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Name: Homework 5.6.L Right Triangle Trig Page 1 of 7 Homework 5.6 Right Triangle Trig MAT 125: Precalculus Fall 2009: Sections 6, 9, and 13 Mr. Jonckheere Due Homework CP 5.6.1. Fill in the blanks to complete each theorem a) If α is an accute angle of a right triangle, opp is the length of the side opposite α, adj is the length of the side adjacent to α, hyp is the length of the hypotenuse, and x = m(α) then opp adj opp sin ( x ) = cos ( x ) = tan ( x ) = hyp hyp adj b) If a right triangle has legs of length a, and b, and a hypotenuse of length c then c2 = a2 + b2. c) The sum of the primary measures of the angles in any triangle is π = 180°. d) If x is in [0, π/2] then arcsin(sin(x)) = x arcos(cos(x)) = x CP 5.6.2. Let α be an accute angle in a right triangle, and x = m(α). Let the side opposite α have length 6, and the side adjecent to α have length 3. Evaluate all six trig functions at x. h2 = 62 + 32 = 36 + 9 = 45 h = 45 = 3 5 opp 6 2 5 sin ( x ) = = = csc ( x ) = hyp 3 5 5 2 adj 3 1 cos ( x ) = = = sec ( x ) = 5 hyp 3 5 5 opp 6 3 1 tan ( x ) = = =2 cot ( x ) = = adj 3 6 2 Name: Homework 5.6.L Right Triangle Trig Page 2 of 7 CP 5.6.3. A right triangle has an angle with measure 68°, and a hypotenuse of length 10. Solve the triangle. x = 68° b c = 10 y = 90° – 68° = 22° y c sin ( x ) = a a c a = c sin ( x ) = 10 sin ( 68° ) x ≈ 9.3 a b b cos ( x ) = c b = c cos ( x ) = 10 cos ( 68° ) ≈ 3.7 Name: Homework 5.6.L Right Triangle Trig Page 3 of 7 CP 5.6.4. A right triangle has legs of lengths 20 and 31. Solve the triangle. b a = 20 b = 31 y 2 2 2 c =a +b c 2 2 2 c = a + b = 20 + 31 = 1361 2 a a tan ( x ) = x b a a x = arctan b b 20 = arctan 31 ≈ 32.8° y = 90° − x ≈ 90° − 32.8° = 57.2° CP 5.6.5. A 40 meter guy wire is attached to the top of a 35 meter atenna and to a point on the ground. What is the primary measure (in degrees) of the angle made by the guy wire and the ground? a sin ( x ) = c a 35 x = arcsin = arcsin = 61° c 40 CP 5.6.6. Two country roads meet at a right angle. A biker travelling on the North-South road and approaching the intersection decides to take a shortcut across the field. When he reaches a large oak, he turns 40° and heads across the field for the East-West road. If he travelled a straight 0.25 miles across the field to get to the East-West road, how far is the intersection from the oak? Round to the nearest 100th of a mile. b cos( x) = c b = c cos( x) = 0.25cos ( 40° ) ≈ 0.19 mi Name: Homework 5.6.L Right Triangle Trig Page 4 of 7 CP 5.6.7. The forest ranger at the top of Kendrick mountain is watching a forst fire spread is her direction. In 10 minutes the angle of depression of the leading edge of the fire changed from 11° to 13°. At what speed is the fire spreading in the direction of the ranger. Assume the ranger is 3430 feet above the fire. a tan ( x1 ) = b1 + b2 a tan ( x2 ) = b2 b1 + b2 b2 b1 cot ( x1 ) − cot ( x2 ) = − = a a a b1 = a ( cot ( x1 ) − cot ( x2 ) ) b1 v= t a ( cot ( x1 ) − cot ( x2 ) ) = t 3430 ( cot (11° ) − cot (13° ) ) = 10 ≈ 279 ft/min Name: Homework 5.6.L Right Triangle Trig Page 5 of 7 Extra Practice CP 5.6.8. A triangle has a hypotenuse of length 100 and a leg of length 10. Solve the triangle. c = 100 b a = 10 c2 = a2 + b2 y b2 = c2 – a2 c b = c −a 2 2 a 2 2 = 100 − 10 = 9900 x = 90 11 a b a sin ( x ) = c a x = arcsin c 10 = arcsin 100 ≈ 5.7° y = 90° – x ≈ 90° – 5.7° = 84.3° Name: Homework 5.6.L Right Triangle Trig Page 6 of 7 CP 5.6.9. One angle of a right triangle measures 39° and the leg adjacent to it is 15 units long. Solve the triangle. b x = 39° b = 15 y y = 90° – x c = 90° – 39° a = 51° b x cos ( x ) = c b a c= b cos ( x ) 15 = cos ( 39° ) ≈ 19.3 a tan ( x ) = b a = b tan ( x ) = 15 tan ( 39° ) ≈ 12.1 CP 5.6.10. A tree casts a shadow which is 6.4 feet long. The primary measue of the angle of elevation from the shadow of the tree’s top to the actual tree’s top is 71°. How tall is the tree to the nearest 100th of a foot? a tan ( x ) = b a = b tan ( x ) = 6.4 tan ( 71° ) ≈ 18.59 Name: Homework 5.6.L Right Triangle Trig Page 7 of 7 CP 5.6.11. A hot-air baloonist spots two mile-posts on the straight road over which he hovers. He measures the angle of depression to mile post 184 to be 17° and the angle of depression to mile post 183 to be 21°. How high is he? a tan ( x1 ) = 1+ b a tan ( x2 ) = b 1+ b b 1 cot ( x1 ) − cot ( x2 ) = − = a a a 1 a= cot ( x1 ) − cot ( x2 ) 1 = cot (17° ) − cot ( 21° ) ≈ 1.5 mi