The Sine Rule The Cosine Rule

The Sine RuleThe Cosine RuleTrigonometrical rules for finding sides and angles in triangles which are not right angledABCFirst, a word about labelling trianglesThe vertices (corners) of a triangle are usually labelled using capital letters, for example A, B, CThe sides of the triangle are usually labelled using lower case letters, in this case, a, b and c, and are positioned opposite the respective vertices. SoSide a will be opposite vertex AaSide b will be opposite vertex BbSide c will be opposite vertex CcIMPORTANT!!Note also that side a could also be called BC as it connects vertex B to vertex C etc.(We wont be using this labelling system in this unit of work)A quick overview of the two rulesWe will look at the two rules very briefly before starting to use them!The Sine RuleABCabcThe Sine Rule states that in any triangle ABC.

This is the general formula for the sine rule. In reality however, 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!The COSine RuleABCabcThe Cosine Rule states that in any triangle ABC.

This formula has c2 as the subject, but the letters can be interchanged, so it can also be written as

or

Study the patterns and locations of the letters in the three formulae closely. More on the cosine rule later!The Sine Rule

Proof of the Sine Rule: Let ABC be any triangle with side lengths a, b, c respectivelyCBAcabNow draw AD perpendicular to BC, and let the length of AD equal hhIn BDC

In ACD

andAs both expressions are equal to h, we can saya sin B = b sin A Dividing through by (sinA)( sinB) this becomes

which is the Sine RuleDExample 1 Use the Sine Rule to find the value of x in the triangle:CAB8812mx m54VERY IMPORTANT!! Take time to study the diagram. Note the positions of the three givens (actual values youre told) the 88, 54 and 12 m, and the one unknown, x. The formula for the sine rule requires three givens (in this case, 88, 54 and 12 m) and one unknown (x)Note that the third angle C and its opposite side c are not used in this problem! two of these givens must be an angle and its opposite side (in this case, the 54 and the 12 m which we will make our A and a). the third given (88) and the unknown (x) must also be an angle and its opposite side.

CAB8812mx54

Now we substitute the 3 givens and the unknown into this formula..A = 54a = 12Remember these two givens must be an angle and its matching opposite sideB = 88b = x

Cross-multiply

Divide through by sin 54 to make the subject

x = 14.82 (to 2 dec. pl)Substituting the values into the formulaFinally, label the x as 14.82 on the diagram and check that your answer fits with the other numbers in the problem! 14.82m (looks OK)These too!Example 2 Use the Sine Rule to find the value of x in the triangle:9535cmx cm22Here, no vertices are labelled so we will have to create our own. But firstStep 1, check that there are 4 labels i.e. 3 givens and 1 unknown. There are a 95, 22, 35 cm and x cm so this fits our requirements.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!Step 3, Allocate letters A, a, B, b (or any other letters of your choice) to matching pairs.A = 95a = 35B = 22b = x

A = 95a = 35B = 22b = x

x = 13.16 (2dec pl)Remember to check that the answer fits the context of the diagram. 9535 cmx cm22AaBbExample 3 Use the Sine Rule to find the value of in the triangle:624.7m5.1mA 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 dont use these.624.7m5.1m

Remember to check that the answer fits the context of the diagram. Example 4 Use the Sine Rule to find the value of x in the triangle:6833x m35.7mLooking at the diagram, it seems we have a problem! Although the 68 and 35.7 form a matching angle and opposite side, the 33 and x do not.Butremembering the angle sum of a triangle is 180, we can work out the 3rd angle to be 180 33 68 = 79. 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.

79

x = 37.80 (2 dec pl)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. AB14m32C110mThis 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:Now extend side AC1 past C1 to the new point C2 where the new length BC2 is the same as it was previously (10m)..AB14m32C110mC210mThe 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. AB14m32C110mANGLE C is obtuseB14m32C1C210mATRIANGLE 1TRIANGLE 2ANGLE C is acuteHow are the two C angles related? (if at all)AB14m32C110mC210mLet angle BC2C1 = . angle BC1C2 = . (isos ) angle BC1A = 180 (straight line)180 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 = 107Back to the question!Draw the triangle with the acute, rather than the obtuse, angle at C.Applying the Sine Rule,

B14m32C210mA

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 = 132.1Ans: = 47.9 or 132.1The 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 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! 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 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.The Sine Rule - SummaryIn 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. abcCBA

These are just re-shaped versions of the original sine rule formulae.The Cosine RuleThere are two variations of this.To find a side usec2 = a2 + b2 2ab cos C To find an angle use

These formulae are just rearrangements of each other. Verify this as an exercise.Proof of the Cosine Rule: Let ABC be any triangle with side lengths a, b, c respectivelyABCacbNow draw AD perpendicular to BC, and let the length of AD equal hhIn ACD

In ABDPythagoras gives

DLet the length CD = x, and so length BD will be a x. xa x

(1)

(2)In ACDPythagoras gives

(3)

The formulae (2) and (3) are both for h2 so we make them equal to each other.NOTE!! The expansion(a x)2 = a2 2ax + x2

Now cancel the x2 on each side and make c 2 the subject

From the first box on the previous slide, taking result (1) x = b cos C(4)and substituting this into (4), we get

which is a version of the Cosine Rule (for finding a side)Cosine Rule Finding a SIDEc 2 = a2 + b2 2ab cos C (1) Note the positions of the letters. If the 2ab cos C were missing, this would just be Pythagoras Theorem, c 2 = 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 writtenb 2 = a2 + c2 2ac cos B a 2 = b2 + c2 2bc cos A c 2 = a2 + b2 2ab cos C Here are the three variations of the formula shown together. Study them closely and note the patterns!Cosine Rule Finding an ANGLEc 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

When do we use the Cosine Rule? First, check to see if you can use the Sine Rule. Its easier! You are told ALL THREE SIDES and asked to find any ANGLEYou can use the Cosine Rule whenOR8m9m10m You are told TWO SIDES and THEIR INCLUDED ANGLE (i.e. the angle between those two sides) and asked to find the THIRD SIDE20 cm4515 cmxHere, we use

Here, we use c 2 = a2 + b2 2ab cos C Example 6 Use the Cosine Rule to find the value of c in the triangle:Note that we have 2 given sides (3 cm and 4 cm) and their included angle (65)654 cmcC3 cmABc 2 = a2 + b2 2ab cos C Leta = 3b = 4C = 65c 2 = 32 + 42 2 3 4 cos 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 cmFinally, check that c = 3.85 fits the diagram.so we can use the Cosine Rule for finding a sideExample 7 Use the Cosine Rule to find the size of C in the triangle:Note that we have 3 given sides and are asked to find angle at C (opposite 7.5)7.5 m8 m?B9 mCALeta = 8b = 9c = 7,5Ans: Angle C is equal to 51.95 (to 2 dec pl) or 5157 (to nearest minute)Finally, check that C = 51.95 fits the diagram.so we can use the Cosine Rule for finding an angle

Caution! Here we MUST make c = 7.5 as it is the side opposite the angle were finding, i.e. C, whereas a and b are interchangeable.

= 0.6163NOTE !! Bracket numerator and denominator when entering into calculator.

Example 8 Use the Cosine Rule to find the value of x in the triangle:Note that we have 2 given sides (10 m and 11 cm) and their included angle (100)10011 mx10 mc 2 = a2 + b2 2ab cos C Leta = 10b = 11c = xC = 100x 2 = 102 + 112 2 10 11 cos 100 x 2 = 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 mFinally, check that x = 16.10 fits the diagram. x is the longest side so this would seem reasonable.so we use the Cosine Rule for finding a sideExample 9 Use the Cosine Rule to find the value of in the triangle:29 mm21 mm40 mmNote that we have 3 given sides and are asked to find angle opposite to 40 mmso we use the Cosine Rule for finding an angle

Leta = 21b = 29c = 40C =

= 105.13Ans: is approx. equal to105.13 (to 2 dec pl) or1058 (to nearest min)remember the bracketsNote the negative cos. This means our angle is obtuse! ALL OBTUSE ANGLES HAVE A NEGATIVE COSINE!Finally, check that = 105 fits the diagram. LOOKS obtuse so this would seem reasonable. Beware you cant always presume the drawings are to scale, so be careful when judging the appropriateness of your answers (in all problems)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

c 2 = a2 + b2 2ab cos C To find a SIDETo find an ANGLEa 2 = 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.The Cosine Rule - Summary To use the Cosine Rule to find an angle you must be given all three sides 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). To use the Cosine Rule to find a side you must be given the other two sides and their included angle. 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. 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.A3571x m16 mE809 cm6 cmC29119x cm12 cmB14 cm10 cm12 cmD67x m11 m13 mF339 cmx cm12 cmA3571x m16 mExample 10First 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

to two dec pl.Ans: the length of side x is 26.38 m approximately.Remember to check appropriateness of your answer!Example 11First check to see if we can use the Sine Rule.We are not given any angle so we cant use the Sine Rule so we have to use the COSINE RULE the angle versionAns: the size of angle is 44.42 or 4425 approx.Remember to check appropriateness of your answer!B14 cm10 cm12 cm

Let.C = c = 10 a = 12b = 14

Example 12First check to see if we can use the Sine Rule.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. BUTthe third angle is 180 119 29 = 32 so we can use the SINE RULE

to two dec pl.Ans: the length of side x is 13.12 cm approximately.Remember to check appropriateness of your answer!C29119x cm12 cm32

Let.a = xA = 32 b = 12B = 29Example 13First check to see if we can use the Sine Rule.We are not given any angle and matching opposite side so we cant use the Sine Rule, so we have to use the COSINE RULE the side versionAns: the size of side x is 13.35 m (to 2 dec places)Remember to check appropriateness of your answer!Let.C = 67c = x a = 11b = 13D67x m11 m13 mc 2 = a2 + b2 2ab cos C x 2 = 112 + 132 2 11 13 cos 67 x 2 = 178.251x = 13.35Example 14First 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. HOWEVERwe 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 5858Remember to check appropriateness of your answer!

Let.a = 6A = b = 9B = 80E809 cm6 cm

Example 15F339 cmx cm12 cmFirst 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 wont work either as the triangles data does not match either of the two configurations for the Cosine Rule. HOWEVERif 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.Part 1 (finding )

Finding = 180 33 46.57 = 100.43Part 2 (finding x)

Note!! Here the diagram is quite out of scale. This becomes apparent on checking the reasonableness of your answer