Math 4 Review for Quarter 2 Cumulative Test

Transcription

1 Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry ( ) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric Rtios (SOH CAH TOA) opposite leg o sinθ = hypotenuse djcent leg o cosθ = hypotenuse opposite leg o tnθ = djcent leg Use the inverse trigonometric functions to find missing ngles. Prctice Problems 1. Find the missing sides or ngles in the right tringles below using SOH CAH TOA.. b. x θ c. d. 7 9 θ 28 x 23

2 e. 7 f. 12 θ 57 5 x 2. A lighthouse is 100 meters tll. A bot is 220 meters wy (horizontlly). Wht is the ngle of elevtion from the bot to the top of the lighthouse? 3. A 6- foot ldder lens ginst the side of house. If the ldder mkes n ngle of 60 with the ground, how fr up the wll does the top of the ldder rech? 4. Chllenge An observer on cliff 100 meters bove se level sights two ships due est. The ngles of elevtion from the ships re 47! nd 32!. Find the distnce between the ships x

3 II. Vectors ( ) Key Fcts Vectors represent direction nd we drw them s rrows. Any vector cn be broken down into its horizontl component (how fr it moves left/right) nd its verticl component (how fr it moves up/down). o Think SOH CAH TOA! o Components re counted s positive or negtive just like on grph. + + horizontl verticl We dd vectors together by connecting them strt to finish (or tip to til ). o When vectors re dded, their horizontl components re dded nd their verticl components re dded. We subtrct vectors by reversing the direction of the second vector nd then connecting them strt to finish (or tip to til ). o When vectors re subtrcted, their horizontl components re subtrcted nd their verticl components re subtrcted. We multiply vectors by constnt (number) by stretching the vector. When you multiply by negtive number, the vector reverses direction. o When vector is multiplied by constnt, both of its components re multiplied by tht constnt. Prctice Problems 5. Ivn pulls sled loded with logs to his cbin. If Ivn pulls with force of 800 N t n ngle of 20 with the horizontl, wht re the horizontl nd verticl components of this force? Horizontl Verticl 6. A cnnonbll is fired t 100 m/s t n ngle of 40 with the horizontl. Wht re the horizontl nd verticl components of its speed? Horizontl Verticl

4 7. Use the vectors shown below to drw the following. Lbel ll your vectors.. x r + y r b. x r z r c. 2y r d. 2z r y r 8.. Find the horizontl nd verticl components of the vectors below b b. Find the horizontl nd verticl components of 2b r r. Horizontl: Verticl:

5 9. A ship trvels 4 km t n ngle of 10 with the horizontl. Then, the ship continues nd trvels 6 km t n ngle of 30 with the horizontl. Find the horizontl nd verticl components of the ship totl distnce trveled. III. Tringle Formuls ( ) Key Fcts To clculte the re of tringle: o A = 1 bh 2 b is the bse nd h is the height of the tringle Bse nd height must be perpendiculr o A = 1 bsinc 2 nd b re two sides of the tringle nd C is the ngle between them Remember tht the lower cse letters stnd for sides. The uppercse letters re the ngles. A is opposite side, B is opposite side b, etc. A Lw of Sines o sin A = b sin B = c sinc o The most importnt prt of this formul is tht you pir up sides nd ngles tht re cross from ech other in ech frction! Lw of Cosines: o c 2 = 2 + b 2 2bcosC o The most importnt prt of this formul is tht the side on the left side of the eqution must be cross from the ngle on the right side of the eqution! If you re sked to solve the tringle, tht mens to find ll missing sides nd ngles. If you know side nd the ngle cross from it, strt with Lw of Sines SSA Ambiguous Cse often there re 2 possible tringles nd you must solve for both possibilities The following formuls will be given to you on the test: o Are = 1 bsinc 2 o o B sin A = b sin B = c OR sinc c 2 = 2 + b 2 2bcosC c sin A b = sin B b C = sinc c

6 Prctice Problems 10.. Find the length of side b. b. Find the re of the tringle in prt MNO hs sides m = 11, n = 9, nd o = 14. Find the mesure of N. b. Find the re of MNO 12. Find the length of side.

7 13 Find the re of ABC. (You might need to find nother piece of info first) 14. DEF hs sides d = 18, f = 15, nd F = 38. There re two possible tringles. Find both set of sides nd ngles.

8 Answers b c d e f feet = m 5. Horizontl: N Verticl: N 6. Horizontl: 76.6 m/s Verticl: m/s r : Horizontl: 6.76 Verticl: b r : Horizontl: Verticl: 7.66 b. Horizontl: Verticl: Horizontl: = 9.14 km Verticl: = 3.69 km 10. b = feet b. Are = ft! b First Tringle: D = E = e = Second Tringle: D = E = 9.63 e = 4.08