The Tangent Ratio. 1. Plan. 1 Using Tangents in Triangles. California Math Background. Lesson Planning and Resources

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1 -. Plan alifornia ontent Standards GEOM.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. hey also know and are able to use elementary relationships between them. GEOM 9.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. alifornia Math ackground What You ll Learn o use tangent ratios to determine side lengths in triangles... nd Why - he angent Ratio o use the tangent ratio to estimate distance to a distant object, as in Eample Using angents in riangles alifornia ontent Standards GEOM.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. Introduce GEOM 9.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Introduce. 0.; 0.;.. 0.; 0.; Lessons - and - Find the ratios,, and. Round answers to the nearest hundredth. lgebra Solve each proportion. 0.; 0.;... = New Vocabulary tangent 0 he Greek mathematicians Hipparchus of Nicaea and laudius Ptolemy created the discipline of trigonometry more than 000 years ago, primarily to study astronomy. Subsequent Indian and rabic mathematicians also developed trigonometry. More Math ackground: p. Lesson Planning and Resources. nswers may vary. Sample: For each &, the ratio is the same no matter how large or small the k. Work in groups of three or four. ctivity: angent Ratios Have your group select one angle measure from {, 0, c, 0 }. hen have each member of your group draw a right triangle, #, where & has the selected measure. Make the triangles different sizes. Measure the legs of each # to the nearest millimeter. heck students work. leg /. ompute the ratio leg adjacent to / and round to two decimal places.. ompare the ratios in your group. Make a conjecture. See left. See p. E for a list of the resources that support this lesson. D, Online, or ransparencies ell Ringer Practice heck Skills You ll Need Use student page, transparency, or PowerPoint. alifornia Standards Daily Review Use transparency 9. Vocabulary ip rigonometry comes from the Greek words trigonon and metria meaning triangle measurement. hapter Right riangles and rigonometry For each pair of complementary angles, & and &, there is a family of similar right triangles. In each such family, the ratio length of leg / length of leg adjacent to / is constant no matter the size of #. his trigonometric ratio is called the tangent ratio. length of leg / tangent of & = length of leg adjacent to / You can abbreviate this equation as tan = adjacent. Leg adjacent to Leg Special Needs L Have students go outside to measure distances indirectly by applying the method in Eample. elow Level L Have students use all the ratios from the ctivity to make a table of tangent values.

2 EXMPLE Writing angent Ratios Write the tangent ratios for & and &U. tan = = UV = adjacent V tan U = = V = adjacent UV a. Write the tangent ratios for &K and &J. ; b. How is tan K related to tan J? hey are reciprocals. s stated on the facing page, the tangent ratio for an acute angle does not depend on leg lengths of a right triangle. o see why this is so, consider the congruent angles, & and &9, in the two right triangles shown here. #OW, 9O9W9 Similarity Postulate OW = OrWr O ror orresponding sides of M triangles are proportional. tan = tan 9 Substitute. You can use the tangent ratio to measure distances that would be difficult to measure directly. J L W O U V K ' W' O'. each Guided Instruction ctivity ecause of measurement errors and rounding, students in a group may find that their ratios are close but not equal. If this occurs, ask students to eplain why the ratios vary. Math ip Point out that this definition of the tangent of an angle cannot be used to find tan 90 because a right angle has no leg. eaching ip fter students complete Eample, ask: What can you conclude about the tangents of complementary angles? hey are reciprocals. not to scale 0 ft EXMPLE Real-World onnection ross-ountry Skiing Your goal in ryce anyon National Park is the distant cliff. bout how far away is the cliff? Step Step Point your compass at a distinctive feature of the cliff and note the reading. urn 90 and stride 0 ft in a straight path. Step urn and point the compass again at the same feature seen in Step. ake a reading. Suppose in Step, you find that m& =. he distance you walked in Step was 0 ft. o find the distance to the cliff use the tangent ratio. tan = 0 Use the tangent ratio. = 0(tan ) Solve for. 0.0 Use a calculator. he cliff is about ft away. Write the tangent ratios for & and &. 0 D, Online, or ransparencies dditional Eamples 9 tan 0 ; tan 0 o measure the height of a tree, lma walked ft from the tree and measured a angle from the ground to the top of the tree. Estimate the height of the tree. Find the value of w to the nearest tenth. a. b..9.0 c.. w w.. w. ft about ft Lesson - he angent Ratio dvanced Learners L Have students eplain how to find tan 0 and tan 0 without using a calculator, and then confirm the values with a calculator. English Learners EL Point out that a directional compass is different from the compass used in geometric constructions. Pass around a directional compass and demonstrate its use.

3 Error Prevention Students may confuse the angle whose tangent is, tan -, with tan -. Point out that tan - = tan, but tan - represents the inverse trigonometric function, or the angle whose tangent is. EXMPLE echnology ip Students should repeat the eample on their own calculators to determine their correct key sequences. For: angent ctivity Use: Interactive etbook, - If you know leg lengths for a right triangle, you can find the tangent ratio for each acute angle. onversely, if you know the tangent ratio for an angle, you can use inverse of tangent, tan -, to find the measure of the angle. EXMPLE Using the Inverse of angent he lengths of the sides of #HX are given. Find m&x to the nearest degree. tan X = = 0. Find the tangent ratio. m&x = tan - (0.) Use the inverse of tangent. N Use a calculator. So m&x <. H X Find m&r to the nearest degree. R D, Online, or ransparencies dditional Eamples EXERISES Standards Practice Find m&y to the nearest degree. P 0 Y For more eercises, see Etra Skill, Word Problem, and Proof Practice. GEOM.0, 9.0 S mlr N 9 Resources Daily Notetaking Guide - L Daily Notetaking Guide - dapted Version L losure Without using a calculator, find the angle whose tangent equals. Eplain. Sample: ; by the onverse of the Isosceles riangle heorem, a triangle has congruent legs, so tan. Practice by Eample GO for Help Eample (page ) Eample (page ) Write the tangent ratios for l and l.... ; ; Find the value of to the nearest tenth Surveying o find the distance from the boathouse on shore to the cabin on the island, a surveyor measures from the boathouse to point X as shown. He then finds m&x with an instrument called a transit. Use the surveyor s measurements to find the distance from the boathouse to the cabin. about 0 yd 0 yd X 9 abin ;. oathouse hapter Right riangles and rigonometry. onsider a k. Let the length of the shorter side be a. hen the length of the longer side, the 0 &, is a ". hus, a" tan 0 a ". a 0 a 0 a

4 Eample (page ) pply Your Skills 0a. 0.; 0.; 0.; 0.; 0.; 0.; 0.; 0.; ;.;.;.;.;.;.;.;. 0c. approaches 0; increases to infinity 0d. nswers may vary. Samples: ;.; Visit: PHSchool.com Web ode: bce-00 Find the value of to the nearest degree Find each missing value to the nearest tenth.. tan j =.. tan = j 9. tan = 0. tan j = 90 0 j he lengths of the diagonals of a rhombus are in. and in. Find the measures of the angles of the rhombus to the nearest degree. and. Pyramids ll but two of the pyramids built by the ancient Egyptians have faces inclined at angles. Suppose an archaeologist discovers the ruins of a pyramid. Most of the pyramid has eroded, but she is able to determine that the length of a side of the square base is m. How tall was the pyramid, assuming its faces were inclined at? Round your answer to the nearest m meter. m. Multiple hoice he legs of a right triangle have lengths and. What is the tangent of the angle the leg with length? D. Writing Eplain why tan 0 =!. Include a diagram with your eplanation.. Eplain why tan -! =.!. See margin.. rectangle is 0 cm long and 0 cm wide. o the nearest degree, find the GPS measures of the angles formed by the diagonals at the center of the rectangle. and Find the value of w, then. Round lengths of segments to the nearest tenth. Round angle measures to the nearest degree w w w ; N. w w N 9; N w N.; N. 0. a. oordinate Geometry omplete the table of values at the right. Give table entries to the nearest tenth. tan b. Plot the points (, tan ) on the coordinate plane. onnect the points with a smooth curve. See margin. c. What happens to the tangent ratio as the angle measure approaches 0? pproaches 90? d. Use your graph to estimate each value. a, c d. See left. tan j = tan = j tan j =. " onsider a --90 k. 0. b.., so we have to " Let the lengths of the show tan. shorter sides be a. hus, his is equivalent to tan a. showing tan. 0 Lesson - he angent Ratio angent Degrees. Practice ssignment Guide - hallenge - Multiple hoice Practice 9, 0 Mied Review - Homework Quick heck o check students understanding of key skills and concepts, go over Eercises,,,,. echnology ip Make sure that students set their calculators in degree mode. Eercise 9 Point out that students can solve a simpler equation by using the ratio tan(90 ) =. Eercises, Students need to apply the Pythagorean heorem before they can find the value of. Eercise Students should sketch and label the triangle, and be careful to find the tangent ratio asked for. Guided Problem Solving Enrichment Reteaching dapted Practice Practice Name lass Date Practice - Write the tangent ratios for le and lf.. E. F D. D Find each missing value. Round your answers to the nearest tenth.. tan =?. tan 9 =. tan = Find the value of. Round your answers to the nearest tenth o the nearest tenth, find the measure of the acute angle that the given line forms with a horizontal line.. y = +. y = +. y = - Find the value of. Round your answers to the nearest degree F E F D 9? he angent Ratio E L L L L L Pearson Education, Inc., publishing as Pearson Prentice Hall. ll rights reserved.

5 . ssess & Reteach Use the figure for Eercises.. Write the tangent ratio for &K.. Write the tangent ratio for &M.. Find m&m to the nearest degree. Find to the nearest whole number D, Online, or ransparencies Lesson Quiz K L 90 M lternative ssessment Have students draw a right triangle, measure one acute angle and the leg adjacent to it, and then use the tangent function to estimate the length of the other leg. Have students draw another right triangle, measure each leg, and use the inverse tangent function to estimate both acute angles of the triangle. Real-World onnection he world s steepest railway is the Katoomba Scenic Railway in ustralia s lue Mountains. PHSchool.com For: Graphing calculator procedures Web ode: bcb- hallenge Problem Solving Hint hink of tan () as the angle whose tangent is. rise Engineering he grade of a road or a railway road bed is the ratio run, usually epressed as a percent. For eample, a railway with a grade of % rises ft for every 0 ft of horizontal distance.. he Katoomba Railway, pictured at left, has a grade of %. What angle does its roadbed make with the horizontal? about. he Johnstown, Pennsylvania, inclined railway was built as a lifesaver after the Johnstown flood of 9. It has a 9-ft run at a % grade. How high does this railway lift its passengers? about 0 ft. he Fenelon Place Elevator railway in Dubuque, Iowa, lifts passengers 9 ft to the top of a bluff. It has an % grade. How long is this railway? about 9 ft. he Duquesne Incline Plane ompany s roadway in Pittsburgh, Pennsylvania, climbs Mt. Washington, located above the mouth of the Monongahela River. It reaches a height of 00 ft with a 9-ft incline. What is its grade? about.% Find the missing value to the nearest tenth..0. =, y = j. =!, y = j. =, y = j =!, y = j 9. = j, y = 0. = j, y = = j,. = j,. = j, y =.0 y = 0. y =.. a. ritical hinking Does tan + tan = tan ( + ) when +, 90? Eplain. a b. See margin. b. Reasoning Does tan - tan = tan ( - ) when -. 0? Use part (a) and indirect reasoning to eplain.. Graphing alculator Use the LE feature of your graphing calculator to study tan X as X gets close to 90. In the Y= screen, enter Y = tan X. a. Use the LSE feature so that X starts at 0 and changes by. ccess the LE. From the table, what is tan X for X = 9?.90 b. Perform a numerical zoom in. Use the LSE feature, so that X starts with 9 and changes by 0.. What is tan X for X = 9.9?.9 c. See margin. c. ontinue to numerically zoom in on values close to 90. What is the greatest value you can get for tan X on your calculator? How close is X to 90? d. Writing Use right triangles to eplain the behavior of tan X found above. See margin.. Graphing alculator Use the LE and graphing features of your graphing calculator to study the product tan X? tan (90 - X). In the Y= screen, enter Y = tan X? tan (90 - X). a. Use the LSE feature so that X starts at and changes by. ccess the LE. What do you notice? Every Y value. b. Press. What do you notice? he graph is that of Y. Proof c. Make a conjecture about tan X? tan (90 - X) based on parts (a) and (b). Write a paragraph proof of your conjecture. See back of book. Use the given information and tan - to find m& to the nearest whole number.. tan = 9.. tan = (tan ) = tan tan = 0. 0 Simplify each epression.. tan (tan - ). tan - (tan X) mlx y hapter Right riangles and rigonometry. a. No; nswers may vary. Sample: tan ± tan 0 N ± 0.., but tan( ± 0) tan N. b. No; assume tan tan tan( ), or tan tan ± tan ( ). Let ±, so by subst., tan( ± ) tan ± tan. his is false by part (a).. c. nswers may vary. Sample: tan X N,9 for X d. nswers may vary. Sample: In a rt. k, as an acute l approaches 90, the opp. side gets longer.

6 oordinate Geometry You can use the slope of a line to find the measure of the acute angle that the line forms with any horizontal line. slope = rise run = y tan = adjacent = m& = tan - () <. rise o the nearest tenth, find the measure of the acute angle that the line forms with a horizontal line. run. y = +.. y = - 0. y. y = -.. y = -. O. - y = y =. alifornia Resources Standards Mastery alifornia Daily Review ransparencies alifornia Review and Practice Workbook Multiple hoice Practice For alifornia Standards and HSEE utorials, visit PHSchool.com. Web ode: bcq-90 GEOM.0 HSEE MG. 9. In #, cos. Use this information to find tan and sin. tan and sin tan and sin tan and sin tan and sin 0. If ony is driving at 0 mph, what is his approimate speed in kilometers per hour? ( mile. kilometers) D.. 9 Mied Review GO for Help Lesson - Lesson - Lesson -. diagonal of a square is units. Find the length of a side of the square. Leave your answer in simplest radical form.! units he lengths of the sides of a triangle are given. lassify each triangle as acute, right, or obtuse..,,.,, 0. 0.,.,. obtuse acute right. For the kite pictured at the right, give the coordinates of the midpoints of its sides. R(a, b); S(a, b); (c, b); V(c, b) V (c, 0) y (0, b) R S (0, -b) (a, 0) nline Visit: PHSchool.com, Web ode: bca-00