1Precalculus – Hollyer Chapter 4 – Lecture A Announcements Quiz 6 AND Quiz 7 BOTH end on 10/9/11 Homeworks: 14 9/29 15 9/30 This material is on Test 3! Re: Test 2 Be sure to finish Quiz 5 and the PT BEFORE you take the test!2 Chapter 4 – Section 1 Right triangles and trigonometric ratios Given a right triangle, The Pythagorean Theorem holds. Apply it to the triangle on the right: Quiz 6, Question 1 TPT can be applied to situations when the unknown is something other than the hypotenuse. Solve for x: Trigonometric Ratios - know by heart! Given a right triangle, the following trigonometric ratios are defined: “SOACAHTOA” AC = 4.00 cmBA = 3.00 cmmBA C = 90.00CBAmBAC = 90.00CBAopp hypsin cschyp oppadj hypcos sechyp adjopp adjtan cotadj opp   x 32cm mBAC = 90.00 CBA9cm3 Note: the alternate form of the tangent equation: sinx/cosx! And a discussion of the other angles:  &  Note:  COMPLEMENTS not supplements The 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 9 A16cm11cm Find the missing side length. Find the tangent for angle A and the secant for angle A. What is tan A? What is csc A? mBAC = 90.00CBA4 Popper 06, Question 15Hint: Quiz 6, Question 10 and HW Suppose  is an acute angle of a right triangle and 35tan7 , find the cosine for . opp hypsin cschyp oppadj hypcos sechyp adjopp adjtan cotadj opp   6 Popper 06, Question 2 For 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!7 Popper 06, Question 3 8 Two 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: 31.7 ) 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? x32xxmACB = 90.00mCA B = 30 .00CABA606cm9 and 45 - 45 - 90 go together in an isosceles right triangle. Note the hypotenuse is the longest side in any right triangle 21.4 sin 45 cos 45 tan 45 Quiz 6, several problems Given an isosceles right triangle with a hypotenuse of 5cm, what is the leg length? x5 cmx2xxmBA C = 45.00mCBA = 45.00ABC10Popper 06, Question 411 MORE 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 quadrant You 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 Chart Count 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] sine cosine tangent12Here’s another reference angle chart for you to work with: angle in deg 0 30 45 60 90 angle in rad sine cosine tangent Count off left to right starting with 0. Count back right to left starting with 0. Square root and divide by 2.13 Reciprocal functions: Quiz 6, Question 10 Suppose B is an acute angle of a right triangle and cot B = 3/2. Find sec B. opp hyp 1sin csc cschyp opp sinadj hyp 1cos sec sechyp adj cosopp adj 1tan cot cotadj opp tan       14 Chapter 4, Section 2 Moving 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) positive rotationterminal sideinitial side15 We will use radian measure. We make use of the fact that 180 is  rads. We then can create two “conversion factors” 180180 1 1180  Radians 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 216 Quiz 6, Question 12 and 13 Find 315 in radian measure. Find 9 in degrees. Popper 06, Question 517135 Quiz 6, Question 14 and HW Arc Length: sr for theta in radian measure or 2360sr for theta in degrees Practice: given a central angle 135 and an arc length of 35 cm what is r?18 Area of a Sector: 212Ar theta in radian measure or 2360Ar theta in degrees Quiz 6, Question 18 and HW on 4.2 Practice: Given an area of 274 and an angle measure of 34, what is the radius of the sector?19Angular speed and Linear speed – units analysis Quiz 6, Questions 19 and 20 and end of 4.2 HW A 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 621 Popper 06, Question 7 2360Ar22 Popper