Trigonometry Equations www mathsrevision com National 5 Trig

Trigonometry Equations www. mathsrevision. com National 5 Trig Function and Circle Connection Solving Trig Equations Negative Cosine Special Trig Relationships Exam Type Questions created by Mr. Lafferty

Starter www. mathsrevision. com National 5 Q 1. How can we tell if two lines are parallel. Q 2. Write down the three ratios connecting the circle , arc length and area of a sector. created by Mr. Lafferty

Solving Trig Equations www. mathsrevision. com National 5 Learning Intention Success Criteria 1. We are investigating the connect between the circle and trig functions. created by Mr. Lafferty 1. Understand the connection between the circle and sine, cosine and tan functions. 2. Solve trig equations using graphically.

Trig and Circle Connection www. mathsrevision. com National 5 Sin +ve All +ve 180 o - xo 1 Sine Graph Construction 2 3 Cosine Graph Construction 4 Tan Graph Construction created by Mr. Lafferty 180 o + xo 360 o - xo Tan +ve Cos +ve Demo

Solving Trig Equations www. mathsrevision. com National 5 Learning Intention Success Criteria 1. We are learning how to solve trig equations of the form a sin xo + 1 = 0 1. a sin xo + 1 = 0 2. created by Mr. Lafferty Use the balancing method to trig equation Realise that there are many solutions to trig equations depending on domain.

Solving Trig Equations Graphically what a sin xo + b are = 0 we trying to solve National 5 www. mathsrevision. com Example : Demo Solving the equation sin xo = 0. 5 in the range 0 o to 360 o sin xo = (0. 5) xo = sin-1(0. 5) xo = 30 o There is another solution 1 2 3 4 created by Mr. Lafferty xo = 150 o (180 o – 30 o = 150 o)

Solving Trig Equations Graphically what a cos xo + b are = 0 we trying to solve National 5 www. mathsrevision. com Example : Demo Solving the equation cos xo = 0. 625 in the range 0 o to 360 o cos xo = 0. 625 xo = cos -1 0. 625 xo = 51. 3 o There is another solution 1 2 3 4 created by Mr. Lafferty (360 o - 53. 1 o = 308. 7 o)

Solving Trig Equations www. mathsrevision. com National 5 Now try N 5 TJ Ex 20. 1 upto Q 4 Ch 20 (Page 198) created by Mr. Lafferty

Solving Trig Equations Graphically what www. mathsrevision. com National 5 a tan xo + b are = 0 we trying to solve Example : Demo Solving the equation tan xo – 2 = 0 in the range 0 o to 360 o tan xo = 2 xo = tan -1(2) xo = 63. 4 o There is another solution 1 2 3 4 created by Mr. Lafferty x = 180 o + 63. 4 o = 243. 4 o

Solving Trig Equations Graphically what a sin xo + b are = 0 we trying to solve National 5 www. mathsrevision. com Example : Demo Solving the equation 3 sin xo + 1 = 0 in the range 0 o to 360 o sin xo = -1/3 Calculate first Quad value xo = 19. 5 o x = 180 o + 19. 5 o = 199. 5 o There is another solution 1 2 3 4 created by Mr. Lafferty ( 360 o - 19. 5 o = 340. 5 o)

Solving Trig Equations Graphically what National 5 www. mathsrevision. com Example : a sin xo + b are = 0 we trying to solve Demo Solving the equation 2 sin xo + 1 = 0 in the range 0 o to 720 o sin xo = -1/2 Calculate first Quad value xo = 30 o xo = 210 o and 330 o There are further solutions at 360 o + 210 o = 570 o 360 o + 330 o = 690 o

o Solving Trig Equations 90 Graphically what Higher www. mathsrevision. com Example to Outcome 3 are we trying A S o 180 solve T C Solving the equation cos 2 x = 1 in the range 0 o to 360 o 270 o cos 2 xo = 1 cos xo = ± 1 cos xo = 1 xo = 0 o and 360 o cos xo = -1 created by Mr. Lafferty xo = 180 o 0 o

Solving Trig Equations www. mathsrevision. com National 5 Now try N 5 TJ Ex 20. 1 Q 4 onwards Ch 20 (Page 198) created by Mr. Lafferty

Starter Questions www. mathsrevision. com Nat 5 54 o 03 -Mar-21 Created by Mr. Lafferty Maths Dept.

Cosine Rule www. mathsrevision. com Nat 5 Learning Intention Success Criteria 1. We are learning what a negative cosine ratio means with respect to the angle. 03 -Mar-21 1. Know what a negative cosine ratio means. 2. Solve REAL LIFE problems that involve finding an angle of a triangle. Created by Mr. Lafferty Maths Dept.

Cosine Rule Works for any Triangle www. mathsrevision. com Nat 5 The Cosine Rule can be used with ANY triangle as long as we have been given enough information. B a c A 03 -Mar-21 Created by Mr Lafferty Maths Dept C b

Cosine Rule Works for any Triangle Nat 5 www. mathsrevision. com How to determine when to use the Cosine Rule. Two questions 1. Do you know ALL the lengths. SAS OR 2. Do you know 2 sides and the angle in between. If YES to any of the questions then Cosine Rule Otherwise use the Sine Rule 03 -Mar-21 Created by Mr Lafferty Maths Dept

Finding Angles Using The Cosine Rule Nat 5 Works for any Triangle www. mathsrevision. com Consider the Cosine Rule again: We are going to change the subject of the formula to cos A o b 2 + c 2 – 2 bc cos Ao = a 2 Turn the formula around: -2 bc cos Ao = a 2 – b 2 – c 2 Take b 2 and c 2 across. Divide by – 2 bc. Divide top and bottom by -1 You now have a formula for finding an angle if you know all three sides of the triangle.

Finding Angles Using The Cosine Rule Works for any Triangle Nat 5 www. mathsrevision. com Example : Calculate the D e f unknown angle Fo. E d F Write down the formula for cos Fo Fo = ? d = 12 e = 10 f = 8 Label and identify Fo and d , e and f. Substitute values into the formula. Cos Fo = 0. 75 Calculate cos Fo. Fo = 41. 4 o Use cos-1 0. 75 to find Fo

Finding Angles Using The Cosine Rule Works for any Triangle Nat 5 www. mathsrevision. com Example : Find the unknown Angle in the triangle: A c b C a B Write down the formula. Ao = y o a = 26 b = 15 c = 13 Identify the sides and angle. Find the value of cos. Ao = - 0. 723 Ao = 136. 3 o negative tells you the angle is obtuse. The

Solving Trig Equations www. mathsrevision. com National 5 Now try N 5 TJ Ex 20. 2 Ch 20 (Page 200) created by Mr. Lafferty

Starter www. mathsrevision. com National 5 created by Mr. Lafferty

Solving Trig Equations www. mathsrevision. com National 5 Learning Intention Success Criteria 1. To explain some special trig relationships sin 2 xo + cos 2 xo = ? and tan xo and sin x cos x created by Mr. Lafferty 1. Know and learn the two special trig relationships. 2. Apply them to solve problems.

Solving Trig Equations www. mathsrevision. com National 5 Lets investigate sin 2 xo + cos 2 xo = ? Calculate value for x = 10, 20, 50, 250 sin 2 xo + cos sin 2 xo = 1 - cos 2 xo = 1 Learn ! cos 2 xo = 1 - sin 2 xo created by Mr. Lafferty

Solving Trig Equations www. mathsrevision. com National 5 Lets investigate tan xo and sin xo cos xo Calculate value for x = 10, 20, 50, 250 tan xo = sin xo cos xo created by Mr. Lafferty Learn !

Given that sin xo 3 =. Find cos xo. 5 cos 2 xo = 1 - sin 2 xo cos 2 xo = 1 - 9 ( ( cos 2 xo = 1 - 3 5 25 cos 2 xo = 16 25 √ cosxo = 4 5 2

Given that cos xo 6 =. Find sin xo and tanxo. 10 sin 2 xo = 1 - cos 2 xo 2 6 sin 2 xo = 1 10 ( ( o sin x tan xo = cos xo = sin 2 xo = 1 - 36 100 sin 2 xo = 64 100 √ sinxo = 8 10 tan xo 8 10 6 10 = 8 = 4 6 3

LHS = sinxsin 2 x + sinxcos 2 x = sinx(sin 2 x + cos 2 x) = 1 = sinx = RHS

= = = sinx cosx sinx 1

LHS 1 – sin 2 A = cos 2 A = = sin 2 A cos 2 A tan 2 A = RHS

Solving Trig Equations www. mathsrevision. com National 5 Now try N 5 TJ Ex 20. 3 Ch 20 (Page 201) created by Mr. Lafferty

Are you on Target ! www. mathsrevision. com Higher • Update you log book • Make sure you complete and correct ALL of the Trigonometry questions in the past paper booklet.