The Sine Rule The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find: 1. An unknown side, when we are given two angles and a side. 2. An unknown angle when we are given two sides and an angle that is not included. a2 = b2 + c2 – 2bcCosA 12 18 75o 19 Cosine Rule ? 12 18 50o Sine Rule 75o 28/08/2018 1

The Sine Rule Deriving the rule C b a B A c This can be extended to Consider a general triangle ABC. Deriving the rule P Draw CP perpendicular to BA This can be extended to or equivalently 28/08/2018 2

To find an unknown side we need 2 angles and a side. The Sine Rule To find an unknown side we need 2 angles and a side. Not to scale a 1. 45o 60o 5.1 cm 2. 63o m 85o 12.7cm 15o 3. p 145o 45 m 28/08/2018 3

To find an unknown angle we need 2 sides and an angle not included. The Sine Rule To find an unknown angle we need 2 sides and an angle not included. Not to scale 1. 60o 5.1 cm 4.2 cm x 2. 63o 12.7cm 11.4cm y 3. 145o 45 m 99.7 m z 28/08/2018 4

The Sine Rule Application Problems A D The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B 35o 25o 10o 36.5 145o 15 m Angle TDA = 180 – 35 = 145o Angle DTA = 180 – 170 = 10o 28/08/2018 5

The Sine Rule A The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C 180 – 115 = 65o Angle BCA = 180 – 110 = 70o Angle ACT = 180 – 70 = 110o Angle ATC = 25o 65o 110o 20o 70o 53.2 m 5o 50 m 28/08/2018 6

The Sine Rule. (Used for Non-Right Angled Triangles) Calculating a Length a = b Sin A Sin B b 580 7 cm A The angles are OPPOSITE their sides. B 390 a x Working Out Let ‘a’ stand for the unknown length. a = 7 a = 9.43 7 Sin 580 Sin 390 Sin 580 Sin 390 a = 7 Sin 580 Sin 390 a = 9.43 cm 7 Menu

Calculate the Missing Lengths. (Answers to 2 d.p.) x = 7.36 cm 3) x 1) 6 cm 1100 8 m 700 x = 4.06 m 500 300 x 2) 4) 800 400 x= 21.70 m x 600 x 7 m x= 4.57 m 500 20 m 8 Answers Menu 8

The Sine Rule. (Used for Non-Right Angled Triangles) Calculating an Angle. a = b Sin A Sin B a b 13.7 cm 6.5 cm B 270 Working Out A x0 13.7 = 6.5 Sin A Sin 270 13.7 Sin 270 = Sin A Sin-1 0.9569 0.9569 13.7 730 A 6.5 = 6.5 Sin A Sin 270 0.9569 = Sin A Sin-1 0.9569 = A 730 = A 9 Menu

2) 1) 3) 4) Calculate The Missing Angles. (Answers to 1 d.p.) 35.30 31.70 2) 1) 800 9 cm 7 cm 8 cm x x 480 15 cm 3) 48.90 78.20 4) 700 8.1 cm 9.5 cm 14 cm x 400 x 25 cm 10 Answers Menu

The Sine Rule The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find: 1. An unknown side, when we are given two angles and a side. 2. An unknown angle when we are given two sides and an angle that is not included. a2 = b2 + c2 – 2bcCosA 12 18 75o 19 Cosine Rule ? 12 18 50o Sine Rule 75o

The Sine Rule Deriving the rule C b a B A c This can be extended to Consider a general triangle ABC. Deriving the rule P Draw CP perpendicular to BA This can be extended to or equivalently

To find an unknown side we need 2 angles and a side. The Sine Rule To find an unknown side we need 2 angles and a side. Not to scale a 1. 45o 60o 5.1 cm 2. 63o m 85o 12.7cm 15o 3. p 145o 45 m

To find an unknown angle we need 2 sides and an angle not included. The Sine Rule To find an unknown angle we need 2 sides and an angle not included. Not to scale 1. 60o 5.1 cm 4.2 cm x 2. 63o 12.7cm 11.4cm y 3. 145o 45 m 99.7 m z

The Sine Rule Application Problems A D The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B 35o 25o 10o 36.5 145o 15 m Angle TDA = 180 – 35 = 145o Angle DTA = 180 – 170 = 10o

The Sine Rule A The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C 180 – 110 = 70o 180 – 70 = 110o Angle ATC = 180 – 115 = 65o Angle BCA = Angle ACT = 25o 65o 110o 20o 70o 53.2 m 5o 50 m

The Sine Rule The Sine Rule is used for cases in which the Cosine Rule cannot be applied. It is used to find: 1. An unknown side, when we are given two angles and a side. 2. An unknown angle when we are given two sides and an angle that is not included. a2 = b2 + c2 – 2bcCosA 12 18 75o 19 Cosine Rule ? 12 18 50o Sine Rule 75o 28/08/2018 17

The Sine Rule Deriving the rule C b a B A c This can be extended to Consider a general triangle ABC. Deriving the rule P Draw CP perpendicular to BA This can be extended to or equivalently 28/08/2018 18

To find an unknown side we need 2 angles and a side. The Sine Rule To find an unknown side we need 2 angles and a side. Not to scale a 1. 45o 60o 5.1 cm 2. 63o m 85o 12.7cm 15o 3. p 145o 45 m 28/08/2018 19

To find an unknown angle we need 2 sides and an angle not included. The Sine Rule To find an unknown angle we need 2 sides and an angle not included. Not to scale 1. 60o 5.1 cm 4.2 cm x 2. 63o 12.7cm 11.4cm y 3. 145o 45 m 99.7 m z 28/08/2018 20

The Sine Rule Application Problems A D The angle of elevation of the top of a building measured from point A is 25o. At point D which is 15m closer to the building, the angle of elevation is 35o Calculate the height of the building. T B 35o 25o 10o 36.5 145o 15 m Angle TDA = 180 – 35 = 145o Angle DTA = 180 – 170 = 10o 28/08/2018 21

The Sine Rule A The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base B T C 180 – 115 = 65o Angle BCA = 180 – 110 = 70o Angle ACT = 180 – 70 = 110o Angle ATC = 25o 65o 110o 20o 70o 53.2 m 5o 50 m 28/08/2018 22

1 1 Trigonometry 5: Sine, cosine and tangent for any angle y As the Point P moves in an anti-clockwise direction around the circumference of the circle, the angle  changes from 0o to 360o. O 1 P (x,y) A y  Consider the right-angled triangle formed by the vertical line PA. x x In this triangle the distance OA = x. The distance OP = y. x O 1 P  A y So point P has co-ordinates (x,y). (cos ,sin ) sin  cos  Therefore x = cos  and y = sin . So the co-ordinates of P are (cos , sin ). 28/08/2018 23

The Sine Rule C. McMinn

SOH/CAH/TOA can only be used for right-angled triangles. The Sine Rule can be used for any triangle: C The sides are labelled to match their opposite angles b a A B c a sinA b sinB c sinC = = The Sine Rule:

A Example 1: Find the length of BC 76º c 7cm b 63º C x B a a sinA c sinC = Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. x sin76º 7 sin63º sin76º × = × sin76º 7 sin63º x = × sin76º x = 7.6 cm

P Example 2: Find the length of PR 82º x r q 43º 55º Q 15cm R p p sinP q sinQ = Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. 15 sin82º x sin43º sin43º × = × sin43º 15 sin82º sin43º × = x x = 10.33 cm

G 1. B 3. 2. F 53º 13 cm 41º x 8.0 35.3 5.5 x A 62º x 28º 130º D E 5 cm 63º 76º C H 26 mm I 4. 10.7 5. 5.2 cm x 61º R 6. P 37º 66º 57º 10 m 35º x 5.2 77º 62º Q 12 cm 6 km 85º 7. x 6.6 65º 86º x 6.9

Remember: Draw a diagram Label the sides Set out your working exactly as you have been shown Check your answers regularly and ask for help if you need it

Finding an Angle The Sine Rule can also be used to find an angle, but it is easier to use if the rule is written upside-down! Alternative form of the Sine Rule: sinA a sinB b sinC c = =

C Example 1: Find the size of angle ABC 6cm a 4cm b 72º x º A B c sinA a sinB b = Draw arrows from the sides to the opposite angles to help decide which parts of the sine rule to use. sin72º 6 sin xº 4 4 × = × 4 sin72º 6 4 × = sin xº sin xº = 0.634 x = sin-1 0.634 = 39.3º

P Example 2: Find the size of angle PRQ 85º q 7cm r x º R p 8.2cm Q sinP p sinR r = sin85º 8.2 sin xº 7 7 × = × 7 sin85º 8.2 7 × = sin xº sin xº = 0.850 x = sin-1 0.850 = 58.3º

1. 7.6 cm 2. 3. 47º 82º 105º 6.5cm 5 cm 8.2 cm xº xº xº 8.8 cm 6 cm 5. 66.6° xº 37.6° xº 45.5° xº 8.8 cm 6 cm 5. 6 km 4. 5.5 cm 31.0° xº 27º 3.5 km 51.1° xº 5.2 cm 33º Slide 10 is incomplete. Try to add slides on applications of Sine Rule 7. 6. 8 m 74º 57.7° xº 70º 9 mm 9.5 m 92.1° xº 52.3º (←Be careful!→) 22.9º 7 mm

Remember: Draw a diagram Label the sides Set out your working exactly as you have been shown Check your answers regularly and ask for help if you need it