UNIT 5 SIMILARITY, RIGHT TRIANGLE TRIGONOMETRY, AND PROOF Unit Assessment

Transcription

1 Unit 5 ircle the letter of the best answer. 1. line segment has endpoints at (, 5) and (, 11). point on the segment has a distance that is 1 of the length of the segment from endpoint (, 5). What are the coordinates of the point? a. (, 7) b. (, 8) c. (, 1) d. (0, ). If EF has vertices with the coordinates (9, ), E ( 6, ), and F (, 6), what are the vertices of EF under a dilation with a scale factor of and the center at the origin, (0, 0)? a. (1.5,.5), E ( 9,.5), F (.5, 9) b. (6, ), E (, ), F (, ) c. (, 6), E (, ), F (, ) d. (.5, 1.5), E (.5, 9), F (9,.5). re the two triangles similar? Why or why not? E a. Yes, they are similar because of the Similarity Statement. b. Yes, they are similar because of the S ongruence Statement. c. No, they are not similar because congruence is not preserved. d. It cannot be determined if the triangles are similar. U5-77

2 . E. What is the length of E? E 9 a. 8.5 units b. 8.5 units c. 9.8 units d. There is not enough information to determine the length of E. 5. is a right triangle. Find the length of..5 x + 5 a. units b. 7.5 units c. 10 units d. There is not enough information to determine the length of. U5-78

3 6. Given two sets of parallel lines in the diagram below, what is the relationship between and 10? E H G F a. No relationship exists. b. 10 because 1 by the lternate Interior ngles Theorem. Then 1 10 by the lternate Exterior ngles Theorem. Therefore, by the Transitive Property, 10. c. and 10 are congruent right angles because all right angles are congruent and two pairs of intersecting parallel lines are always perpendicular. d. is supplementary to 10 because 1 by the lternate Interior ngles Theorem. Then 1 is supplementary to 10 by the lternate Exterior ngles Theorem. This means that m 1+ m 10 = 180 and, by substitution, m + m 10 = 180. Therefore, by the definition of supplementary angles, is supplementary to 10. U5-79

4 7. What is the measure of? a. 1 b. 6 c d. There is not enough information to determine the measure of. U5-70

5 8. If = 5x + and XZ = x 5, what is the actual length of? (5x + ) Y X (x 5) Z a. 1.5 units b. units c. 11 units d. units 9. lassify a quadrilateral as precisely as possible given four vertices: (, ), (5, 0), E ( 6, ), and F (, 6). a. rhombus b. square c. rectangle d. kite U5-71

6 10. is a right triangle. One of the acute angles measures º. What is the cosine of the other acute angle? a b..6 c. 0.5 d monument is 75 meters high. t an information booth, an observer notices the angle of elevation to the top of the monument is 6º. How far is the observer from the base of the building? a. about 18 m b. about 10 m c. about 9 m d. about 55 m 1. If 0º < θ < 90º and sin θ = 9, find cos θ using a Pythagorean identity. a c b. 81 d Use the information given in each problem to complete problems 1 and bluebird on a 0-meter-tall flagpole spots a worm on the ground below at an angle of depression of 5º. a. What equation can be written to determine the distance of the worm from the base of the flagpole? b. How far away from the base of the flagpole is the worm? U5-7

7 1. Given: is a parallelogram. Prove: H H H = HE H F E Use Pythagorean identities, reciprocal identities, and the Pythagorean Theorem to complete the problem that follows. 15. recent storm downed a tree in your backyard. The top of the tree is now leaning on the roof of your shed. The shed s roof is 8 feet tall and the tree is feet long. The angle that the tree makes with the ground is θ. a. Find all of the trigonometric functions of θ, using Pythagorean and reciprocal identities. b. Find the measure of angle θ. c. How far is the base of the tree from the base of the shed? U5-7

8 HONORS Unit 5 ircle the letter of the best answer. 1. n angle measures a. 10 b radians. What is the measure of the angle in degrees? 6 c. 10 d. 0. Find the value of x if the point x, 1 a. x = b. x = 1 lies on the unit circle at an angle θ such that < θ<. c. x = d. x =. Which trigonometric expression is equivalent to sin θ sec θ + sin θ csc θ? a. 1 + sin θ c. sec θ b. csc θ d. tan θ. Lola proved the trigonometric identity cos θ + cos θ tan θ = 1 by using just two identities. Which two types of identities did she use? a. even-odd and reciprocal b. even-odd and ratio c. Pythagorean and ratio d. Pythagorean and double-angle 5. What is the exact value of a. b. 6 6 cos +? c. d Which of the following is not an equivalent expression for cos θ? a. 1 sin θ c. cos θ 1 b. cos θ sin θ d. sin θ + 1 U5-7

9 Use what you have learned about the unit circle, trigonometric ratios, and trigonometric identities to complete all parts of the following problems. 7. traveler drives a car in a straight line along a course defined by an angle θ, such that θ = 0 represents due east. He starts at the origin (0, 0) and continues in a straight line along θ = If the traveler continues along this course and stops 100 miles to the east of the origin, how many total miles has the traveler covered? a. etermine the quadrant in which θ = 11 lies. 6 b. What is the reference angle for θ = 11 in radians and degrees? 6 c. Find how many total miles the traveler covered, if he stopped the car 100 miles to the east of the origin. Round the answer to the nearest tenth of a mile. 8. Write a two-column proof of the trigonometric identity cos( θ) sin θ = cscθ tanθ. sinθcos θ 9. Fernando is standing on level ground and flying a kite attached to a 50-foot string. The kite s angle of elevation with the ground is θ = 5 1. a. Write θ = 5 as the sum of two special known angles. Write these angles in radians and 1 in degrees. b. If the string breaks and the kite falls straight down, how far from Fernando will the kite land? Round the answer to the nearest tenth of a foot. c. If the string breaks and the kite falls straight down, how many feet will the kite have fallen? Round the answer to the nearest tenth of a foot. d. Name the formulas/identities used to find the solutions in parts b and c. U5-75

10 Lesson HONORS: Proving the ddition and Subtraction Formulas Lesson HONORS Pre-, p. U c. a Warm-Up HONORS, p. U fter the Ferris wheel s first 0 rotation, the height of Quentin s seat has increased by about 10 feet, and after the Ferris wheel s second 0 rotation, the height of his seat has increased by an additional 7.5 feet.. No, doubling the angle does not double the height of the car.. sin (0 + 0 ) is not equal to sin 0 + sin 0. However, both sin (0 + 0 ) and sin 60 are equal to reinforces the answer for problem.. This Practice HONORS: Proving the ddition and Subtraction Formulas, pp. U5-7 U cos 10 = ; sin10 = ; tan10 =. cos = ; sin = + ; tan = + 6. Use the angle difference formula for cosine with α = θ and β =. Statements 1. cos (α β) = cos α cos β + sin α sin β. cos [(θ) ()] = cos (θ) cos () + sin (θ) sin (). cos (θ ) = cos θ( 1) + sin θ(0) Reasons 1. ngle difference formula for cosine. cos (θ ) = cos θ + 0. Simplify. 5. cos (θ ) = cos θ 5. Simplify.. Substitute θ for α and for β.. Substitute 1 for cos and 0 for sin.. Use the angle sum formula for tan (α + β) with α = θ and β = θ. Statements tanα+ tanβ 1. tan( α+ β) = 1 tanαtanβ tan( θ) + tan( θ). tan [( θ) + ( θ) ] = 1 tan( θ)tan( θ) Reasons 1. ngle sum formula for tan (α + β). Substitute θ for α and θ for β. tanθ. tan( θ )=. Simplify. 1 tan θ 5. sin α cos β 6. sin β 7. In the diagram, the hypotenuse of the leftmost right triangle containing angle α + β is equal to 1, so the side opposite to angle α + β has length sin (α + β). The hypotenuse of the central right triangle containing angle β is also 1, so the side adjacent to angle β has length cos β, and the side opposite to angle β has length sin β. The hypotenuse of the rightmost right triangle containing angle α is not equal to 1, but you can use the definition opposite tanα = and the opposite length sin β to find an adjacent expression for the length of the side adjacent to α. Statements opposite 1. tanα = adjacent sinβ. tanα = adjacent Reasons 1. efinition of tan α. Substitute sin β for the opposite side. sinβ. adjacent =. Solve for the hypotenuse. tanα. sinβ adjacent = cosα cosα 5. adjacent = sinβ. Rewrite using the ratio identity tanα =. cosα 5. To divide by a fraction, multiply by its reciprocal. 8. To find an expression for sin (α + β), look for a trigonometric ratio that involves the side of length sin (α + β). onsider the largest right triangle, which contains the angle α. The opposite side of this triangle has length sin (α + β), and the hypotenuse has length sinβcosα cosβ +. Substitute these into the definition of sin α. U5-759

11 Statements opposite = hypotenuse sin α β = + sinβcosα cosβ + Reasons efinition of sine Substitute sin (α + β) for the opposite side and sinβcosα cosβ + for the hypotenuse. sinβcosα cosβ+ sin( α β) = + ross multiply to eliminate fractions. sinβcosα sin( α+ β) = cosβ+ pply the Symmetric Property of Equality. sinβcosα sin( α+ β) = ( cosβ) + istribute sin α. sin (α + β) = sin α cos β + sin β cos α Simplify feet 10. If due east is 0 and north is 90, then northeast is 5, which is halfway between north and east. The angle halfway between northeast (5 ) and east (0 ) is θ =.5, or in radians,. To find cos.5, use the half-angle 8 5 identity for sine with.5 =. Statements θ 1 cosθ sin =± ( sin 5 ) 1 cos( 5 ) =± sin.5 =± sin.5 =± 1 Reasons Half-angle identity for sine Substitute 5 for θ. Substitute Simplify. for cos 5. Lesson HONORS Progress, p. U c. d. b. a. cosθ = 6 ; sinθ = b. approximately 1.9 feet pp. U5-77 U c. b. a. a 5. c 6. d 1. nswers: b 8. d 9. c 10. d 11. b 1. c 0 a. tan 5º = x b. The worm is approximately 6 meters away from the base of the flagpole. 1. ecause they are vertical angles, m H = m EH. ecause E, m H = m EH. These two pairs of equal angles indicate that the Similarity Postulate can be used: H HE. ecause they are corresponding H H sides of similar triangles, =. Using the property HE H H H of proportions, we conclude that = H HE. 15. nswers: a. sin θ = 1 ; cos θ = ; tan θ = sec θ = ; cot θ = b. θ 19.8º c. approximately.6 feet ; csc θ = ; sin.5 =± Rationalize the denominator. U5-760