The Tangent Ratio
9.4The Tangent RatioEssential QuestionHow is a right triangle used to find the tangentof an acute angle? Is there a unique right triangle that must be used?BoppositeLet ABC be a right triangle with acute A.The tangent of A (written as tan A) is definedas follows.BClength of leg opposite Atan A ——— —length of leg adjacent to A ACACadjacentCalculating a Tangent RatioWork with a partner. Use dynamic geometry software.— to form righta. Construct ABC, as shown. Construct segments perpendicular to ACtriangles that share vertex A and are similar to ABC with vertices, as A(0, 0)B(8, 6)C(8, 0)Anglem BAC 36.87 8b. Calculate each given ratio to complete the table for the decimal value of tan A foreach right triangle. What can you conclude?ATTENDINGTO PRECISIONTo be proficient in math,you need to expressnumerical answers witha degree of precisionappropriate for theproblem AH—PIAI—QJAJ—tan AUsing a CalculatorWork with a partner. Use a calculator that has a tangent key to calculate the tangentof 36.87 . Do you get the same result as in Exploration 1? Explain.Communicate Your Answer3. Repeat Exploration 1 for ABC with vertices A(0, 0), B(8, 5), and C(8, 0).—Construct the seven perpendicular segments so that not all of them intersect ACat integer values of x. Discuss your results.4. How is a right triangle used to find the tangent of an acute angle? Is there aunique right triangle that must be used?Section 9.4int math2 pe 0904.indd 541The Tangent Ratio5411/30/15 11:38 AM
9.4 LessonWhat You Will LearnUse the tangent ratio.Solve real-life problems involving the tangent ratio.Core VocabulVocabularylarrytrigonometric ratio, p. 542tangent, p. 542angle of elevation, p. 544READINGRemember the followingabbreviations.tangent tanopposite opp.adjacent adj.Using the Tangent RatioA trigonometric ratio is a ratio of the lengths of two sides ina right triangle. All right triangles with a given acute angle aresimilar by the AA Similarity Theorem. So, JKL XYZ,KL JLand you can write — —. This can be rewritten asYZ XZKL YZ— —, which is a trigonometric ratio. So, trigonometricJLXZratios are constant for a given angle measure.KJLYZXThe tangent ratio is a trigonometric ratio for acute anglesthat involves the lengths of the legs of a right triangle.Core ConceptTangent RatioLet ABC be a right triangle with acute A.Blegopposite AThe tangent of A (written as tan A) is definedas follows.BClength of leg opposite Atan A ——— —length of leg adjacent to A AChypotenuseCleg adjacentto AAIn the right triangle above, A and B are complementary. So, B is acute. You canuse the same diagram to find the tangent of B. Notice that the leg adjacent to A isthe leg opposite B and the leg opposite A is the leg adjacent to B.ATTENDING TOPRECISIONFinding Tangent RatiosUnless told otherwise, youshould round the valuesof trigonometric ratiosto four decimal placesand round lengths to thenearest tenth.SFind tan S and tan R. Write each answer as afraction and as a decimal rounded to four places.8218SOLUTIONT80RRT 80 40opp. Stan S — — — — 4.4444adj. to S ST 189opp. RST 189tan R — — — — 0.2250adj. to R RT 80 40Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind tan J and tan K. Write each answer as a fraction and as a decimal roundedto four places.1.K40J542Chapter 9int math2 pe 0904.indd 5423224L2.L815J17KRight Triangles and Trigonometry1/30/15 11:38 AM
Finding a Leg LengthFind the value of x. Round your answer to the nearest tenth.11SOLUTION32 xUse the tangent of an acute angle to find a leg length.opp.tan 32 —adj.11tan 32 —xx tan 32 11USING TOOLSSTRATEGICALLYYou can also use the Table ofTrigonometric Ratios availableat BigIdeasMath.com to findthe decimal approximationsof trigonometric ratios. Write ratio for tangent of 32 .Substitute.Multiply each side by x.11x —tan 32 x 17.6Divide each side by tan 32 .Use a calculator.The value of x is about 17.6.STUDY TIPThe tangents of all 60 angles are the sameconstant ratio. Any righttriangle with a 60 anglecan be used to determinethis value.You can find the tangent of an acute angle measuring 30 , 45 , or 60 by applying whatyou know about special right triangles.Using a Special Right Triangle to Find a TangentUse a special right triangle to find the tangent of a 60 angle.SOLUTIONStep 1 Because all 30 -60 -90 triangles are similar, you can simplify yourcalculations by choosing 1 as the length of the shorter leg. Use the 30 -60 -90 Triangle Theorem to find the length of the longer leg. —longer leg shorter leg 3 — 1 3— 330 -60 -90 Triangle TheoremSubstitute.1Simplify.60 3Step 2 Find tan 60 .opp.tan 60 —adj.— 3tan 60 —1—tan 60 3Write ratio for tangent of 60 .Substitute.Simplify.—The tangent of any 60 angle is 3 1.7321.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comFind the value of x. Round your answer to the nearest tenth.3.x4.61 x1356 225. WHAT IF? In Example 3, the length—of the shorter leg is 5 instead of 1. Show thatthe tangent of 60 is still equal to 3 .Section 9.4int math2 pe 0904.indd 543The Tangent Ratio5431/30/15 11:38 AM
Solving Real-Life ProblemsThe angle that an upward line of sight makes with a horizontal line is called the angleof elevation.Modeling with MathematicsYou are measuring the height of a spruce tree. You stand 45 feet from the base of thetree. You measure the angle of elevation from the ground to the top of the tree to be59 . Find the height h of the tree to the nearest foot.h ft59 45 ftSOLUTION1. Understand the Problem You are given the angle of elevation and the distancefrom the tree. You need to find the height of the tree to the nearest foot.2. Make a Plan Write a trigonometric ratio for the tangent of the angle of elevationinvolving the height h. Then solve for h.3. Solve the Problemopp.tan 59 —adj.htan 59 —4545 tan 59 h Write ratio for tangent of 59 .Substitute.Multiply each side by 45.74.9 hUse a calculator.The tree is about 75 feet tall.4. Look Back Check your answer. Because 59 is close to 60 , the value of h shouldbe close to the length of the longer leg of a 30 -60 -90 triangle, where the lengthof the shorter leg is 45 feet. —longer leg shorter leg 3 — 45 3 77.930 -60 -90 Triangle TheoremSubstitute.Use a calculator.The value of 77.9 feet is close to the value of h.Monitoring Progressh in. Help in English and Spanish at BigIdeasMath.com6. You are measuring the height of a lamppost. You stand 40 inches from the base of70 40 in.544Chapter 9int math2 pe 0904.indd 544the lamppost. You measure the angle of elevation from the ground to the top of thelamppost to be 70 . Find the height h of the lamppost to the nearest inch.Right Triangles and Trigonometry1/30/15 11:38 AM
Exercises9.4Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE The tangent ratio compares the length of to the length of .2. WRITING Explain how you know the tangent ratio is constant for a given angle measure.Monitoring Progress and Modeling with MathematicsIn Exercises 3–6, find the tangents of the acute anglesin the right triangle. Write each answer as a fractionand as a decimal rounded to four decimal places.(See Example 1.)3.4.RT5.6.G 1 J5FL18tan 55 —11.011.055 21.9CIn Exercises 13 and 14, use a special right triangleto find the tangent of the given angle measure.(See Example 3.)13. 45 14. 30 5321830 A25DSB72445 E532812.15. MODELING WITH MATHEMATICSJHK34In Exercises 7–10, find the value of x. Round youranswer to the nearest tenth. (See Example 2.)7.8.1527 12x41 x9.16. MODELING WITH MATHEMATICS Scientists can10.22A surveyor is standing 118 feetfrom the base of the WashingtonMonument. The surveyormeasures the angle of elevationfrom the ground to the top ofthe monument to be 78 . Findthe height h of the WashingtonMonument to the nearest foot.(See Example 4.)x637 x58 measure the depths of craters on the moon by lookingat photos of shadows. The length of the shadow castby the edge of a crater is 500 meters. The angle ofelevation of the rays of the Sun is 55 . Estimate thedepth d of the crater.ERROR ANALYSIS In Exercises 11 and 12, describe theSun’s rayerror in the statement of the tangent ratio. Correct theerror if possible. Otherwise, write not possible.11. 55 55 D12E3735F35tan D —37d500 m17. USING STRUCTURE Find the tangent of the smalleracute angle in a right triangle with side lengths 5, 12,and 13.Section 9.4int math2 pe 0904.indd 545The Tangent Ratio5451/30/15 11:38 AM
18. USING STRUCTURE Find the tangent of the larger24. THOUGHT PROVOKING To create the diagramacute angle in a right triangle with side lengths 3, 4,and 5.below, you begin with an isosceles right triangle withlegs 1 unit long. Then the hypotenuse of the firsttriangle becomes the leg of a second triangle, whoseremaining leg is 1 unit long. Continue the diagramuntil you have constructed an angle whose tangent1is —— . Approximate the measure of this angle. 619. REASONING How does the tangent of an acuteangle in a right triangle change as the angle measureincreases? Justify your answer.20. CRITICAL THINKING For what angle measure(s) is thetangent of an acute angle in a right triangle equal to 1?greater than 1? less than 1? Justify your answer.11121. MAKING AN ARGUMENT Your family room has asliding-glass door. You want to buy an awning for thedoor that will be just long enough to keep the Sun outwhen it is at its highest point in the sky. The angle ofelevation of the rays of the Sun at this point is 70 ,and the height of the door is 8 feet. Your sister claimsyou can determine how far the overhang shouldextend by multiplying 8 by tan 70 . Is your sistercorrect? Explain.25. PROBLEM SOLVING Your class is having a classpicture taken on the lawn. The photographer ispositioned 14 feet away from the center of the class.The photographer turns 50 to look at either end ofthe class.Sun’s ray8 ftt10 14 ft50 50 10 70 70 a. What is the distance between the ends of the class?b. The photographer turns another 10 either way tosee the end of the camera range. If each studentneeds 2 feet of space, about how many morestudents can fit at the end of each row? Explain.22. HOW DO YOU SEE IT? Write expressions for thetangent of each acute angle in the right triangle.Explain how the tangent of one acute angle is relatedto the tangent of the other acute angle. What kind ofangle pair is A and B?26. PROBLEM SOLVING Find the perimeter of the figure,where AC 26, AD BF, and D is the midpointof —AC .BaCcAbBAHE23. REASONING Explain why it is not possible to find thetangent of a right angle or an obtuse angle.50 DF35 GCMaintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsFind the value of x. (Section 9.2)27.x3546Chapter 9int math2 pe 0904.indd 54628.29.730 560 x45 xRight Triangles and Trigonometry1/30/15 11:38 AM
Solve real-life problems involving the tangent ratio. Using the Tangent Ratio A trigonometric ratio is a ratio of the lengths of two sides in a right triangle. All right triangles with a given acute angle are similar by the AA Similarity Theorem. So, JKL XYZ, and you can write
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1 SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMS Tangent Problem Definition (Secant Line & Tangent Line) Example 1 Find the equation of the tangent lines to the curve 1 3x 2 y at the
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