Trigonometry 4103 Trigonometry triangle measure A little bit

Trigonometry (4103)

Trigonometry “triangle measure”

A little bit of review. . .

The 3 angles from a triangle ALWAYS equal 180 o a + b + c = 180 o a b c

Find the total of the other angles a 30◦

Find the total of the other angles a = 90◦ 30◦

Find the total of the other angles Total angles = 180◦ 90◦ + 30◦ + a = 180◦ 120◦ + a = 180◦ – 120◦ a = 60◦ a = 90◦ 30◦

Equilateral triangle All sides are the same length

Equilateral triangle All angles are the same (180 o ÷ 3 = 60 o)

Isosceles triangle Two sides are the same length

Isosceles triangle Two angles are the same

Scalene triangle No sides are the same length

Scalene triangle No angles are the same

Right-angled triangle side hypotenuse side

Right-angled triangle side opposite to angle A hypotenuse A side adjacent (next to) angle A

Right-angled triangle hypotenuse (c) side (a) 90 o side (b)

Pythagorean Theorem c 2 = a 2 + b 2 side (a) hypotenuse (c) side (b)

What if you switch a and b? c 2 = a 2 + b 2 side (a) hypotenuse (c) side (b)

What if you switch a and b? c 2 = a 2 + b 2 side (b) hypotenuse (c) side (a) Doesn’t matter, they’re both sides!

Right-angled triangle B side adjacent to angle B hypotenuse A side opposite to angle B

What is the length of the hypotenuse? c 2 = a 2 + b 2 side (a) 3 cm hypotenuse (c) x cm side (b) 4 cm

What is the length of the hypotenuse? c 2 = a 2 + b 2 side (a) 3 cm hypotenuse (c) x cm side (b) 4 cm x 2 = 3 2 + 4 2 x 2 = 9 + 16 x 2 = 25 x = 5 cm

What is the length of the side? c 2 = a 2 + b 2 side (a) x cm hypotenuse (c) 10 cm side (b) 5 cm

What is the length of the side? c 2 = a 2 + b 2 side (a) x cm hypotenuse (c) 10 cm side (b) 5 cm 102 = x 2 + 52 100 = x 2 + 25 100 – 25 = x 2 75 = x 2 = 75 x = 8. 7 cm

Trigonometric ratios sine cosine tangent depend on which angle is used

Sine ratio (SOH) sin A = opposite hypotenuse side opposite to angle A hypotenuse A side adjacent to angle A

Sine ratio (SOH) sin B = opposite hypotenuse side adjacent to angle B B hypotenuse A side opposite to angle B

Cosine ratio (CAH) cos A = adjacent hypotenuse side opposite to angle A hypotenuse A side adjacent to angle A

Cosine ratio (CAH) cos B = adjacent hypotenuse side adjacent to angle B B hypotenuse A side opposite to angle B

Tangent ratio (TOA) tan A = opposite adjacent side opposite to angle A hypotenuse A side adjacent to angle A

Tangent ratio (TOA) tan B = opposite adjacent side adjacent to angle B B hypotenuse A side opposite to angle B

Trigonometric ratios SOH CAH TOA sin θ = opp cos θ = adj tan θ = opp hyp adj hyp

Find the lengths of the missing sides and angle (right triangle) B 7 cm A 35 o C

Find the lengths of the missing sides and angle (right triangle) B 7 cm A 35 o 90 o C

Step 1. List the information given, and what is needed B 7 cm 35 o A 90 o C What we know: m. BC = 7 cm A = 35 o C = 90 o What we need: m. AB = ? m. AC = ? B=?

Step 2. Find the missing side AB B hyp 7 cm (opp) 35 o A 90 o C Look at the triangle from A: m. BC = opposite m. AB = hypotenuse

Step 2. Find the missing side AB B hyp 7 cm (opp) 35 o A 90 o C Look at the triangle from A: m. BC = opposite m. AB = hypotenuse ? = opp hyp

Step 2. Find the missing side AB B hyp 7 cm (opp) 35 o A 90 o C Look at the triangle from A: m. BC = opposite m. AB = hypotenuse sin θ = opp hyp

Step 2. Find the missing side AB B hyp 7 cm (opp) 35 o A 90 o C sin θ = opp hyp sin 35 o = opp hyp

Step 2. Find the missing side AB B hyp 7 cm (opp) 35 o A 90 o C sin θ = opp hyp sin 35 o = opp hyp 0. 574 = 7 cm hyp

Step 2. Find the missing side AB sin θ = opp hyp B hyp 7 cm (opp) 35 o A 90 o C sin 35 o = opp hyp 0. 574 = 7 cm hyp 0. 574 (hyp) = 7 cm

Step 2. Find the missing side AB sin θ = opp hyp B hyp 7 cm (opp) 35 o A 90 o C sin 35 o = opp hyp 0. 574 = 7 cm hyp 0. 574 (hyp) = 7 cm 0. 574

Step 2. Find the missing side AB sin θ = opp hyp B hyp 7 cm (opp) 35 o A 90 o C sin 35 o = opp hyp 0. 574 = 7 cm hyp 0. 574 (hyp) = 7 cm hyp = 12. 2 cm

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C Look at the triangle from A: m. BC = opposite m. AB = hypotenuse m. AC = adjacent

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C Look at the triangle from A: m. BC = opposite m. AB = hypotenuse m. AC = adjacent Since we have two sides, we have a choice of trig ratios!

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C cos θ = adj hyp or tan θ = opp adj

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C cos θ = adj hyp cos 35 o = adj hyp

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C cos θ = adj hyp cos 35 o = adj hyp 0. 819 = adj 12. 2 cm

Step 3. Find the missing side AC cos θ = adj hyp B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C cos 35 o = adj hyp 0. 819 = adj 12. 2 cm 0. 819 (12. 2 cm) = adj

Step 3. Find the missing side AC cos θ = adj hyp B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C cos 35 o = adj hyp 0. 819 = adj 12. 2 cm 0. 819 (12. 2 cm) = adj = 10 cm

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C tan θ = opp adj tan 35 o = opp adj

Step 3. Find the missing side AC B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C tan θ = opp adj tan 35 o = opp adj 0. 700 = 7 cm adj

Step 3. Find the missing side AC tan θ = opp adj B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C tan 35 o = opp adj 0. 700 = 7 cm adj 0. 700 (adj) = 7 cm

Step 3. Find the missing side AC tan θ = opp adj B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o (adj) C tan 35 o = opp adj 0. 700 = 7 cm adj 0. 700 (adj) = 7 cm 0. 700

Step 3. Find the missing side AC tan θ = opp adj B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o 10 cm (adj) C tan 35 o = opp adj 0. 700 = 7 cm adj 0. 700 (adj) = 7 cm adj = 10 cm

Step 4. Find the missing angle B B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o 10 cm (adj) C

Step 4. Find the missing angle B 180 o = A + B + C B 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o 10 cm (adj) C

Step 4. Find the missing angle B 180 o = A + B + C B 180 o = 35 o + B + 90 o 12. 2 cm (hyp) 7 cm (opp) 35 o A 90 o 10 cm (adj) C

Step 4. Find the missing angle B 180 o = A + B + C B 180 o = 35 o + B + 90 o 12. 2 cm (hyp) o = B + 125 o 180 7 cm (opp) 35 o A 90 o 10 cm (adj) C

Step 4. Find the missing angle B 180 o = A + B + C B 180 o = 35 o + B + 90 o 12. 2 cm (hyp) o = B + 125 o 180 7 cm (opp) 35 o A 90 o 10 cm (adj) C B = 180 o – 125 o

Step 4. Find the missing angle B 180 o = A + B + C B 12. 2 cm (hyp) 55 o 180 o = 35 o + B + 90 o o = B + 125 o 180 7 cm (opp) 35 o A 90 o 10 cm (adj) C B = 180 o – 125 o B = 55 o

Steps to completing a right triangle Step 1. List the information given, and what is needed Step 2. Find the missing side(s) Step 3. Find the missing angle(s)

Find the length of the missing side and angles A 25 cm 19 cm 30 o B C

Step 1. List the missing information, and what is needed A 25 cm 19 cm 30 o B C What we know: m. AB = 25 cm m. AC = 19 cm B = 30 o What we need: m. BC = ? A=? C=?

Step 2. Create a 90 o angle by cutting the triangle in two A 25 cm 19 cm 30 o B H C Start at the top angle and continue until it hits the bottom of the triangle at a 90 o angle

Step 2. Create a 90 o angle by cutting the triangle in two Name the point of intersection H A 25 cm 19 cm 30 o B H C

Step 2. Create a 90 o angle by cutting the triangle in two Name the point of intersection H A 25 cm 19 cm 30 o B H C Now find the missing information for each new triangle!

Step 3. Find the length BH Look at the new triangle from B: m. BH = adjacent m. AB = hypotenuse A 25 cm 19 cm 30 o B H C

Step 3. Find the length BH Look at the new triangle from B: m. BH = adjacent m. AB = hypotenuse A 25 cm 19 cm ? = adj hyp 30 o B H C

Step 3. Find the length BH Look at the new triangle from B: m. BH = adjacent m. AB = hypotenuse A 25 cm 19 cm cos θ = adj hyp 30 o B H C

Step 3. Find the length BH cos θ = adj hyp A 25 cm 19 cm 30 o B H C cos 30 o = adj hyp

Step 3. Find the length BH cos θ = adj hyp A 25 cm 19 cm 30 o B H C cos 30 o = adj hyp 0. 866 = adj 25 cm

Step 3. Find the length BH cos θ = adj hyp A 25 cm 30 o B H cos 30 o = adj 19 cm hyp 0. 866 = adj 25 cm (0. 866)(25 cm) = adj C

Step 3. Find the length BH cos θ = adj hyp A 25 cm 30 o B 21. 7 cm H cos 30 o = adj 19 cm hyp 0. 866 = adj 25 cm (0. 866)(25 cm) = adj C adj = 21. 7 cm

Step 4. Find the angle A A 25 cm 180 o = A + B + H 19 cm 30 o B 21. 7 cm H C

Step 4. Find the angle A A 180 o = A + B + H 180 o = A + 30 o + 90 o 25 cm 19 cm 30 o B 21. 7 cm H C

Step 4. Find the angle A A 180 o = A + B + H 180 o = A + 30 o + 90 o 25 cm 19 cm 30 o B 21. 7 cm H C 180 o = A + 120 o

Step 4. Find the angle A A 180 o = A + B + H 180 o = A + 30 o + 90 o 25 cm 19 cm 30 o B 21. 7 cm H C 180 o = A + 120 o A = 180 o – 120 o

Step 4. Find the angle A A 180 o = A + B + H 180 o = A + 30 o + 90 o 25 cm 60 o 19 cm 21. 7 cm A = 180 o – 120 o A = 60 o 30 o B 180 o = A + 120 o H C

Step 5. Find the length AH A 25 cm 60 o 19 cm 30 o B 21. 7 cm H C There are many different ways to find m. AH: – Pythagoras – tan A or tan B – cos A – sin B

Step 5. Find the length AH sin θ = opp hyp A 25 cm 60 o 19 cm 30 o B 21. 7 cm H C

Step 5. Find the length AH sin θ = opp hyp A 25 cm 60 o 19 cm 30 o B 21. 7 cm H C sin 30 o = opp hyp

Step 5. Find the length AH sin θ = opp hyp A 25 cm 60 o 19 cm 30 o B 21. 7 cm H C sin 30 o = opp hyp 0. 500 = opp 25 cm

Step 5. Find the length AH sin θ = opp hyp A 25 cm 60 o 30 o B 21. 7 cm H sin 30 o = opp 19 cm hyp 0. 500 = opp 25 cm (0. 500)(25) = opp C

Step 5. Find the length AH sin θ = opp hyp A 25 cm 60 o 12. 5 cm 30 o B 21. 7 cm H sin 30 o = opp 19 cm hyp 0. 500 = opp 25 cm (0. 500)(25) = opp C hyp = 12. 5 cm

Step 6. Find the angle C Look at the new triangle from C: m. AH = opposite m. AC = hypotenuse A 25 cm 60 o 19 cm 12. 5 cm ? = opp hyp 30 o B 21. 7 cm H C

Step 6. Find the angle C Look at the new triangle from C: m. AH = opposite m. AC = hypotenuse A 25 cm 60 o 19 cm 12. 5 cm sin θ = opp hyp 30 o B 21. 7 cm H C

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm H C

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm H C sin θ = opp hyp

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm H C sin θ = opp hyp sin θ = 12. 5 cm 19 cm

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm H C sin θ = opp hyp sin θ = 12. 5 cm 19 cm sin θ = 0. 66

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm H C sin θ = opp hyp sin θ = 12. 5 cm 19 cm sin θ = 0. 66 sin-1(0. 66) = θ

Step 6. Find the angle C sin θ = opp hyp A 25 cm 60 o 19 cm 12. 5 cm 41. 1 o 30 o B 21. 7 cm H C sin θ = opp hyp sin θ = 12. 5 cm 19 cm sin θ = 0. 66 sin-1(0. 66) = θ θ = 41. 1 o

Step 7. Find the length CH A 25 cm 60 o 19 cm 12. 5 cm 41. 1 o 30 o B 21. 7 cm H C There are many different ways to find m. CH: – Pythagoras – cos C – tan C

Step 7. Find the length CH cos θ = adj hyp A 25 cm 60 o 19 cm 12. 5 cm 41. 1 o 30 o B 21. 7 cm H C

Step 7. Find the length CH cos θ = adj hyp A 25 cm 60 o cos 41. 1 o = adj hyp 19 cm 12. 5 cm 41. 1 o 30 o B 21. 7 cm H C

Step 7. Find the length CH cos θ = adj hyp A 25 cm 60 o 12. 5 cm 41. 1 o 30 o B 21. 7 cm cos 41. 1 o = adj hyp 19 cm 0. 754 = adj 19 cm H C

Step 7. Find the length CH cos θ = adj hyp A 25 cm 60 o 12. 5 cm 30 o B 21. 7 cm H cos 41. 1 o = adj hyp 19 cm 0. 754 = adj 19 cm 41. 1 o 0. 754 (19 cm) = adj C

Step 7. Find the length CH cos θ = adj hyp A 25 cm 60 o 12. 5 cm 30 o B 21. 7 cm cos 41. 1 o = adj hyp 19 cm 0. 754 = adj 19 cm 41. 1 o 0. 754 (19 cm) = adj H 14. 3 cm C adj = 14. 3 cm

Step 8. Find the angle A 180 o = A + C + H A 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C

Step 8. Find the angle A 180 o = A + C + H A 180 o = A + 41. 1 o + 90 o 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C

Step 8. Find the angle A 180 o = A + C + H A 180 o = A + 41. 1 o + 90 o 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C 180 o = A + 131. 1 o

Step 8. Find the angle A 180 o = A + C + H A 180 o = A + 41. 1 o + 90 o 25 cm 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C 180 o = A + 131. 1 o A = 180 o – 131. 1 o

Step 8. Find the angle A 180 o = A + C + H A 48. 9 o 25 cm 60 o 180 o = A + 41. 1 o + 90 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C 180 o = A + 131. 1 o A = 180 o – 131. 1 o A = 48. 9 o

Step 9. Complete triangle A = 60 o + 48. 9 o = 108. 9 o 25 cm 180 o = A + B + C A 48. 9 o 60 o 19 cm 12. 5 cm 30 o B 21. 7 cm 41. 1 o H 14. 3 cm C

Step 9. Complete triangle A = 60 o + 48. 9 o = 108. 9 o 25 cm 30 o B 21. 7 cm 180 o = A + B + C A 108. 9 o 180 o = 108. 9 o + 30 o + 41. 1 o 19 cm 41. 1 o H 14. 3 cm C

Step 9. Complete triangle A = 60 o + 48. 9 o = 108. 9 o 25 cm 30 o B 21. 7 cm 180 o = A + B + C A 108. 9 o 180 o = 108. 9 o + 30 o + 41. 1 o 19 cm 41. 1 o H 14. 3 cm C The angles in the original triangle ABC add up to 180 o

Step 9. Complete triangle m. BC = m. BH + m. CH A 108. 9 o 25 cm 30 o B 21. 7 cm 19 cm 41. 1 o H 14. 3 cm C

Step 9. Complete triangle m. BC = m. BH + m. CH A 108. 9 o 25 cm 30 o B 21. 7 cm 19 cm 41. 1 o H 14. 3 cm C m. BC = 21. 7 + 14. 3

Step 9. Complete triangle m. BC = m. BH + m. CH A 108. 9 o 25 cm 19 cm 41. 1 o 30 o B m. BC = 21. 7 + 14. 3 36 cm C m. BC = 36 cm

Steps to complete a non-right angle triangle Step 1. List the missing information, and what is needed Step 2. Create 90 o angles by cutting the triangle in two Step 3. Looking at the first triangle, solve for missing angle(s) and/or side(s) Step 4. Looking at the second triangle, solve for missing angle(s) and/or side(s) Step 5. Put the halves of sides and angles together into the one original triangle

So far, there are two ways to solve a right-angled triangle:

So far, there are two ways to solve a right-angled triangle: Pythagoras (c 2 = a 2 + b 2) Trigonometric ratios (SOH CAH TOA)

Isn’t there another way to solve a non-right angled triangle?

Isn’t there another way to solve a non-right angled triangle? Yes! Sin Law and Cos Law

Sine Law Uses the sine ratio

Sine Law a = b = c sin A sin B sin C

Sine Law lengths a = b = c sin A sin B sin C angles

Find the length of the missing side and angles A 25 cm 19 cm 30 o B C

Find the length of the missing side and angles Remember: Capital letters = angles Lower-case letters = sides A c 25 cm b 19 cm 30 o B a C

Find the length of the missing side and angles angle A ↔ side a Remember: Capital letters = angles Lower-case letters = sides A c 25 cm b 19 cm Angles and sides with the same letters are opposite each other 30 o B a C

Find the length of the missing side and angles angle B ↔ side b Remember: Capital letters = angles Lower-case letters = sides A c 25 cm b 19 cm Angles and sides with the same letters are opposite each other 30 o B a C

Find the length of the missing side and angles angle C ↔ side c Remember: Capital letters = angles Lower-case letters = sides A c 25 cm b 19 cm Angles and sides with the same letters are opposite each other 30 o B a C

Step 1. List the missing information, and what is needed A c 25 cm b 19 cm 30 o B a C What we know: m. AB = c = 25 cm m. AC = b = 19 cm B = 30 o What we need: m. BC = a = ? A=? C=?

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ We have both angle B and side b A c 25 cm b 19 cm 30 o B a C We can use these to fill out the C ‘pair’

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm b = c sin B sin C A b 19 cm 30 o B a C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm A b 19 cm 30 o B a C b = c sin B sin C 19 cm = 25 cm sin 30 o sin C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm A b 19 cm 30 o B a C b = c sin B sin C 19 cm = 25 cm sin 30 o sin C 19 (sin C) = sin 30 o (25)

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm A b = c sin B sin C 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 30 o B a C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm A b = c sin B sin C 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 19 (sin C) = 12. 5 30 o B a C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm 30 o B a A b = c sin B sin C 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 19 (sin C) = 12. 5 19 19 C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm 30 o B a A b = c sin B sin C 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 19 (sin C) = 12. 5 19 19 sin C = 0. 658 C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm 30 o B a A b = c sin B sin C 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 19 (sin C) = 12. 5 19 19 sin C = 0. 658 C sin-1 (0. 658) = C

Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ a = b = c sin A sin B sin C c 25 cm 30 o B a b = c sin B sin C A 19 cm = 25 cm sin 30 o sin C b 19 (sin C) = sin 30 o (25) 19 cm 19 (sin C) = (0. 5)(25) 19 (sin C) = 12. 5 19 19 o 41. 1 sin C = 0. 658 C sin-1 (0. 658) = C C = 41. 1 o

Step 3. Find the last angle (A) 180 o = A + B + C A c 25 cm b 19 cm 41. 1 o 30 o B a C

Step 3. Find the last angle (A) 180 o = A + B + C A 180 o = A + 30 o + 41. 1 o c 25 cm b 19 cm 41. 1 o 30 o B a C

Step 3. Find the last angle (A) 180 o = A + B + C A 180 o = A + 30 o + 41. 1 o c 25 cm b 19 cm 41. 1 o 30 o B a C 180 o = A + 71. 1 o

Step 3. Find the last angle (A) 180 o = A + B + C A 180 o = A + 30 o + 41. 1 o c 25 cm b 19 cm 180 o = A + 71. 1 o 180 o – 71. 1 o = A 41. 1 o 30 o B a C

Step 3. Find the last angle (A) 108. 9 o c 25 cm 180 o = A + B + C A 180 o = A + 30 o + 41. 1 o b 19 cm 180 o = A + 71. 1 o 180 o – 71. 1 o = A B A = 108. 9 o 41. 1 o 30 o a C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm b 19 cm 41. 1 o 30 o B a C

Step 4. Find the last ‘pair’ (A) a = b sin A sin B a = b = c sin A sin B sin C A 108. 9 o c 25 cm b 19 cm 41. 1 o 30 o B a C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm b 19 cm 41. 1 o 30 o B a C a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm 41. 1 o 30 o B a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o b a = 19 cm 0. 946 0. 5 a C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm 41. 1 o 30 o B a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o b a = 19 cm 0. 946 0. 5 a (0. 5) = 0. 946 (19) a C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm 30 o B a a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o b a = 19 cm 0. 946 0. 5 a (0. 5) = 0. 946 (19) a (0. 5) = 17. 97 41. 1 o C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm 30 o B a a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o b a = 19 cm 0. 946 0. 5 a (0. 5) = 0. 946 (19) a (0. 5) = 17. 97 41. 1 o 0. 5 C

Step 4. Find the last ‘pair’ (A) a = b = c sin A sin B sin C A 108. 9 o c 25 cm 30 o B a 36 cm a = b sin A sin B a = 19 cm sin 108. 9 o sin 30 o b a = 19 cm 0. 946 0. 5 a (0. 5) = 0. 946 (19) a (0. 5) = 17. 97 41. 1 o 0. 5 C a = 36 cm

Done! 108. 9 o c 25 cm b 19 cm 41. 1 o 30 o B A a 36 cm C

Steps to complete a triangle using Sine Law Step 1. List the missing information, and what is needed Step 2. Find one ‘pair’, and use it to fill in another ‘pair’ Step 3. Find the last angle Step 4. Find the last ‘pair’

Cos Law Uses the cos ratio Also uses ‘pairs’

Looking for a Cos Law other two lengths 2 a = 2 b + 2 c – 2 bc(cos. A) pair you’re looking for

Looking for b Cos Law other two lengths 2 b = 2 a + 2 c – 2 ac(cos. B) pair you’re looking for

Looking for c Cos Law other two lengths 2 c = 2 b + 2 a – 2 ab(cos. C) pair you’re looking for

3 variations of Cos Law 2 a 2 b 2 c = + – 2 bc(cos. A) 2 2 2 b = a + c – 2 ac(cos. B) 2 2 2 c = b + c – 2 ab(cos. C)

Find the length of b A c 25 cm b 30 o B a 36 cm C

Find the length of b A c 25 cm To use Cos Law, you have to know: - One of the values of the pair you need (angle or length) - The two other lengths b 30 o B a 36 cm C

Step 1. List the missing information, and what is needed What we know: m. AB = c = 25 cm m. BC = a = 36 cm B = 30 o A c 25 cm b What we need: m. AC = b = ? 30 o B a 36 cm C

Step 2. Choose the variation of Cos Law that you need a 2 = b 2 + c 2 – 2 bc(cos. A) A b 2 = a 2 + c 2 – 2 ac(cos. B) c 25 cm c 2 = b 2 + c 2 – 2 ab(cos. C) b 30 o B a 36 cm C

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B)

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o)

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o) b 2 = 362 + 252 – 2(36)(25)(0. 866)

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o) b 2 = 362 + 252 – 2(36)(25)(0. 866) b 2 = 1296 + 625 – 2(36)(25)(0. 866)

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o) b 2 = 362 + 252 – 2(36)(25)(0. 866) b 2 = 1296 + 625 – 1558. 8

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o) b 2 = 362 + 252 – 2(36)(25)(0. 866) b 2 = 1296 + 625 – 1558. 8 b 2 = 362. 2

Step 3. Solve the equation b 2 = a 2 + c 2 – 2 ac(cos. B) b 2 = 362 + 252 – 2(36)(25)(cos 30 o) b 2 = 362 + 252 – 2(36)(25)(0. 866) b 2 = 1296 + 625 – 1558. 8 b 2 = 362. 2 b = 19 cm

Done! A c 25 cm b 19 cm 30 o B a 36 cm C

Steps to complete triangles using Cos Law Step 1. List the missing information, and what is needed Step 2. Choose the variation of Cos Law that you need Step 3. Solve the equation

How do you know which to use? Use Sin Law if: Use Cos Law if: • Given 2 sides, 1 angle opposite one of the sides • Given 3 sides a a A b • Given 2 angles, 1 side opposite one of the angles B b • Given 1 angle, 2 sides adjacent to that angle a c C b

Summary of trigonometry Right-angled triangle Non-right angled triangle • If you only have sides --Pythagoras • If you have a pair (a + A) • If you have sides and angles • If you have to fill in a pair (looking for angle or side) – SOH CAH TOA – Sin Law – Cos Law – a 2 = b 2 + c 2 – 2 ac(cos A)