1 / 38

The Sine Rule The Cosine Rule

The Sine Rule The Cosine Rule. Trigonometrical rules for finding sides and angles in triangles which are not right angled. First, a word about labelling triangles……. A. c. B. a. b. The vertices (corners) of a triangle are usually labelled using capital letters, for example A, B, C. C.

kiefer
Download Presentation

The Sine Rule The Cosine Rule

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Sine RuleThe Cosine Rule Trigonometrical rules for finding sides and angles in triangles which are not right angled

  2. First, a word about labelling triangles…… A c B a b The vertices (corners) of a triangle are usually labelled using capital letters, for example A, B, C C The sidesof the triangle are usually labelled using lower case letters, in this case, a, b and c, and are positioned opposite the respective vertices. So IMPORTANT!! Side a will be opposite vertex A Note also that side a could also be called BC as it connects vertex B to vertex C etc…. (We won’t be using this labelling system in this unit of work) Side b will be opposite vertex B Side c will be opposite vertex C

  3. A quick overview of the two rules We will look at the two rules very briefly before starting to use them!

  4. The Sine Rule c A B a b The Sine Rule states that in any triangle ABC…. C This is the general formula for the sine rule. In realityhowever, you will use only two of the three fractions at any one time. So the rule we will be using is More on this later!

  5. The COSine Rule c A B The Cosine Rule states that in any triangle ABC…. a b C This formula has c2 as the subject, but the letters can be interchanged, so it can also be written as Study the patterns and locations of the letters in the three formulae closely. More on the cosine rule later! or

  6. The Sine Rule

  7. Proof of the Sine Rule: Let ABC be any triangle with side lengths a, b, c respectively h C D a b Now draw AD perpendicular to BC, and let the length of AD equal h A B c In BDC In ACD and asin B = b sin A As both expressions are equal to h, we can say which is the Sine Rule Dividing through by (sinA)( sinB)this becomes

  8. Example 1 – Use the Sine Rule to find the value of x in the triangle: 54 x m C A VERY IMPORTANT!! Take time to study the diagram. Note the positions of the three “givens” (actual values you’re told) – the 88, 54 and 12 m, and the one “unknown”, x. 12m 88 B The formula for the sine rule requires • three “givens” (in this case, 88, 54 and 12 m) and one unknown (x) • two of these givens must be an angle and its opposite side (in this case, the 54 and the 12 mwhich we will make our A and a). • the third given (88) and the unknown (x) must also be an angle and its opposite side. Note that the third angle C and its opposite side c are not used in this problem!

  9. 14.82m (looks OK) 54 x C A Now we substitute the 3 givens and the unknown into this formula….. 12m Remember these two “givens” must be an angle and its matching opposite side A = 54 Substituting the values into the formula a = 12 88 B B = 88 These too! b = x Finally, label the x as 14.82 on the diagram and check that your answer fits with the other numbers in the problem! Cross-multiply Divide through by sin 54 to make the subject  x = 14.82 (to 2 dec. pl)

  10. Example 2 – Use the Sine Rule to find the value of x in the triangle: x cm 95 Here, no vertices are labelled so we will have to create our own. But first… 35cm Step 1, check that there are 4 “labels” – i.e. 3 givens and 1 unknown. There are a 95, 22, 35 cm and xcm so this fits our requirements. 22 Step 2, check that 2 of the 3 givens are a matching angle and opposite side. 95 and 35 cm fit this. Also check that the remaining given and the unknown form another matching angle and opposite side (22 and x cm). They do! All our requirements are in place so we can now use the Sine Rule! A = 95 a = 35 B = 22 b = x Step 3, Allocate letters A, a, B, b (or any other letters of your choice) to matching pairs.

  11. A = 95 a = 35 B = 22 b = x A b x cm 95 a 35 cm B 22 x = 13.16(2dec pl) Remember to check that the answer fits the context of the diagram.

  12. Example 3 – Use the Sine Rule to find the value of  in the triangle: 62 4.7m  5.1m • A quick check indicates everything is in place to use the Sine Rule…. • 3 givens and one unknown • One pair of givens (5.1 and 62) form a matching angle and opposite side; and • The other pair (4.7 and ) form the second matching angle and opposite side. Note the third side and angle are unmarked – we don’t use these.

  13. 62 4.7m  5.1m Remember to check that the answer fits the context of the diagram.

  14. Example 4 – Use the Sine Rule to find the value of x in the triangle: 35.7m 33 Looking at the diagram, it seems we have a problem!  79 Although the 68 and 35.7 form a matching angle and opposite side, the 33 and x do not. x m But…remembering the angle sum of a triangle is 180, we can work out the 3rd angle to be 180 – 33 – 68 = 79. 68 So now we use the 79 as the matching angle for the x and proceed as usual, ignoring the 33 which plays no further part. x = 37.80 (2 dec pl)

  15. Example 5 – The “Ambiguous Case”. Draw two different shaped triangles ABC in which c = 14m, a = 10m and A = 32. Hence find the size(s) of angle C. B This process (drawing triangles from verbal data and no diagram) takes time and practice. You need to access these types of problems and practise them thoroughly. Below is one possible diagram: 14m 10m 32 Now extend side AC1 past C1 to the new point C2 where the new length BC2 is the same as it was previously (10m)….. C1 A B The new ABC2 has the same given properties as the original ABC1 . Both triangles have c = 14, a = 10 and A = 32 . But note the angles at C are different! One is acute and the other obtuse. 14m 10m 10m 32 C1 C2 A

  16. TRIANGLE 1 TRIANGLE 2 B 14m B 10m 14m 32 10m C2 C1 A ANGLE C is obtuse 32 ANGLE C is acute C1 A How are the two C angles related? (if at all) Let angle BC2C1 = .  angle BC1C2 = . (isos ) B  angle BC1A = 180 –  (straight line) 10m 14m  180 –  10m  A C2 32 C1 Conclusion: The (green) acute angle at C2 and the (blue) obtuse angle at C1 are supplementary. Thus, for example if one solution is 73 then the other solution is 180 – 73 = 107

  17. Back to the question! B Draw the triangle with the acute, rather than the obtuse, angle at C. 14m 10m 32 C2 A  Applying the Sine Rule, One solution (the acute angle which is the only one given by the calculator) is therefore 47.9 and the second solution (the obtuse angle) is 180 – 47.9 = 132.1 Ans:  = 47.9 or 132.1

  18. The Sine Rule - Summary • The Sine Rule can be used to find unknown sides or angles in triangles. • The Sine Rule formula is • To use the Sine Rule, you must have • A matching angle and opposite side pair (two givens) • A third given and an unknown, which also make an angle and opposite side pair • When asked to find the size of an ANGLE, first check whether the problem could involve the ambiguous case (see Example 5). In that case, the two answers are supplementary – i.e. add to 180 • When confronted with a problem where you have to decide whether to use the Sine Rule or the Cosine Rule, always try for the Sine Rule first, as it is easier. We will have this discussion later! • In every triangle, the largest side is always opposite the largest angle. The side lengths are in the ratio of the sines of their opposite angles.

  19. The Sine Rule - Summary • In every triangle, • The largest side is always opposite the largest angle. • The middle sized side is always opposite the middle sized angle, and • The smallest side is always opposite the smallest angle • The ratio of any two side lengths is always equal to the ratio of the sines of their respective opposite angles. A b c These are just re-shaped versions of the original sine rule formulae. B C a

  20. The Cosine Rule There are two variations of this…. To find a side use To find an angle use c2 = a2 + b2– 2ab cosC These formulae are just rearrangements of each other. Verify this as an exercise.

  21. Proof of the Cosine Rule: Let ABC be any triangle with side lengths a, b, c respectively NOTE!! The expansion (a – x)2 = a2 – 2ax + x2 h A x D a – x c b Now draw AD perpendicular to BC, and let the length of AD equal h C B a Let the length CD = x, and so length BD will be a – x. In ACD In ABD Pythagoras gives In ACD Pythagoras gives (3) (1) (2) The formulae (2) and (3) are both for h2 so we make them equal to each other.

  22. Now cancel the x2on each side and make c2 the subject… (4) From the first box on the previous slide, taking result (1) x = b cosC and substituting this into (4), we get which is a version of the Cosine Rule (for finding a side)

  23. Cosine Rule – Finding a SIDE c 2 = a2 + b2 – 2ab cos C (1) Note the positions of the letters. If the 2ab cosC were missing, this would just be Pythagoras’ Theorem, c2 = a2 + b2 . If the triangle were right angled, then C would be 90 and as cos 90 = 0, it becomes Pythagoras’ Theorem! (2) When c2 is the subject, the only angle in the formula is C (the angle opposite to side c). Note A and B are absent from the formula. (3) The above formula is to find a side length. The letters can be swapped around and the same formula can be written Here are the three variations of the formula shown together. Study them closely and note the patterns! a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c 2 = a2 + b2 – 2ab cos C

  24. Cosine Rule – Finding an ANGLE c 2 = a2 + b2 – 2ab cos C (4) This formula can be rearranged to make cos C the subject, i.e. This is the version of the Cosine Rule to use when FINDING AN ANGLE. (5) Again, the letters can be swapped around and the same formula can be written

  25. When do we use the Cosine Rule? • First, check to see if you can use the Sine Rule. It’s easier! You can use the Cosine Rule when • You are told TWO SIDES and THEIR INCLUDED ANGLE (i.e. the angle between those two sides) and asked to find the THIRD SIDE • You are told ALL THREE SIDES and asked to find any ANGLE OR 8m x 10m  20 cm 9m Here, we use 15 cm Here, we use c 2 = a2 + b2 – 2ab cos C 45

  26. Example 6 – Use the Cosine Rule to find the value of c in the triangle: C Finally, check that c = 3.85 fits the diagram. 65 4 cm 3 cm A B c Note that we have 2 given sides (3 cm and 4 cm) and their included angle (65) so we can use the Cosine Rule for finding a side… c 2 = a2 + b2 – 2ab cos C c 2 = 32 + 42 – 2 × 3× 4 ×cos65 Let a = 3 b = 4 C = 65 c 2 = 14.857 (do in one step on calculator) c = 3.85 (to two dec pl) Ans: The length of the required side is 3.85 cm

  27. Example 7 – Use the Cosine Rule to find the size of Cin the triangle: B Finally, check that C = 51.95 fits the diagram. 8 m 7.5 m C ? 9 m A Note that we have 3 given sides and are asked to find angle at C (opposite 7.5) so we can use the Cosine Rule for finding an angle… Let a = 8 b = 9 c = 7,5 Caution! Here we MUST make c = 7.5 as it is the side opposite the angle we’re finding, i.e. C, whereas a and b are interchangeable. NOTE !! Bracket numerator and denominator when entering into calculator. = 0.6163 Ans: Angle C is equal to 51.95 (to 2 dec pl) or 5157’ (to nearest minute)

  28. Example 8 – Use the Cosine Rule to find the value of x in the triangle: 10 m 100 Finally, check that x = 16.10 fits the diagram. x is the longest side so this would seem reasonable. 11 m x Note that we have 2 given sides (10 m and 11 cm) and their included angle (100) so we use the Cosine Rule for finding a side… c 2 = a2 + b2 – 2ab cos C Let a = 10 b = 11 c = x C = 100 x2 = 102 + 112 – 2 × 10 × 11 ×cos100 x2 = 259.2 (do in one step on calculator) x = 16.10 (to two dec pl) Ans: The length of the required side is 16.10 m

  29. Example 9 – Use the Cosine Rule to find the value of  in the triangle: Finally, check that  = 105 fits the diagram.  LOOKS obtuse so this would seem reasonable. Beware – you can’t always presume the drawings are to scale, so be careful when judging the appropriateness of your answers (in all problems) 40 mm 29 mm  21 mm Note that we have 3 given sides and are asked to find angle opposite to 40 mm so we use the Cosine Rule for finding an angle… Let a = 21 b = 29 c = 40 C =  remember the brackets Note the negative cos. This means our angle is obtuse! ALL OBTUSE ANGLES HAVE A NEGATIVE COSINE! Ans:  is approx. equal to 105.13 (to 2 dec pl) or 1058’ (to nearest min) = 105.13

  30. The Cosine Rule - Summary • The Cosine Rule can be used to find unknown sides or angles in triangles. • There are two versions of the Cosine Rule formula and three variations within each of these, depending on what is required as the subject To find a SIDE To find an ANGLE c 2 = a2 + b2 – 2ab cos C a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B Make sure you familiarise yourself with how the PATTERNS in these configurations work. Also remember each formula on the left is just a rearrangement of its corresponding formula on the right.

  31. The Cosine Rule - Summary • To use the Cosine Rule to find an angle you must be given all three sides • To use the Cosine Rule to find a side you must be given the other two sides and their included angle. • When deciding whether to use the Sine Rule or the Cosine Rule, always try the Sine Rule first, as it is easier (only one formula to deal with). • When dealing with angles in the range 90 <  < 180, i.e. OBTUSE ANGLES, remember that their cosines are negative. This does not apply to their sines – they are still positive.

  32. Mixed examples – which rule to use? Study each of these diagrams and determine which rule to use – Sine Rule or Cosine Rule? If Cosine Rule, which version? Answers & working on next slides. 10 cm 12 cm 16m B x m 14 cm 12 cm A 119 71 C  x cm 35 29 33  x m 6 cm x cm 9 cm 12 cm E D 13m 80 F 11m 67 9 cm

  33. Example 10 First check to see if we can use the Sine Rule. We have a given angle and opposite side (35 and 16m), and the unknown x and the other given (71) also form a matching angle and opposite pair. So we can use the SINE RULE 16m x m A 71 35 Ans: the length of side x is 26.38 m approximately. Remember to check appropriateness of your answer! to two dec pl.

  34. Example 11 First check to see if we can use the Sine Rule. We are not given any angle so we can’t use the Sine Rule so we have to use the COSINE RULE – the angle version 10 cm B 14 cm 12 cm  Ans: the size of angle  is 44.42 or 4425’ approx. Let…. C =  c = 10 a = 12 b = 14 Remember to check appropriateness of your answer!

  35. Example 12 First check to see if we can use the Sine Rule. 32 We have a given angle and opposite side (29 and 12cm), but the unknown x and the other given (119) are NOT a matching angle and opposite pair. BUT…the third angle is 180 – 119 – 29 = 32 so we can use the SINE RULE 12 cm 119 C x cm 29 Ans: the length of side x is 13.12 cm approximately. Let…. a = x A = 32 b = 12 B = 29 Remember to check appropriateness of your answer! to two dec pl.

  36. Example 13 First check to see if we can use the Sine Rule. We are not given any angle and matching opposite side so we can’t use the Sine Rule, so we have to use the COSINE RULE – the side version c 2 = a2 + b2 – 2ab cos C x2 = 112 + 132 – 2 × 11 × 13 × cos67 x2 = 178.251 Let…. C = 67 c = x a = 11 b = 13 Ans: the size of side xis 13.35 m (to 2 dec places) x m x = 13.35 D 13m 11m 67 Remember to check appropriateness of your answer!

  37. Example 14 First check to see if we can use the Sine Rule. We have a given angle and opposite side (80 and 9 cm), but the unknown  and the other given (6 cm) are NOT a matching angle and opposite side. HOWEVER…we can use the SINE RULE to find the third angle  (which forms a matching pair with the 6cm) then use the 180 rule to find   Ans: the size of angle  is approx. 58.96 or 5858’ Let…. a = 6 A =  b = 9 B = 80  6 cm 9 cm E 80 Remember to check appropriateness of your answer!

  38. Example 15 First check to see if we can use the Sine Rule. We have a given angle and opposite side (33 and 9 cm), but the unknown x and the other given (12 cm) are insufficient data for Sine Rule. The Cosine Rule won’t work either as the triangle’s data does not match either of the two configurations for the Cosine Rule. HOWEVER…if we let be the angle opposite the 12cm we then have a second matching pair and can begin with using the SINE RULE to find angle . (This is PART 1)   NOW FOR PART 2 …..Once we know  we can then find the third angle  (which is opposite to x) and then apply the Sine Rule a second time to find x. 33 Part 1 (finding ) Part 2 (finding x) x cm Finding  12 cm  = 180 – 33 – 46.57 F  = 100.43 Note!! Here the diagram is quite out of scale. This becomes apparent on checking the reasonableness of your answer 9 cm

More Related