Precalculus – HollyerChapter 4 – Lecture AAnnouncementsQuiz 6 AND Quiz 7 BOTH end on the same day!Homeworks:1415This material is on Test 3!Re: Test 2 Be sure to finish Quiz 5 and the PT BEFORE you take the test!1Chapter 4 – Section 1 Right triangles and trigonometric ratiosGiven a right triangle, The Pythagorean Theorem holds.Apply it to the triangle on the right:Quiz 6, Question 1TPT can be applied to situations when the unknownis something other than the hypotenuse. Solve for x:Trigonometric Ratios - know by heart!Given a right triangle, the following trigonometric ratios are defined:“SOACAHTOA”2A C = 4.00 c mBA = 3.00 cmmBAC = 90.00CBAmBAC = 90.00CBAopp hypsin cschyp oppadj hypcos sechyp adjopp adjtan cotadj oppqq= q=qqq= q=qq qq= q=q qx cmmBAC = 90.00CBA9cmNote: the alternate form of the tangent equation: sinx/cosx!And a discussion of the other angles:  & Note:  COMPLEMENTS not supplementsThe angles and are ACUTE angles of the right triangle. A is the right angle. There can only be one right angle or obtuse angle in a triangle.What is a formula for the measure of angle  (n.b. It’s not independent of !)Quiz 6, Question 3 and Question 93mBAC = 90.00CBAFind the missing side length. Find the tangent for angle A and the secant for angle A. What is tan A? What is csc A?4Popper 06, Question 15Hint:Quiz 6, Question 10 and HWSuppose  is an acute angle of a right triangle and 3 5tan7b =, find the cosine for .6opp hypsin cschyp oppadj hypcos sechyp adjopp adjtan cotadj oppqq= q=qqq= q=qq qq= q=q qPopper 06, Question 2For today, leave your answer as an improper fraction! Note that the homework and quizzes report only PROPER fractions, no radicals in the denominator on those!7Popper 06, Question 38Two Special Triangles:Note that 30 - 60 - 90 go together in a right triangle. Isosceles right triangles are the second special triangles.Mnemonic: 30 - 60 - 90 :: small, medium, big :: x , 3x, 2x(note: 3 1.7�)sin 30 sin 60cos 30 cos 60tan 30° tan 60°Quiz 6, several problems. If you have an isosceles triangle with side length 6 cm, how can a 30-60-90 triangle help you with the length of the altitude?What is the height of the triangle?9x32xxmA CB = 90.00mCAB = 30.00CABA606cmand 45 - 45 - 90 go together in an isosceles right triangle.Note the hypotenuse is the longest side in any right triangle2 1.4�sin 45 cos 45tan 45Quiz 6, several problemsGiven an isosceles right triangle with a hypotenuse of 5cm, what is the leg length?x5 cm10x2xxmBA C = 45.00mCBA = 45.00ABCPopper 06, Question 411MORE VOCABULARY:The angles 30, 45, and 60 are all called “reference angles”. 0, 90, 180, 270, and 360 are all called “quadrantal angles”.Angles ON the axes, not IN a particular quadrantYou have to know ALL of the special angle material by heart. BUT, happily, I have a nice little mnemonic device for you below.Ms. Leigh’s Famous ChartCount off left to right starting with 0.Count back right to left starting with 0.Square root and divide by 2.angle in deg 0 30 45 60 90angle in rad[for later]sinecosinetangent12Here’s another reference angle chart for you to work with:angle in deg 0 30 45 60 90angle in radsinecosinetangentCount off left to right starting with 0.Count back right to left starting with 0.Square root and divide by 2.13Reciprocal functions:Quiz 6, Question 10Suppose B is an acute angle of a right triangle and cot B = 3/2.Find sec B.14opp hyp 1sin csc cschyp opp sinadj hyp 1cos sec sechyp adj cosopp adj 1tan cot cotadj opp tanqq= q= q=q qqq= q= q=q qq qq= q= q=q q qChapter 4, Section 2Moving on to radian measure and some new formulas:Recall: The Unit Circle version of Trig. If not, see me or a tutor!The trigonometric convention: positive and negative rotations on the unit circle(x, y) = (cosine, sine)15positive rotationterminal sideinitial sideWe will use radian measure. We make use of the fact that 180 is  rads.We then can create two “conversion factors”180180 1 1180p=p = =pRadians are actually unit-free so I won’t be writing “rad” very often.Find 0, 90, 180, 270, and 360 on the unit circle. Mark them in radian measure.Now find 90 and 270 in radian measure.Find 3 angles Co-terminal with 2pQuiz 6, Question 12 and 13Find 315 in radian measure.16Find 9p in degrees.Popper 06, Question 517135Quiz 6, Question 14 and HWArc Length: s rq=for theta in radian measure or 2360s rpq=� for theta in degreesPractice:given a central angle 135 and an arc length of 35 cm what is r?18Area of a Sector:212A r q=theta in radian measure or2360A rpq=�theta in degreesQuiz 6, Question 18 and HW on 4.2Practice: Given an area of 274p and an angle measure of 34p, what is the radius of the sector?19Angular speed and Linear speed – units analysisQuiz 6, Questions 19 and 20 and end of 4.2 HWA bicycle has wheels with an 8” radius. Each wheel turns at 2 revolutions per second.Find the angular speed in terms of radians per second.A truck has wheels with a 16” radius. The wheels are turning with a speed of 4 revolutions per second. How fast is the truck moving in units of inches per second?A bicycle has wheels with an 11” radius. Each wheel turns at 5 revolutions per second.Find the angular speed in terms of radians per second.A truck has wheels with a 14” radius. The wheels are turning with a speed of 7 revolutions per second. How fast is the truck moving in units of inches per second?20Popper 06, Question 621Popper 06, Question 72360A rpq=�22Popper 06, Question 823Popper 06, Question 9hint:24opp hypsin cschyp oppadj hypcos sechyp adjopp adjtan cotadj oppqq= q=qqq= q=qq qq= q=q qPopper 06, Question