Math 2201 Unit 3: Acute Triangle Trigonometry. Ch. 3 Notes

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1 Rea Learning Goals, p. 17 text. Math 01 Unit 3: ute Triangle Trigonometry h. 3 Notes 3.1 Exploring Sie-ngle Relationships in ute Triangles (0.5 lass) Rea Goal p. 130 text. Outomes: 1. Define an aute triangle. See notes. Make a onjeture about the relationship between the length of the sies of an aute triangle an the sine of the opposite angles. p Derive the Law of Sines (Sine Law). p. 130 This hapter is about solving aute triangles. This means fining the lengths of the missing sies an/or the measures of the missing angles. We will examine two ways to solve aute triangles the Law of Sines (Sine Law) an the Law of osines (osine Law). In Math 101, we labele the sies of a right triangle with respet to the angles in the triangle. With respet to aute angle, these sies were alle the ajaent sie an the opposite sie. The sie opposite the right angle was alle the hypotenuse. hypotenuse opposite ajaent Reall from Math 101 that you were introue to three speial ratios that were obtaine from the lengths of the three sies of a right triangle an that we gave them speial names. Ratio opposite hypotenuse ajaent hypotenuse opposite ajaent Speial Name sine ratio (sin) osine ratio (os) tangent ratio (tan) 1

Up to now, we have use the Pythagorean Theorem an the primary trigonometri ratios to solve a right triangle. E.g.: Solve the right triangle below.

2 In the right triangle on page 1, remember the name of the Inian below.. You an remember these ratios if you n Def Solving a triangle means fining the length(s) of the missing sie(s) an/or the measure(s) of the missing angle(s). Up to now, we have use the Pythagorean Theorem an the primary trigonometri ratios to solve a right triangle. E.g.: Solve the right triangle below. 104m 40m x Sine we know the lengths of two sies of the triangle an sine it is a right triangle, we an use the Pythagorean Theorem to fin the length of the thir sie. Now we must use the trig ratios to fin one of the missing angles. We will use the lengths of the two given sies just in ase we mae a mistake fining the length of the missing sie. With respet to angle, we are given the lengths of the opposite sie an the hypotenuse so we will use the sine ratio. Make sure your graphing alulator is set to egree moe an not raian moe.

3 40 sin sin Sine m.6, then m Solving a triangle using the Pythagorean Theorem an primary trigonometri ratios only works with a RIGHT triangle. How o we solve triangles that o NOT have a right angle? Triangles that o not have a right angle are either aute triangles or obtuse triangles. In this hapter we will eal with aute triangles only. n Def :n aute triangle is a triangle in whih all the angles have a measure less than 90. Labeling ny Triangle a b We label eah vertex with an UPPERSE letter an the sie opposite the vertex with the orresponing lowerase letter. The Law of Sines (Sine Law) an be use to solve SOME aute triangles. onjeture about the Relationship between the Length of the Sies of an ute Triangle an the Sine of the Opposite ngles. Measure eah sie to the nearest tenth of a entimeter an eah angle to the nearest tenth of a egree an omplete the table. 3

4 Measure (1 eimal plae) Sie a Sie b Sie Length (m) (1 eimal plae) alulate (4 eimal plaes) sin a sin b sin Make a onjeture about the relationship between the length of the sies of an aute triangle an the sine of the opposite angles. onjeture: In an aute triangle. Derivation of the Law of Sines (Sine Law) Let s raw any a ute triangle with an altitue (h) rawn from a vertex. h b a D h Using the sine ratio we an write sin or h sin. Similarly, we an write h sin or h bsin. b Sine we have two expressions equal to h, we an substitute to get sin bsin. Sine b 0, we an ivie both sies by b to get sin b sin b b sin sin b 4

5 Now let s use the sa e triangle above but raw a ifferent altitue. h b a D h Using the sine ratio we an write sin or h asin. Similarly, we an write a h sin or h bsin. b Sine we have two expressions equal to h, we an substitute to get asin bsin. Sine ab 0, we an ivie both sies by ab to get a sin b sin ab ab sin sin b a Sine sin sin an b sin sin then b a sin sin sin ************* ***************** a b This is the Sine Law. 5

6 3. pplying the Sine Law ( lasses) Rea Goal p. 13 text. Outomes: 1. Use the Sine Law to fin the length of a missing sie in an aute triangle. p Use the Sine Law to fin the measure of a missing angle in an aute triangle. p Solve problems using the Sine Law. pp E.g.: Solve the triangle below m b 39 Substituting into sin sin gives, a sin39 sin5 11. ross multiplying an solving gives sin sin 5 sin sin 5 sin39 sin m Sine m 39 an m 5, then m Substituting into sin sin gives, a b sin 39 sin b ross multiplying an solving gives b sin sin 89 b sin sin 89 sin39 sin39 b 17.8m 6

7 Generally, there are two ases when you an use the Sine Law. E.g.: Fin the value of x in the iagram to the right. Substituting into sin sin gives a b sin 63 sin x x sin 63 10sin 49 x sin 63 10sin 49 sin 63 sin 63 x 8.5m Do # s a, 3 a, b,, 8, pp text in your homework booklet. E.g.: Fin the value of x in the iagram to the right. Substituting into sin sin gives a b 7

sin 50 sin x 11 7 7sin 50 11sin x 7sin 50 11sin x 11 11 sin x 0.48748874... x 9. Do # s b, 3, e, f, 7, 11, pp. 138-140 text in your homework booklet.

8 sin 50 sin x sin 50 11sin x 7sin 50 11sin x sin x x 9. Do # s b, 3, e, f, 7, 11, pp text in your homework booklet. Problem Solving Using the Sine Law. We nee to fin the values of a an b. Substituting into sin sin gives b sin 47 sin 68 a 46 46sin 47 a sin 68 46sin 47 a sin 68 sin 68 sin 68 a 36.3m Substituting into sin sin gives a 8

sin 65 sin 68 b 46 46sin 65 b sin 68 46sin 65 b sin 68 sin 68 sin 68 b 45.0m So 46 36.3 45.0 17.3m of hain-link fene is neee to enlose the entire park.

9 sin 65 sin 68 b 46 46sin 65 b sin 68 46sin 65 b sin 68 sin 68 sin 68 b 45.0m So m of hain-link fene is neee to enlose the entire park. Substituting into sin Q sin R gives q r sin 60 sin R sin sin R 13sin sin R sin R R 35.3 m P h sin h 184.5sin ft 9

10 Fin the value of x. First let s look at the triangle to the right. Substituting into sin Z z sin 30 sin10 z 50 50sin 30 z sin10 50sin 30 z sin10 sin10 sin10 z 144.0ft sin gives feet 10 z Using right triangle trigonometry, we an write x os x 144os ft Do # s 4, 10, 13-15, pp text in your homework booklet. Do # s 4-7, 9 a, p. 143 text in your homework booklet. 10

11 3.3 Deriving an pplying the osine Law ( lasses) Rea Goal p. 144 text. Outomes: 1. Illustrate an explain situations where the Sine Law annot be use to fin a missing sie or a missing angle in a triangle. p Derive the Law of osines (osine Law). p. 144 The Law of Sines N be use given two sies an the non-inlue angle. The Law of Sines NNOT be use in the following situations: 1. Given two sies an the inlue angle m sin 5 sin m This is impossible to solve sine there are two unknowns in the equation.. Given three sies sin sin This is impossible to solve sine there are two unknowns in the equation. 3. Given three angles. 5 a 35 sin93 sin5 a This is impossible to solve sine there are two unknowns in the equation. 93 Therefore, we nee another formula that will enable us to solve a triangle in two of those situations. This law is the Law of osines. 11

12 Derivation of the Law of osines (osine Law) h a P b Using right triangle P an the Pythagorean Theorem, we an write Similarly, using right triangle P an the Pythagorean Theorem, we an write Sine the expressions on the left han sie (LHS) of equation 1 an equation equal to eah other. y substitution,, they must be equal Solving for gives ut m is not a sie of the triangle. Using right triangle P again an the trig ratios, we an write Therefore, substituting for m in we get, This is the Law of osines. ******* Note that the Law of osines an be written in three ways

13 E.g.: Solve the following triangle. This is not a right triangle an we annot use the Law of Sines so we must use the Law of osines to solve the triangle. Now we have the hoie of using the Law of Sines or the Law of osines to fin the measure of angle. We will use the Law of osines again for pratie. The measure of angle is Like the Sine Law, we an use the osine Law to fin the length of a missing sie or the measure of a missing angle. 13

E.g.: How long is the tunnel in the iagram to the right? Substituting into a b b os gives 300 00 300 00 os80 90000 40000 10000os80 130000 10000os80 130000 0837.7813... 10916.187... 330.4 330.

14 E.g.: How long is the tunnel in the iagram to the right? Substituting into a b b os gives os os os ft So the length of the tunnel is about 330.4ft long. Draw a triangle whih illustrates the information in the workings to the right. Do # s, 4 a, 6 a, p. 151 text in your homework booklet. E.g.: Fin the value of. Substituting into a b b os gives os os os os os os os os Do # s 3, 5 a, 6, 7 a,, pp text in your homework booklet. 14

Problem Solving Using the osine Law. E.g.: irraft 1 flies at 400km/h an airraft flies at 350km/h. If the angle between their paths is 49, how far apart are the airraft after h?

15 Problem Solving Using the osine Law. E.g.: irraft 1 flies at 400km/h an airraft flies at 350km/h. If the angle between their paths is 49, how far apart are the airraft after h? Substituting into a b b os gives os os os km The airraft are about 68.66km apart. E.g.: How far apart are the two trees in the iagram to the right. Substituting into a b b os gives os os os m The trees are about 53.89m apart. 15

16 E.g.: What is the angle between the ft an the 18ft sies? Substituting into a b b os gives os os os b os os os os os The angle between the ft an 18ft sies is about

17 Errors Using the osine Law E.g.: Fin the error in the solution below an orret it. Step 1: os35 Step : os35 Step 3: os35 Step 4: 65os35 Step 5: Step 6: Step 7:.63 Step 8:.63m The error ours in Step where. In this step,. Do # s 9, 13, pp text in your homework booklet. Do # s 1, 5, 6, 7, 9, 13, pp text in your homework booklet. Do # s 1-7, 9, 10 p. 168 text in your homework booklet. Formulae Pythagorean Theorem a b Primary Trigonometri Ratios sin o ; os a ; tan o h h a sin sin sin Sine Law a b a b b os osine Law b a a os a b ab os 17

Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using

Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric