Trigonometric Functions (Chapters 6 & 7, 10.1, 10.2) E. Law of Sines/Cosines May 21-12:26 AM May 22-9:52 AM 1
degree measure May 22-9:52 AM Measuring in Degrees (360 degrees) is the angle obtained when rotating a Right Angle Jan 21-11:11 PM 2
θ, in radians. Jan 16-9:55 PM Jan 16-10:26 PM 3
"in radians" radians: Jan 16-10:27 PM Ex: Convert to degrees Jan 22-10:23 PM 4
Jan 22-10:25 PM May 22-9:42 PM 5
Jan 22-10:32 PM Jan 22-10:27 PM 6
Ex 2: Find the angle of the sector when: a). radius is 4.3 m and arc length is 2.95m b). radius is 10 cm and area is 30 cm 2 Jan 7-10:04 AM Angular & Linear Speed rpm (revolutions per minute) Angular speed If the wheel rotates an angle on the rotating wheel travels a distance in May 23-9:26 PM 7
Jan 22-10:58 PM Recall: Ex: Find the missing side. Remark: Since the right triangle has a 90 angles must be. Jan 3-2:55 PM 8
Hypotenuse Opposite Jan 3-2:55 PM Six Trigonometric Functions Jan 3-7:20 PM 9
Hypotenuse Opposite Jan 3-3:16 PM SOH-CAH-TOA Jan 3-3:16 PM 10
Jan 3-7:43 PM 1. 30 0-60 0-90 0 Right Triangle Proof: Start with any equilateral triangle. Why? 1 1 h h 1 h Jan 7-10:59 AM 11
1. 30 0-60 0-90 0 Right Triangle Consequences: Jan 7-10:59 AM csc30 Jan 3-9:50 PM 12
Jan 3-9:57 PM 2. 45 0-45 0-90 0 Right Triangle Proof: Start with any isosceles right triangle. Why? x x 1 x h x Jan 7-10:59 AM 13
2. 45 0-45 0-90 0 Right Triangle x h Consequences: x Jan 7-10:59 AM We found 45º 45º 45º 45º csc45 Jan 3-8:04 PM 14
/6 /4 /3 May 23-10:40 PM 2) cos53 5) sec53 Jan 3-10:09 PM 15
May 27-10:11 PM May 27-10:16 PM 16
Jan 3-10:15 PM Jan 8-11:34 AM 17
Jan 8-12:02 PM Ex: (Application) The base of the tree is 65 feet from point P & the angle of elevation to the top of the tree is 41º. Find the height of Jan 8-12:04 PM 18
base b is given by b θ. Jan 8-12:05 PM Jan 8-12:14 PM 19
Jan 8-12:19 PM C. Trigonometry of Angles: Negative Angle (Clockwise) May 25-12:05 AM 20
May 25-7:40 PM Remark: Just as we discussed clockwise would represent a Jan 8-2:34 PM 21
sinθ = sinθ = May 25-7:34 PM where it intersects unit circle. Jan 8-2:46 PM 22
The terminal pt ( 3/5, 4/5) is a point on the unit circle that corresponds to an angle Jan 8-6:04 PM θ (in standard position) meets the unit circle. Use info to find the Jan 22-10:59 PM 23
Jan 8-6:06 PM Jan 8-6:38 PM 24
sin π/6 π/4 π/3 π/2 sin Jan 8-6:54 PM Unit Circle in degrees and radians Jan 8-6:47 PM 25
Tip to memorize Jan 8-6:51 PM Tip to memorize Jan 8-6:51 PM 26
Tip to memorize Jan 8-6:51 PM Tip to memorize Jan 8-6:51 PM 27
Unit Circle in degrees and radians Jan 8-6:47 PM θ = π θ = -π/2 Jan 22-11:19 PM 28
Jan 8-6:49 PM θ. Jan 8-6:49 PM 29
Jan 8-8:10 PM Jan 8-8:17 PM 30
Jan 8-8:47 PM References to Jan 22-11:38 PM 31
Jan 23-10:57 PM 1) Reference of 2 cos(2 2) Reference of -5 cos(-5 Jan 23-10:39 PM 32
Jan 23-11:05 PM The Complete Unit Circle - Memorize this! Jan 16-12:33 PM 33
& Recall: other trig functions" (Need to memorize!) May 27-10:21 PM trigonometric functions. 0 0, find remaining Jan 8-8:49 PM 34
trigonometric functions. 0 0, find remaining May 27-10:51 PM trigonometric functions. 0 0, find remaining May 27-10:51 PM 35
Jan 8-8:59 PM Remark: Given Ex. Derive similar identities for the other four trig functions. May 27-11:03 PM 36
Trigonometric Proofs How to 'do' proofs: We use a combination of algebra and basic identities along with the following guidelines: Guidelines for Proving Trig Identities 1) Start with one side. - Start w/ most complicated side - You may work with both sides if stuck. 2) Use known Identities Jan 29-10:02 PM A(1 + cot Guidelines for Proving Trig Identities 1) Start with one side. - Start w/ most complicated side - You may work with both sides if stuck. 2) Use known Identities Jan 29-10:02 PM 37
Jan 8-9:05 PM Jan 8-9:02 PM 38
= 2 + 2cot Jan 29-10:03 PM Trigonometric Algebra Ex: Simplify Jan 29-9:05 PM 39
1 + cos Jan 29-9:59 PM cot csc Jan 29-10:00 PM 40
Recall: The Complete Unit Circle May 21-12:18 AM Skill: Can you think "backwards" from the unit circle? Ex. Find the angle that satisfies the trig value: May 21-12:18 AM 41
θ = 0.2345 2) cosθ = 0.8654 Aug 5-7:42 PM May 21-12:18 AM 42
1) (SAA) May 25-11:13 AM 2) (ASA) May 25-11:31 AM 43
May 25-9:54 PM 4) (SSA) a = 5 May 25-10:18 PM 44
3) (SSA) a = 7 May 25-10:18 PM May 25-11:34 AM 45
+ c - 2bccosA + c - 2accosB - 2abcosC May 25-10:20 PM 1) (SSS) May 25-10:59 PM 46
2) (SAS) May 25-11:03 PM EX 2: The end wall of a building has the shape illustrated, where the center of arc AB is at C. Find: a). to 4 significant digits b) to 4 significant digits c) the area of the wall Jan 7-10:12 AM 47