and Degree Measure s r = 1 radian One radian the measure of the angle when the arc length = the radius 1 revolution (360) = 2 radians (~6.28 arc lengths) revolution (180) = radians revolution (90) = radians 1/3 revolution (60) = radians 1/8 revolution (45) = radians
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• Ex1) Find the acute angle equivalent to Ex2) Find the negative angle equivalent to Ex3) Find the positive angle equivalent to
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• Ex4) Find the complement and supplement angles to a) b)
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• Conversions:Conversion Factor:
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• Ex5) Convert: a) 135 b) -270 c) d) 2 rad (Quiz on conv in 2 days)Ch4.1A p318 5-19odd,45-55odd
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• Ch4.1A p318 5-19odd,45-55odd Quiz tomorrow! (On conversions)
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• Ch4.1B Arc Length (Quiz tomorrow!) r
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• Ch4.1B Arc Length r Length of a circular arc: s = r. ( must be in radians) Ex1) A circle has a radius of 4inches. What is the arc length intercepted by a central angle of 240.
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• Linear speed: distance traveled Angular speed: angle swept out timetime omega ( must be in radians) Ex3) The second hand of a clock is 10.2cm long. Find the speed of the second hand.
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• Ex3) A lawn roller is 30in in diameter and makes 1 revolution every 5/6 sec. a) Find the angular speed b) How fast does it move across the lawn?
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Do #35 and #39 in class) (Quiz tomorrow on conversions!)
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class) (Quiz today on conversions!)
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class) (Quiz today on conversions!)
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class) (Quiz today on conversions!)
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class) (Quiz today on conversions!)
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• Ch4.1B p319 35-41odd, 52,54,71-81odd,91,95 (Did #35 and #39 in class) (Quiz today on conversions!)
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• Ch4.1 QuizName___________ Convert radians to degrees: A B C D Convert degrees to radians: A B C D 4. 270 4. 904. 1804. 360 5. 60 5. 305. 1205. 150 6. 210 6. 2406. 3306. 300
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• Ch4.2 The Unit Circle x 2 + y 2 = 1 Ex1) 45 = _____ rad x = _____ y = _____
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• Ex2) 60 = _____ rad x = _____ y = _____ Ex3) 30 = _____ rad x = _____ y = _____Ex5) 90 = _____ rad Ex4) 0 = _____ radx = _____ x = _____ y = _____ y = _____
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• Ex6) x = _____ y = _____ x = _____ y = _____
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• Trig Functions (sine)(cosine)(tangent) sin t = ycos t = xtan t = Ex7) Eval 3 trigs for: a) b) c) d) Ch4.2A p328 1-39odd (only sin,cos,tan)
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• Ch4.2B Trig Functions sin t = y(cosecant) csc t = cos t = x(secant) sec t = tan t = (cotangent) cot t = Ex8) Eval 6 trigs for:
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• sin t = ycsc t = cos t = xsec t = tan t = cot t = Ex9) Eval: a) b)
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• x = cos t y = sin t Domains: (what you put in for t) Ranges: (what you get out for x or y)
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• Types of functions: 1. x = cos t is an even function (So is secant) 2. y = sin t is an odd function (So is tan, csc, and cot) Ex10) Use calc: a) sin 76.4 (must be in degree mode.) b) cot 1.5 (must be in radian mode.) Ch4.2B p328 57, 2-38 (eoe)
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• Ch4.2B p328 57, 2-38 (every other even)
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• Ch4.3 Right Triangle Trig Quiz tomorrow on this! SOH-CAH-TOA sin =cos = tan = hypotenuse adjacent opposite
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• Ch4.3 Right Triangle Trig SOH-CAH-TOA sin =cos = tan = csc = sec = cot = Ex1) Eval 6 trigs for: 5 4 3 hypotenuse adjacent opposite
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• Ch4.3 Right Triangle Trig SOH-CAH-TOA sin =cos = tan = csc = sec = cot = Ex2) Find the value of sin45, cos45, tan45 hypotenuse adjacent opposite 45
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• Ex3) Use the equilateral triangle to find the value of sin60, cos60, sin30, cos30
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• Sine, Cosine, and Tangent of Special Angles sin30 =cos30 = tan30 = sin45 = cos45 = tan45 = 1 sin60 = cos60 = tan60 = HW#8) Find exact values of 6 trigs for: 3 6 Ch4.3A p338 1-22(a,b) Quiz tomorrow would u like 2 c a sample?
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• Sine, Cosine, and Tangent of Special Angles sin30 =cos30 = tan30 = sin45 = cos45 = tan45 = 1 sin60 = cos60 = tan60 = HW#8) Find exact values of 6 trigs for: 3 6 Ch4.3A p338 1-22(a,b) Quiz tomorrow would u like 2 c a sample?
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• Trigonometry & Vector Components SOHCAHTOASOHCAHTOA in pp yp os dj yp an pp dj sin = cos = tan = opp hyp adj hyp opp adj
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• Ch4.3A p338 1-22(a,b) Quiz today!
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• Ch4.3A p338 1-22(a,b)
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• Ch4.2 QuizName___________ Find exact values: A B C D 1. sin 301. cos 301. sin 601. cos 60 2. tan 30 2. tan 602. sin 302. cos 30 3. cos 603. sin 603. tan 453. tan 30 4. tan 45 4. cos 45 4. cos 60 4. sin 60 5. sin 60 5. cos 60 5. tan 30 5. tan 45 6. cos 45 6. tan 45 6. cos 45 6. sin 45
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• Ch4.3B Trig Identities Reciprocals: sin =cos = tan = csc = sec = cot = Combos: Pythag:Quiz in 2 days on these identities.
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• Ex4) Let be acute angle, with sin = 0.6, find: a) cos b) tan Ex5) Let be acute, with tan = 3, find: a) cot b) sec
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• Ex6) Use a calc to eval: a) cos 28 b) sec 28 c) sec 540 Ex7) Find the value of in radians and degrees: a) sin =b) cos = b) csc = 2 Ex8) Use calc to find in: a) degrees for cos = 0.3746 b) radians for sin = 0.3746 Ch4.3B p339 23-31odd,37-40all(a only),47-55all(a only) Quiz in 2 days
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• Ch4.3B p339 23-31odd,37-40all(a only), 47-55all(a only)
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• Ch4.3C Trig Word Problems (Quiz on IDs tomorrow!) Ex7) A surveyor is standing 50ft from the base of a large tree. The surveyor measures the angle of elevation to the top of the tree as 71.5. How tall is the tree? h = ? =71.5 | --------50ft-----------|
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• Ex8) A person is 200yds straight away from a river. The person walks at an angle, going 400yds til he gets to the rivers edge. At what angle did he walk? 200yds 400yds
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• Ex9) A 12 meter flagpole casts a shadow 9 meters long. What is the angle of elevation to the sun? Ch4.3C p33924-32even,68-70all,81,82 Quiz tomorrow. Would u like to c an example?
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• Quiz example: =
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• Ch4.3C p33924-32even,68-70all,81,82
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• Ch4.3 Identities QuizName__________ Reciprocals A BC D 1. 2. Combinations 3. Pythag 4. 5.
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• Ch4.4A Trig Functions of Any Angle (x,y)
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• Ch4.4A Trig Functions of Any Angle (x,y) sin = cos = tan = r csc = sec = cot = Reminder:Ex1) Let -3,4 be a point on the QIIQIterminal side of an angle . sin = sin = find sin,cos,tan. cos = tan = QIIIQIV sin = cos = tan =
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• Ex2) Given tan =. and cos > 0. Find sin and sec . Ex3) Evaluate sin and tan at
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• Reference Angles - any given angle has an equivalent angle where 0 > > 90, or 0 > > /2. Ex4) Find ref angle: a) 300 b) 2.3 c) -135 Ex5) Evaluate: a) cosb) tan (-210) c) csc Quiz in 2 days (Ex?)Ch4.4A p349 2-42eoe (all 6 trigs)
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• Sample Quiz for Ch4.2/4.3 1. 2. 3. 4. 5. 6. 7. 8.
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• Lab4.1 Heights and Lengths (angles measured)
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• Ch4.4A p349 2-42eoe (all 6 trigs)
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• Lab4.1 Heights and Lengths (angles measured)
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• Ch4.4B More Trigs at Any Angle Ex6) Let be an angle in QII, such that sin = Find cos and tan using trig identities. Ex7) Use a calculator to evaluate: a) cot (410) b) sin (-7) c) Solve for :tan = 4.812, where 0 < < 2 Ch4.4B p350 43-81odd (a only) Quiz tomorrow!
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• Ch4.4B More Trigs at any Angle HW#43) Eval sin,cos,tan for: a) 225 #63) Find 2 values in degrees and radians a) sin = , where 0 < < 2 Ch4.4B p350 43-81odd (a only) Quiz tomorrow!
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• Ch4.4B p350 43-81odd (a only)
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• Ch4.5A Graphs of Sine and Cosine (Quiz first) Ex1) What are the values of sine and cosine at: sin cos
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• Ch4.5A Graphs of Sine and Cosine (Quiz first) Ex1) What are the values of sine and cosine at: sin cos Ex2) Graph on a # line: y = sin x Ex3) Graph on a # line: y = cos x
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• y = a. sin x y = a. cos x Amplitude stretches and shrinks graph vertically Ex4) Sketch y = 2. sin x 2. sinx
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• y = a. sin x y = a. cos x Amplitude stretches and shrinks graph vertically Ex5) Sketch y = . cos x . cos x
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• Period of Sine and Cosine (Normally its 2) y = a. sin (bx) y = a. cos (bx) b determines the period Ex6) Sketch y = sin(x)vs y = sin (x) vs y = sin(2x) 1 1 Ch4.5A p361 113odd, 43,45
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• Ch4.5B Translations y = a. sin(bx c) y = a. cos(bx c) a = amplitude c = horizontally shifts b determines the periodthe period start: bx c = 0 end: bx c = 2 Ex7) Sketch y = . sin(x )
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• Ex8) y = 3. cos(2x + 1) Use a calc to find its period
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• Ex9) Sketch y = 2. cos(2x ) + 1 Ch4.5B p361 1521odd,4753odd (lets do 17 and 47 in class)
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• Ch4.5B p361 1521odd,4753odd
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• Ch4.6A Graphs of Other Functions Ex1) Sketch the graph of y = tan x tan
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• Ch4.6A Graphs of Other Functions Ex1) Sketch the graph of y = tan x tan To find asymptotes for tangent:
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• Ex2) Sketch the graph of y = tanTo find asymptotes tan for tangent:
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• Ex3) Sketch the graph of y = 3tan2xTo find asymptotes tan for tangent:
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• Ex4) Sketch the graph of y = cot x cot
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• Ex5) Sketch the graph of y = 2cotTo find asymptotes cot for cotangent: Ch4.6A p372 9-12,21,22,24
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• Ch4.6B Graphs of Other Functions Ex6) Sketch the graph of y = csc x sin x csc x To find asymptotes for cosecant: anywhere sin x = 0
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• Ex7) Sketch the graph of y = sec x cos x sec x To find asymptotes for secant: anywhere cos x = 0
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• HW#13) Sketch the graph of y = -sec x
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• HW#19) Sketch the graph of y = csc Ch4.6B p372 13-20all,23,26
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• Ch4.7 Inverse Trig Functions Ex1) Graph y = sin x 1
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• Ch4.7 Inverse Trig Functions Ex1) Graph y = sin x 1 The inverse function of sin x is called sin -1 x or arcsin x - its domain is [-1,1], and its range is. Graph arcsin x
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• If sin = then sine is taking an angle and giving us the ratio of the side opposite to the hypotenuse. If sin -1 = then inverse sine is taking the ratio of the sides and giving us the angle Ex2) a) Find the exact value of arcsin(- ) b) Find the exact value of arcsin( ) c) Find the exact value of arcsin(2)
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• Ex3) Graph y = cos x 1 Graph y = arccos x Ex4) Find the exact value of arccos() Find the exact value of arccos ( )
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• Ex5) Graph y = tan x 1 Graph y = tan -1 x Ex6) Find the approx value of tan -1 (.7042) Ch4.7A p383 7-19odd (a,b),23,25
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• Ch4.7B Inverse Trig Functions cont Ex6) Graph y = sin 1 Graph y = sin -1
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• Inverse properties sin(arcsin x) = x arcsin(sin x) = x cos(arccos x) = x arccos(cos x) = x tan(arctan x) = x arctan(tan x) = x Ex7) Find the exact value of tan(arctan(-5)) Find the exact value of arcsin(sin )
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• Ch4.7B p383 8-20even (a only),27-41odd
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• Ch4.7C Inverse Trig Functions cont Ex7) Find the exact value of a. tan(arccos( ) b. cos(arcsin( )
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• Ex8) Write each as an algebraic expression: a. sin(arccos(3x) 0 < x < 1/3 b. cot(arccos(3x) 0 < x < 1/3
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• Ch4.7C p385 34-42even,43-51odd,71,75
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• 34.2 c b = 19.4 a Ch4.8A Applications Ex1) Solve the triangle: B C A
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• 72 c =110ft h = ? Ex2) The maximum angle for a ladder is 72. If a fire depts longest ladder is 110ft, what is the max rescue height?
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• 35 h = ? Ex3) At a point 200ft from the base of a building, the angle of elevation to the bottom of a smoke stack is 35. The angle of elevation to the top of the smoke stack is 53. Find the height of the smoke stack. 53
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• 3.9m 20m Ex4) Find the angle of depression to the bottom of a pool. Ch4.8A p394 1-11odd,18 1.3m
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• Ch4.8A p394 1-11odd,18
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• Ch4.8B Applications Ex5) A ship leaves a port with a heading of N54W traveling at 20mph. Ship 2 leaves port at the same tjme with a heading N36E traveling at 30mph. After 2 hours how far apart are they?
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• HW#35) An observer in a lighthouse 350ft above sea level observes 2 ships directly offshore. The angle of depression to the ships are 4 and 6.5. How far apart are the ships? Ch4.8B p395 17-37odd 44 6.5
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• Lab4.2 Finding Angles
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• Ch4.8B p395 17-37odd
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• Ch4.8C Simple Harmonic Motion (SHM) 10cm 0cm -10cm
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• 10cm 0cm -10cm d = a. sint or d = a. cost |a| = amplitude = period = frequency
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• Ex6) a) Write an equation for the SHM of a ball attached to a spring, that is pushed up 10cm, and oscillates with a period of 4sec. b) Find the frequency.
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• Ex7) Given a spring in SHM described by: d = 6cos[ t] find: a) Period b) Frequency c) Where is the ball located when t = 4? d) Find 2 times where d = 0.
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• Lab4.3 Simple Harmonic Motion Go over HW quickly HW: Finish lab questions + Ch4 Rev#1 p401 1-53eoe Ch4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107
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• Ch4.8C p398 20,49-56all,58 (lets do 52 and 56 in class #20 on next slide)
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• Ch4 Rev#1 p401 1-53eoe Ch4 Rev#2p402 61,65,69,73,75,77, 87,93,97,99,103,104,105,107
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• Ch4 Rev#1 p401 1-53eoe
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• Ch4 Rev#2p402 61,65,69,73,75,77,87,93,97,99,103,104,105,107
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• 6 5 4 3 2 1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2-1 1 2 3 4 5 6