4.2 Proving and Applying

Transcription

1 YOU WILL NEED alulator ruler EXPLORE 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles An isoseles obtuse triangle has one angle that measures 120 and one side length that is 5 m. What ould the other side lengths be? GOAL Eplain steps in the proof of the sine and osine laws for obtuse triangles, and apply these laws to situations that involve obtuse triangles. INVESTIGATE the Math In Lesson 3.2, you analyzed en s proof of the sine law for aute triangles. en wanted to adjust his proof to show that the sine law also applies to obtuse triangles. Consider en s new proof: Step 1 I drew obtuse triangle AC with height AD. Step 2 I wrote equations for sin (180 2 / AC ) and sin C using the two right triangles. In ^ AD, opposite sin (180 2 / AC) 5 hypotenuse A D b In ^ ACD, opposite sin C 5 hypotenuse a C sin (180 2 / AC) 5 AD sin (180 2 / AC) 5 AD sin / AC 5 AD sin C 5 AD b b sin C 5 AD Step 3 oth epressions for AD equal eah other (transitive property), so: sin / AC 5 b sin C 1 sin /AC2 5 b sin C sin C 5 b sin /AC Step 4 I drew a new height, h, from to base b in the triangle. In ^ AE, In ^ CE, sin A 5 h sin C 5 h a sin A 5 h a sin C 5 h D Step 5 oth epressions for h equal eah other, so: sin A 5 a sin C sin C 5 a sin A A b E h a C 164 Chapter 4 Oblique Triangle Trigonometry NEL

2 ? I have already shown that sin C 5 b sin /AC, so sin C 5 b sin /AC 5 a sin A How an you eplain what en did to prove the sine law for obtuse triangles? A. Why did en hoose to write epressions for the sin (180 2 / AC) and sin C?. In step 3, en mentions the transitive property. What is this property, and how did he use it in this step? C. In step 4, en drew a new height in ^ AC. Why was this neessary? D. Why was en able to equate all three side angle ratios in step 5? Refleting E. Compare the proof above to en s original proof in Lesson 3.2, pages 118 to 119. How is the proof of the sine law for obtuse triangles the same as that for aute triangles? How is it different? F. If en started his proof by writing epressions for sin (180 2 / CA) and sin A, where would he have drawn the height in step 1? APPLY the Math eample 1 Use reasoning and the sine law to determine the measure of an obtuse angle In an obtuse triangle, / measures 23.0 and its opposite side, b, has a length of 40.0 m. Side a is the longest side of the triangle, with a length of 65.0 m. Determine the measure of / A to the nearest tenth of a degree. ijan s Solution C b 40.0 m a 65.0 m 23 A I drew an obtuse triangle to represent ^ AC. I knew that the longest side is always opposite the largest angle, so the 65.0 m side must be opposite the obtuse angle, / A. Sine ^ AC is not a right triangle, I knew that I ould not use the primary trigonometri ratios to determine the measure of /A. NEL 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles 165

3 sin A a sin A sin b 5 sin a sin A 23 b 5 asin b65.0 sin A / A 5 sin 1 ( ) / A / A / A / A measures I notied that the diagram has two side-angle pairs with only one unknown, /A. I deided to use the sine law. The measure of an angle is the unknown, so I used the form of the sine law that has the angles in the numerator. I isolated sin A. I used the inverse sine to determine the measure of / A. My reasoning suggests that / A must be the obtuse angle. I used the relationship sin A 5 sin (180 2 A). The measure of the angle seems appropriate, aording to my diagram. Your Turn Determine the length of side A in ^ AC above, to the nearest tenth of a entimetre. eample 2 Solving a problem using the sine law Colleen and Juan observed a tethered balloon advertising the opening of a new fitness entre. They were 250 m apart, joined by a line that passed diretly below the balloon, and were on the same side of the balloon. Juan observed the balloon at an angle of elevation of 7 o while Colleen observed the balloon at an angle of elevation of 82 o. Determine the height of the balloon to the nearest metre. Colleen s Solution J m C D 82 I drew a diagram to represent the situation. The height of the balloon is represented by D. I need to determine the length of C in order to determine the length of D. I an use the sine law in JC. 166 Chapter 4 Oblique Triangle Trigonometry NEL

4 /CJ /CJ 5 98 /JC /JC 5 75 C sin /JC 5 JC sin /JC C sin sin C 5 sin 17 2 a C sin 1/CD2 5 D C sin (82 ) 5 ( ) (sin (82 )) 5 D m 5 D D sin b The advertising balloon is 31 m above the ground. I determined the supplement of 82 to determine the measure of a seond angle in ^ JC. This is an obtuse triangle. I determined the measure of the third angle in JC. This gave me a known side, JC, and a known angle opposite this side, /JC, in this triangle. I used the sine law to write an equation that involved C and the known side-angle pair. I substituted the known information into the equation and solved for C. I wrote an equation that involved D, C, and the known angle in CD. I substituted the known information into the equation and solved for D. Your Turn Determine the distane between Juan and the balloon. eample 3 Use reasoning to demonstrate the osine law for obtuse triangles Show that the osine law holds for obtuse triangles, using ^ AC. a Hyun Yoon s Solution A b C A a h AC b C D 180 AC I etended the base of the triangle to D. This reated two overlapping right triangles, ^CD and ^ AD, with height D. It also reated two angles at C, / AC and /DC, suh that /DC / AC. NEL 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles 167

5 In ^ AD In ^CD h (b 1 ) 2 h 2 5 a (b 1 ) 2 5 a (b 1 ) 2 1 a b 2 1 2b a a 2 1 b 2 1 2b os (180 2 / AC) 5 a a os (180 2 / AC) a 2 1 b 2 1 2b[a os (180 2 / AC)] 2 5 a 2 1 b 2 2 2ab os / AC I have demonstrated the osine law. I used the Pythagorean theorem to write two epressions for h 2, using the two right triangles. The epressions that equal h 2 equal eah other (transitive property). I solved for 2. The aute angle in ^CD has a measure of / AC. I used the osine ratio to write an epression for. I substituted my epression for into my equation. To write an equation that ontained only measures found in the original triangle, I used the following fat: os (180 2 / AC) 5 2os / AC Your Turn Review the proof of the osine law for aute triangles in Lesson 3.3, pages 130 to 131. Eplain how Hyun Yoon modified this proof to deal with an obtuse triangle. eample 4 Using reasoning and the osine law to determine the measure of an obtuse angle The roof of a house onsists of two slanted setions, as shown. A roofing ap is being made to fit the rown of the roof, where the two slanted setions meet. Determine the measure of the angle needed for the roofing ap, to the nearest tenth of a degree ft roofing ap 33.5 ft 20.3 ft Maddy s Solution: Substituting into the osine law and then rearranging a 17.0 ft 33.5 ft b 20.3 ft I skethed a triangle to represent the problem situation. The largest angle is u, beause it is opposite the longest side. 2 5 a 2 1 b 2 2 2ab os u Three side lengths are given, so I knew that I ould use the osine law. 168 Chapter 4 Oblique Triangle Trigonometry NEL

6 (33.5) 2 5 (17.0) 2 1 (20.3) 2 2 2(17.0)(20.3) os u (33.5) 2 2 (17.0) 2 2 (20.3) (17.0)(20.3) os u I substituted the known values into the formula for os u the osine law and isolated u os u os 1 a b 5 u u An angle of is needed for the roofing ap. My answer is reasonable, given the diagram. Georgia s Solution: Rearranging the osine law before substituting a 17.0 ft 33.5 ft b 20.3 ft I skethed a triangle to represent the problem situation. 2 5 a 2 1 b 2 2 2ab os u 2 1 2ab os u 5 a 2 1 b 2 2 2ab os u 1 2ab os u ab os u 5 a 2 1 b ab os u 5 a2 1 b ab 2ab os u5 a2 1 b ab os u os u u 5 os 21 ( ) u I knew the lengths of all three sides, so I used the osine law. Sine I wanted to solve for u, I rearranged the formula to isolate os u. I substituted the values of a, b, and into the rearranged formula. The angle for the roofing ap should measure Your Turn Determine the angle of elevation for eah roof setion, to the nearest tenth of a degree. NEL 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles 169

7 In Summary Key Idea The sine law and osine law an be used to determine unknown side lengths and angle measures in obtuse triangles. Need to Know The sine law and osine law are used with obtuse triangles in the same way that they are used with aute triangles. Use the sine law when you know - the lengths of two sides and the measure of the angle that is opposite a known side Use the osine law when you know - the lengths of two sides and the measure of the ontained angle - the measures of two angles and the length of any side - the lengths of all three sides or e areful when using the sine law to determine the measure of an angle. The inverse sine of a ratio always gives an aute angle, but the supplementary angle has the same ratio. You must deide whether the aute angle, u, or the obtuse angle, u, is the orret angle for your triangle. eause the osine ratios for an angle and its supplement are not equal (they are opposites), the measures of the angles determined using the osine law are always orret. CHECK Your Understanding 1. There are errors in eah appliation of the sine law or osine law. Identify the errors. a) b) m 12 m m 5 sin sin (12)() os Chapter 4 Oblique Triangle Trigonometry NEL

8 2. Whih law ould be used to determine the unknown angle measure or side length in eah triangle? For your answer, hoose one of the following: sine law, osine law, both, neither. Eplain your hoie. a) d) 12 m in. 15 m 35 b) e) 7 m 5 m 3 m ) 25 m m PRACTISING 3. Determine the unknown side length in eah triangle, to the nearest tenth of a entimetre. a) b) ) 24.0 m m 2.0 m m 1.4 m 4. Determine the unknown angle measure in eah triangle, to the nearest degree. a) 44 m b) ) m 106 m 68 m 5 m 180 m m NEL 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles 171

9 5. Determine eah unknown angle measure to the nearest degree and eah unknown side length to the nearest tenth of a entimetre. a) L ) A 5 m C 7.5 m 105 M 11.2 m N 8 m 10 m b) R d) X 25.6 m S T Y m 35 Z 6. A triangle has side lengths of 4.0 m, 6.4 m, and 9.8 m. a) Sketh the triangle, and estimate the measure of the largest angle. b) Calulate the measure of the largest angle to the nearest tenth of a degree. ) How lose was your estimate to the angle measure you alulated? How ould you improve similar estimates in the future? 7. Wei-Ting made a mistake when using the osine law to determine the unknown angle measure below. Identify the ause of the error message on her alulator. Then determine u to the nearest tenth of a degree. Q 12.8 m R 20.5 m 10.2 m S (10)(12) os u os u os u os u os u os 1 (100) 5 u <error!> 5 u 8. In ^QRS, q m, r m, and s m. Solve ^QRS by determining the measure of eah angle to the nearest tenth of a degree. T 23 H 9. While golfing, Sahar hits a tee shot from T toward a hole at H. Sahar hits the ball at an angle of 23 to the hole and it lands at. The soreard says that H is 295 yd from T. Sahar walks 175 yd to her ball. How far, to the nearest yard, is her ball from the hole? 172 Chapter 4 Oblique Triangle Trigonometry NEL

10 10. The posts of a hokey goal are 6 ft apart. A player attempts to sore by shooting the puk along the ie from a point that is 21 ft from one post and 26 ft from the other post. Within what angle, u, must the shot be made? Epress your answer to the nearest tenth of a degree. 11. In ^DEF, /E 5 136, e m, and d m. Solve the triangle. Round eah angle measure or side length to the nearest tenth. F E 68.4 m m D 12. A 15.0 m telephone pole is beginning to lean as the soil erodes. A able is attahed 5.0 m from the top of the pole to prevent the pole from leaning any farther. The able is seured 10.2 m from the base of the pole. Determine the length of the able that is needed if the pole is already leaning 7 from the vertial. 13. A building is observed from two points, P and Q, that are m apart. The angles of elevation at P and Q measure 40 and 32, as shown. Determine the height, h, of the building to the nearest tenth of a metre. 14. A surveyor in an airplane observes that the angles of depression to points A and, on opposite shores of a lake, measure 32 and 45, as shown. Determine the width of the lake, A, to the nearest metre m A able 10.2 m 5.0 m 15.0 m h Q P m Closing 15. In ^PQR, /Q is obtuse, /R 5 12, q m, and r m. Eplain to a lassmate the steps required to determine the measure of /Q. Etending 16. Two roads interset at an angle of 15. Darryl is standing on one of the roads, 270 m from the intersetion. a) Create a problem that must be solved using the sine law. Inlude a sketh and a solution. b) Create a problem that must be solved using the osine law. Inlude a sketh and a solution. 17. The interior angles of a triangle measure 120, 40, and 20. The longest side of the triangle is 10 m longer than the shortest side. Determine the perimeter of the triangle, to the nearest entimetre. NEL 4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles 173