Objective: Manipulate trigonometric properties to verify, prove, and understand trigonmetric relationships.

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1 Objective: Manipulate trigonometric properties to verify, prove, and understand trigonmetric relationships. Apr 21 4:09 AM Warm-up: Determine the exact value of the following (without a calculator): sin π/3 cos -7π/6 tan 7π/4 sin 13π/6 Apr 21 4:10 AM 1

2 Notation: sin(x) sin(x) = (sin(x)) 2 = sin 2 x (cos(x)) 2 = cos 2 x (tan(x)) 2 = tan 2 x 1 1 = csc(x) csc(x) = csc 2 x sin(x) sin(x) Similarly... sec 2 x... cot 2 x Apr 21 4:14 AM However: sinx 2 = sin (x 2 )...so be careful...where is the 2? Determine the values sin 2 (π/6) sin (30º) 2 1/4 0 Apr 21 4:21 AM 2

3 We use identities for three basic tasks: 1. express one function in terms of another 2. simplify expressions 3. verify identities These three tasks build our skills so that we can solve trigonometric equations later on. Apr 21 4:27 AM We've seen & Used the Reciprocal Identities: We've used one of the Tangent-Cotangent Identities: Apr 21 4:27 AM 3

4 Pythagorean Identities: (cos, sin) Apr 21 4:33 AM For the other 2 Pythagorean Identities, manipulate the first one: instead of sin 2 θ make csc 2 θ Apr 21 4:35 AM 4

5 You will have a quiz on the identities... Apr 21 4:38 AM Let's manipulate some... express sin θ in terms of cos θ sin 2 θ + cos 2 θ = 1 Apr 21 4:39 AM 5

6 express tan θ in terms of sin θ 1 + cot 2 θ = csc 2 θ Apr 21 4:41 AM Homework: Pg 385 # odd Apr 21 4:43 AM 6

7 Objectives: Manipulate trig identities. Evaluate values using circular trig definitions. Solve application problems. Apr 22 4:23 PM Warm up: Write down the Fundamental Trig Identities Reciprocal: Cotangent-Tangent: Pythagorean: Apr 22 4:25 PM 7

8 at the end of class you will have a quiz on the Fundamental Identities... use them in class...keep them handy Homework Review: Pg 385 # odd Apr 22 4:26 PM Apr 23 7:32 AM 8

9 Apr 23 7:37 AM Verify the identity by transforming the left hand side into the right hand side. tan 2 θ cosθ = sin 2 θ secθ Apr 22 4:11 PM 9

10 Verify the identity by transforming the left hand side into the right hand side. secθ - cosθ = tanθ sinθ Apr 22 4:11 PM Verify the identity by transforming the left hand side into the right hand side. log csc θ = -log sin θ Apr 22 4:11 PM 10

11 express sin θ in terms of cot θ Apr 21 7:56 AM The point P(3, 4) is on the terminal ray of an angle in standard position. What are the values of the trig functions of this angle? Apr 22 4:19 PM 11

12 The point P( 15, 8) is on the terminal ray of an angle in standard position. What are the values of the trig functions of this angle? Apr 22 4:08 PM If sin Θ = 1/2, and the terminal side lies in Quadrant II, find cosθ, tan Θ, cscθ, cot Θ, and sec Θ. If sin Θ =, find possible values for x, y, & r. Then possible values for cosθ, tan Θ. Apr 22 4:19 PM 12

13 An angle θ is in standard position with its terminal ray in quadrant III on the line y = 3x. What are the values of the trig functions of the angle? Apr 22 4:09 PM If cos θ > 0 and sin θ < 0, in what quadrant does the terminal ray of θ lie? Apr 22 4:10 PM 13

14 If sin θ = 3/5 and tan θ < 0, find the values of the other trig functions. Apr 22 4:18 PM There are many other applications that require trig solutions. For example, surveyors use special instruments to find the measures of angles of elevation and angles of depression. An angle of elevationis the angle between a horizontal line and the line of sight from an observer to an object at a higher level. An angle of depressionis the angle between a horizontal line and the line of sight from the observer to an object at a lower level. Mar 7 8:36 AM 14

15 A tower 250 meters high casts a shadow 176 meters long. Find the angle of elevation of the sun to the nearest minute. Mar 7 8:36 AM You just crossed a bridge 112' across from one cliff to another. Your angle of depression to the river below is 39 o. How far away are you from the river? Mar 7 8:37 AM 15

16 Solve the right triangle ABC. Round angle measure to the nearest degree and side measures to the nearest tenth. A 5 c C 12 B Mar 7 8:30 AM You can use right triangle trig to solve problems involving geometric figures. The apothem of a regular polygon is the measure of a line segment from the center of the polygon to the midpoint of one of its sides. Example 2: A regular hexagon is inscribed in a circle with diameter 7.52 cm. Find the apothem. Round the answer to the nearest hundredth cm a Mar 7 8:34 AM 16

Example 1: The longest truck-mounted ladder used by the Dallas Fire Department is 108 feet long and consists of four hydraulic sections. Gerald Travis, aerial expert for the dept.

17 Example 1: The longest truck-mounted ladder used by the Dallas Fire Department is 108 feet long and consists of four hydraulic sections. Gerald Travis, aerial expert for the dept., indicates that the optimum operating angle of this ladder is 60. Outriggers, with an 18 ft. span between each, are used to stabilize the ladder truck and permit operating angles greater than 60, allowing the ladder truck to be closer to buildings in downtown Dallas. Assuming the ladder is mounted 8 ft off the ground, how far from an 84-ft burning building should the ladder be placed to achieve the optimum operating angle of 60? How far should the ladder be extended to reach the roof? Mar 7 8:37 AM In class identities practice: Pg 385 # 39 (hint sum of cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ), even Homework: Pg 385 #s 3, 4, 9 (rt triangle review) 11, 13, 15 (special rt triangles, or trig values) 23, 25, 26 (angle of elevation) 35, 37 (pyth. id) 41 (simplify) odd (verify identities) Next class review & quiz in addition to the homework, know: identities, exact values (unit circle values), right triangle trig, aka:sohcahtoa Apr 21 4:46 AM 17

18 Apr 23 8:35 AM Apr 25 7:26 AM 18

19 Apr 25 7:32 AM Apr 25 7:36 AM 19

DAY 1 - GEOMETRY FLASHBACK

DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ =