MMS10_L1.qxp 1/21/10 10:34 AM Page 1 CHAPTER 2 Trigonometry What You’ll Learn • Determine the measure of an acute angle in a right triangle using the lengths of two sides. • Determine the length of a side in a right triangle using the length of another side and the measure of an acute angle. • Solve problems that involve more than one right triangle. Why It’s Important Trigonometric ratios are used by: • surveyors, to determine the distance across a river or a very busy street • pilots, to determine flight paths and measure crosswinds • forestry technicians, to calculate the heights of trees Key WordsDRAFT tangent ratio cosine ratio angle of inclination angle of elevation indirect measurement angle of depression sine ratio 1 MMS10_L1.qxp 1/21/10 10:34 AM Page 2 2.1 Skill Builder Similar Triangles Similar triangles have: • the measures of matching angles equal OR • the ratios of matching sides equal E 12 cm P S ° Q 30 5 cm 8 cm F 6 cm 4 cm A 110° T R U 2 cm 40° These triangles are not similar because the 40° ratios of matching sides are different. 110° 30° PQ 12 C B D ϭϭ2.4 ST 5 These triangles are similar because QR 6 ϭϭ3 Compare the longest sides, matching angles are equal. TU 2 compare the shortest sides, ЄA ϭ ЄD ϭ 40° RP 8 then compare the third ϭϭ2 pair of sides. ЄB ϭ ЄE ϭ 30° US 4 ЄC ϭ ЄF ϭ 110° Check 1. Which triangles in each pair are similar? B 1.0 cm Y a) M b) K 3 cm E F 3.0 cm 4 cm 5 cm 8 cm 3.5 cm 5.0 cm 10 cm 4.0 cm G DRAFT W J 6 cm D C X 2.0 cm Compare the ratios of matching sides. Compare the ratios of matching sides. DB _______ 5.0 _______ FG ______ 5 ϭ Џ ______1.4 ϭϭ______0.5 ______MJ _______3.5 _______XY ______10 CD _______ 2.0 _______ EF _____ 3 ______2 ϭϭ______0.5 ______KM _______1.0 _______WX _____6 BC _______ 4.0 _______ GE _____ 4 ϭϭ______1.3 ϭϭ______0.5 ______JK _______3.0 _______YW _____8 The triangles ____________are not similar. The triangles ____________are similar. 2 MMS10_L1.qxp 1/21/10 10:34 AM Page 3 2.1 The Tangent Ratio FOCUS Use the tangent ratio to find an angle measure. The Tangent Ratio A An acute angle is less than 90°. side adjacent to ∠A C B side opposite ∠A length of side opposite ЄA If ЄA is an acute angle in a right triangle, then tan A ϭ length of side adjacent to ЄA Example 1 Finding the Tangent Ratio Find the tangent ratio for ЄG. 18 E F 10 G Solution Draw an arc at ЄG. opposite 18 The side opposite ЄG is EF. E F The side adjacent to ЄG isDRAFT GE. 10 length of side opposite Є G adjacent tan G ϭ length of side adjacent to Є G G EF tan G ϭ Substitute: EF ϭ 18 and GE ϭ 10 GE 18 The side opposite the right angle tan G ϭ is always the hypotenuse. 10 tan G ϭ 1.8 3 MMS10_L1.qxp 1/21/10 10:34 AM Page 4 Check 1. a) Find tan P. b) Find tan Q. P The side opposite ЄP is ________QR . The side opposite The side adjacent to ЄP is ________RP . ЄQ is ________RP . 8 Q length of side opposite Є P The side adjacent 10 ϭ ______________ tan P to ЄQ is QR . R length of side ______________ adjacent to Є P ________ length of side opposite Є Q QR tan Q ϭ tan P ϭ _____________________________________length of side adjacent to Є Q _________RP 10 ϭ RP tan P ϭ tan Q _________8 _______QR ϭ 8 tan P _________1.25 tan Q ϭ _______10 tan Q ϭ _______0.8 To find the measure of an angle, use the tanϪ1 key on a scientific calculator. Example 2 Using the Tangent Ratio to Find the Measure of an Angle Find the measure of ЄA to the nearest degree. 16 CB 7 A Solution The side opposite ЄA is BC. The side adjacent to ЄA is AB.DRAFT length of side opposite Є A tan A ϭ length of side adjacent to Є A If you are using a different BC tan A ϭ Substitute: BC ϭ 16 and AB ϭ 7 calculator, consult the user’s AB manual. 16 tan A ϭ 7 To find ЄA using a TI-30XIIS calculator, enter: tan-1(16/7) 66.37062227 %@.3W4E<, ЄA Џ 66° 4 MMS10_L1.qxp 1/21/10 10:34 AM Page 5 Check 1. Find the measure of each indicated angle to the nearest degree. a) ЄF F 10 G 13 H The side opposite ЄF is ________GH . The side adjacent to ЄF is ________FG . length of side _______________ opposite Є F tan F ϭ length of side _______________ adjacent to Є F G H tan F ϭ _____________FG 13 tan F ϭ _____________10 tan F ϭ ___________1.3 Use a calculator. Ϫ1 ЄF ϭ tan ___________(1.3) ЄF Џ ___________52° b) ЄE The side opposite ЄE is ________CD . 5 The side adjacent to ЄE is ________DE . CD length of side opposite ЄE tan E DRAFTlength of side adjacent to ЄE ________________________________ 9 CD tan E _____________DE 5 tan E ϭ E _____________9 5 Ϫ ЄE ϭ tan 1___________aa 9bb ЄE Џ ___________29° 5 MMS10_L1.qxp 1/21/10 10:34 AM Page 6 Example 3 Using the Tangent Ratio to Find an Angle of Inclination A guy wire is fastened to a cell-phone tower 8.5 m above the ground. The wire is anchored to the ground The angle the wire makes 14.0 m from the base of the tower. What angle, to the with the ground is called the angle of inclination. nearest degree, does the wire make with the ground? Solution A Draw a diagram. Є The angle the wire makes with the ground is B. Assume the tower is 8.5 m To find ЄB, use the tangent ratio. perpendicular to the ground. B C 14.0 m length of side opposite Є B The side opposite ЄB is CA. tan B ϭ length of side adjacent to Є B The side adjacent to ЄB is BC. CA tan B Substitute: CA ϭ 8.5 and BC ϭ 14.0 BC 8.5 tan B ϭ Use a calculator. 14.0 ЄB Џ 31° The angle between the ground and the wire is about 31°. Check 1. A ladder leans against a house. The top of the ladder is 2.4 m above the ground. F Its base is 0.9 m from the wall. What angle, to the nearest degree, does the ladder make with the ground? Label the given triangle FGH. 2.4 m Label G where the ladder meets the ground. Label F where it meets the wall. We want to find the measureDRAFT of ЄG. G H 0.9 m The side opposite ЄG is __________.HF The side adjacent to ЄG is __________GH . HF opposite tan G ϭ tan G ϭ ________________GH adjacent 2.4 tan G ϭ ________________0.9 ЄG ________________69° The angle between the ground and the ladder is about _______69° . 6 MMS10_L1.qxp 1/21/10 10:34 AM Page 7 Practice 1. Label the hypotenuse, opposite, and adjacent sides of each right triangle in relation to the given angle. a) ЄH b) ЄP G M adjacent opposite opposite hypotenuse P hypotenuse N F H adjacent 2. Find the tangent ratio for each indicated angle. Leave the ratio in fraction form. a) Z b) V 7 16 15 X 3 UW Y The side opposite ЄY is _______ZX . The side opposite ЄW is _______UV . The side adjacent to ЄY is _______XY . The side adjacent to ЄW is _______VW . length of side _______________ opposite Є Y length of side opposite ЄW tan Y ϭ tan W length of side _______________ adjacent to Є Y ________________________________length of side adjacent to ЄW ZX UV tan Y ϭ tan W ϭ ______________XY ______________VW 7 tan Y ϭ tan W ϭ 16 ______________3 DRAFT______________15 3. Find the measure of ЄA for each value of tan A. Give your answer to the nearest degree. 5 a) tan A ϭ 0.5 b) tan A ϭ 6 Є ϭ Ϫ1 5 A tan (_________0.5 ) Use a calculator. tan؊1 ЄA ϭ ________________aa 6bb ЄA Џ ________27° ЄA Џ ________40° 7 MMS10_L1.qxp 1/21/10 5:24 PM Page 8 4. Find the measure of ЄB to the nearest degree. 13 B C The side opposite ЄB is ___________CD . The side adjacent to ЄB is ___________BC . 12 length of side _________________ opposite Є B tan B ϭ length of side _________________ adjacent to Є B CD D tan B ϭ _________________BC 12 tan B ϭ _________________13 ЄB Џ _________________43° 5. A telephone pole is supported by a wire, as shown. M What angle, to the nearest degree, does the wire make with the ground? We want to find the measure of ЄN. 18 m Use the tangent ratio. length of side _________________ opposite Є N tan _________N ϭ Є N P length of side _________________ adjacent to N 8 m PM tan _________N ϭ _________NP 18 tan _________N ϭ __________8 Є_________N Џ _________66° The angle between the ground and the wire is about _________66° . 6. Victor is building a wheelchair ramp to an entranceway that is 3 m above the sidewalk.