Pythagorean Theorem: Trigonometry Packet #1 S O H C A H T O A. Examples Evaluate the six trig functions of the angle θ. 1.) 2.)

Transcription

1 Trigonometry Packet #1 opposite side hypotenuse Name: Objectives: Students will be able to solve triangles using trig ratios and find trig ratios of a given angle. S O H C A H T O A adjacent side θ Right Triangle Definitions of Trig Functions b c sinθ = cscθ = a Pythagorean Theorem: cosθ = tanθ = secθ = cotθ = Mar 26 1:51 PM Examples Evaluate the six trig functions of the angle θ. 1.) 5 θ sinθ = cscθ = 13 cosθ = secθ = tanθ = cotθ = 2.) sinθ = cscθ = 5 θ 5 2 cosθ = secθ = tanθ = cotθ = Mar 26 6:39 PM 1

2 Example: Let θ be an acute angle of a right triangle. Find the values of the other five trig functions of θ. tanθ = 7 sinθ = cscθ = 3 cosθ = secθ = cotθ = Example: Find x and y. x 4 30 o y Mar 26 6:41 PM Example: Solve ΔABC. Note: This means to find all of the missing angles measures and side lengths. B c a A 28 o 15 C Example: A tree casts a shadow as shown. What is the height of the tree? 31 o 25 ft Mar 26 7:02 PM 2

3 Objectives: Students will be able to work with angles in standard position, convert between radians and degrees and use the unit circle to solve problems. standard position: Examples: Draw an angle with the given measure in standard position. 1.) 240 o 2.) 500 o 3.) -50 o Apr 7 9:55 AM coterminal angles: Examples: Find one positive angle and one negative angle that are coterminal with the given angles. 1.) 45 o 2.) -380 o Angles can also be measured in. There are radians in a full circle. radians = 360 o, so radians = 180 o. -To convert degrees to radians, multiply by π. -To convert radians to degrees, multiply by 180. Apr 7 10:18 AM 3

4 Examples: 1.) Convert 125 o to radians. 2.) Convert -π to degrees. Degree measure Radian measure 0 o 30 o π/4 60 o π/2 2π/3 135 o 150 o 180 o 7π/6 5π/4 240 o 270 o 5π/3 315 o 11π/6 360 o Apr 7 10:30 AM Fill in the ratios using O = opposite, A = adjacent and H = hypotenuse. General Definitions of Trig Functions Let θ be an angle in standard position, and let (x,y) be the point where the terminal side of θ intersects the circle x 2 + y 2 = r 2. The six trig functions of θ are as follows: (x,y) r θ Apr 7 10:44 AM 4

5 Example: Let (-4,3) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ. The Unit Circle : the circle x 2 + y 2 = 1, which has center (0,0) and radius 1. (x,y) 1 θ Apr 7 10:53 AM Example Use the unit circle to evaluate the six trig functions of θ=270 o. Reference Angles Acute angles formed by the terminal side of θ and the x-axis. Recall: 30 o = 45 o = 60 o = 1 60 o 45 2 o o 45 o 3 1 Apr 7 11:02 AM 5

6 Examples: Evaluate the six trig functions of θ. Simplify and rationalize. 1.) θ = π/3 2.) θ = 7π/6 Apr 7 11:12 AM 3.) θ=7π/4 4.) θ=2π/3 Apr 7 11:15 AM 6

7 Objectives: Students will be able use inverse trig functions to solve for angles. So far, we've learned how to evaluate trig functions of a given angle. Now, we'll study how to reverse the problem - find an angle that corresponds to a given value of a trig function. Example sinθ = 1 Note: There are many θs that could satisfy the above equation. For this reason, we must make some restrictions. Inverse Trig Functions: -Sine Inverse: -90 o θ 90 o -Tangent Inverse: -90 o θ 90 o Cosine Inverse: 0 o θ 180 o Apr 7 11:24 AM Examples Evaluate the expression in both radians and degrees. 1.) cos ) sin Apr 7 11:40 AM 7

8 Trig Packet Notes #1.notebook Examples Find the measure of angle θ. 1.) θ ) A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp? Apr 7 11:43 AM Some More Application Problems 1.) The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles has an angle of elevation of 30o. The length of the escalator is 152 feet. What is the height of the escalator? 2.) A fire truck has a 100 ft. ladder whose base is 10 feet above the ground. A firefighter extends a ladder toward a burning building to reach a window 90 ft. above the ground. Draw a diagram. At what angle should the firefighter set the ladder? Apr 7 11:55 AM 8

9 Find the length of the arc. Feb 25 8:44 AM Feb 25 8:55 AM 9

10 Feb 25 8:56 AM Feb 25 8:56 AM 10

11 Trig Homework #1 Name: 1.) Find all 6 trig functions for 30 o, 45 o and 60 o and fill in the table below. Make sure to rationalize all values o 45 2 o o 45 o θ sinθ cosθ tanθ cscθ secθ cotθ 1 30 o 45 o 60 o Mar 26 7:19 PM 2.) Evaluate the six trig functions of θ. sinθ = cscθ = 15 θ 17 cosθ = secθ = tanθ = cotθ = 3.) Let θ be an acute angle of a right triangle. Find the values of the other 5 trig functions of θ. cotθ = 6 11 sinθ = cscθ = cosθ = secθ = tanθ = cotθ = Mar 26 7:38 PM 11

12 4.) Solve ΔABC. A b 35 o 16 C a B B = b = a = Mar 26 7:43 PM 5.) Find the length, x, of the prop holding open the piano. 25 o x 150 cm 6.) A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48 o. Estimate the parasailer's height above the boat. 300 ft 48 o Mar 26 7:46 PM 12

13 7.) On an analog clock, the minute hand has moved 128 o from the hour. What number will it pass next? 8.) It is 2:46 p.m. What is the measure of the angle that the minute hand swept through since 2:00 p.m? Feb 25 8:41 AM Draw an angle with the given measure in standard position. 9.) 110 o 10.) 450 o 11.) -3π/2 Find one positive angle and one negative angle that are coterminal with the given angles. 12.) -87 o 13.) 120 o Apr 7 12:44 PM 13

14 14.) Let (-3,-4) be a point on the terminal side of an angle θ in standard position. Evaluate the six trig functions of θ. 15.) Let (-6,9) be a point on the terminal side of an angle θ. Find all the trig ratios. Simplify and rationalize all values. Apr 7 12:50 PM Evaluate the six trig functions of θ. Simplify and rationalize. 16.) θ = π 17.) θ = 4π/3 Apr 7 12:53 PM 14

15 Evaluate the six trig functions of θ. Simplify and rationalize. 18.) θ = -7π/6 19.) θ = -π/4 Apr 7 12:53 PM Evaluate the expressions in both radians and degrees. 20.) cos -1 (1/2) 21.) tan -1 (-1) 22.) sin -1 ( 2/2) 23.) A crane has a 200 ft. arm with a lower end that is 5 ft. off the ground. The arm has to reach to the top of the building that is 160 ft. high. At what angle θ should the arm be set? Apr 7 12:56 PM 15

16 24.) A geostationary satellite is positioned 35,800 km above the Earth's surface. It takes 24 hours to complete one orbit. The radius of the Earth is about 6,400 km. What distance does the satellite travel in 1 hour? 3 hours? After how many hours has the satellite travelled 200,000 km? Feb 25 8:50 AM 25.) Suppose a windshield wiper arm has a length of 22 inches and rotates through an angle of 110 o. What distance does the tip of the wiper travel as it moves once across the windshield? 26.) A CD with a diameter of 12 cm spins in a CD player. Calculate how much farther a point on the outside edge of the CD travels in one revolution than a point 1 cm closer to the center of the CD. Feb 25 8:52 AM 16