Higher Unit 2 www mathsrevision com Higher Unit

Higher Unit 2 www. mathsrevision. com Higher Unit 2 EF 1. 2 N 5 Trig Exact Values and Trig Identities Connection between Radians and degrees & Exact values Trigonometry identities of the form sin(A+B) Double Angle formulae Application & Exam Type Questions Wave function format ksin(x ± α) Wave function format ksin(2 x ± α) www. mathsrevision. com

Exact Values www. mathsrevision. com Higher Unit 2 EF 1. 2 Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values 2 60º 2 2 3 60º 30º 2 1 This triangle will provide exact values for sin, cos and tan 30º and 60º

Exact Values Unit 2 EF 1. 2 www. mathsrevision. com Higher x Sin xº 0º 0 Cos xº 1 Tan xº 0 30º 60º 90º ½ 3 2 1 3 ½ 0 3 2 45º

Exact Values Unit 2 EF 1. 2 Higher www. mathsrevision. com Consider the square with sides 1 unit 1 1 2 45º 1 We are now in a position to calculate exact values for sin, cos and tan of 45 o

Exact Values Unit 2 EF 1. 2 www. mathsrevision. com Higher x Sin xº 0º 0 Cos xº 1 Tan xº 0 30º 45º 60º 90º ½ 1 2 3 2 1 3 1 2 ½ 0 1 3 Undefined 2

Exact value table and quadrant rules. Unit 2 EF 1. 2 www. mathsrevision. com Higher tan 150 o = - tan(180 - 150) o = - tan 30 o = -1/ √ 3 (Q 2 so neg) cos 300 o = cos(360 - 300) o = cos 60 o (Q 4 so pos) sin 120 o = sin(180 - 120) o = sin 60 o (Q 2 so pos) tan 300 o = - tan(360 -300)o = - tan 60 o (Q 4 so neg) = 1/ 2 =√ = 3/2 -√ 3

Extra Practice www. mathsrevision. com Higher Unit 2 EF 1. 2 HHM Ex 4 D & Ex 4 E

Trig Identities Unit 2 EF 1. 2 Higher www. mathsrevision. com An identity is a statement which is true for all values. eg 3 x(x + 4) = 3 x 2 + 12 x eg (a + b)(a – b) = a 2 – b 2 Trig Identities (1) sin 2θ + cos 2 θ = 1 (2) sin θ = tan θ cos θ θ ≠ an odd multiple of π/2 or 90°.

Trig Identities Unit 2 EF 1. 2 Higher www. mathsrevision. com Reason c a θo b (1) a 2 +b 2 = c 2 sin 2θo + cos 2 θo = sinθo = a/c cosθo = b/c

Trig Identities www. mathsrevision. com Higher Unit 2 EF 1. 2 Simply rearranging we get two other forms sin 2θ + cos 2 θ = 1 sin 2 θ = 1 - cos 2 θ = 1 - sin 2 θ

Extra Practice www. mathsrevision. com Higher Unit 2 EF 1. 2 HHM Page 352 Ex 17 Q 2

Radians Unit 2 EF 1. 2 www. mathsrevision. com Higher Radian measure is an alternative to degrees and is based upon the ratio of arc Length radius L θ θ- theta (angle at the centre) r So, full circle 360 o 2π radians

Radians Copy Table 90 o π 2 π o 60 3 π o 45 4 30 o π 6 360 o 2π 270 o 3π 2 2π o 120 3 3π o 135 4 240 o 5π 6 210 o 150 o 180 o 4π 3 5π o 225 4 7π 6 π 5π 3 7π o 315 4 300 o 330 o 11π 6

Converting Unit 2 EF 1. 2 www. mathsrevision. com Higher For any values ÷ 180 then X π degrees radians ÷ π then x 180 Drill

Converting Unit 2 EF 1. 2 www. mathsrevision. com Higher Ex 1 72 o = Ex 2 330 o = 330/180 X π = 11 π /6 Ex 3 2π /9 = 2π /9 ÷ π x 180 o = Ex 4 72/ 180 X π = 2π /5 2/ 9 23π/18 = 23π /18 ÷ π x 180 o = X 180 o = 40 o 23/ 18 X 180 o = 230 o Drill

Exact value table and quadrant rules. Unit 2 EF 1. 2 www. mathsrevision. com Higher Find the exact value of cos 2(5π/6) – sin 2(π/6) cos(5π/6) = cos 150 o = cos(180 - 150)o = - cos 30 o = - √ 3 / (Q 2 so neg) sin(π/6) = sin 30 o = 1/2 cos 2(5π/6) – sin 2(π/6) = (- √ 3 / 2) 2 – ( 1/ 2 ) 2 = ¾ - 1/ 4 = 1/ 2 2

Exact value table and quadrant rules. Unit 2 EF 1. 2 Higher www. mathsrevision. com Prove that sin(2 π /3) = tan (2 π /3) cos (2 π /3) sin(2π/3) = sin 120 o = sin(180 – 120)o = sin 60 o = √ 3/ 2 cos(2 π /3) = cos 120 o = cos(180 – 120)o = - cos 60 o = -1/2 tan(2 π /3) = tan 120 o = tan(180 – 120)o = -tan 60 o = - √ 3 2π/ ) sin( 3 3 / ÷ - 1/ LHS = = √ 2 2 cos (2 π /3) = - √ 3 = tan(2π/3) = RHS = √ 3/ 2 X -2

Trig Identities Unit 2 EF 1. 2 sin θ = 5/13 where 0 < θ < π/2 Higher www. mathsrevision. com Example 1 Find the exact values of cos θ and tan θ. cos 2 θ = 1 - sin 2 θ = 1 – (5/13)2 = 1– = 25/ 144/ 169 cos θ = √(144/169) = 12/ 13 or -12/13 Since θ is between 0 < θ < π/2 then cos θ > 0 So cos θ = tan θ = sinθ = cos θ 5/ = 5/ tan θ = 5/ 13 13 12 ÷ X 12/ 13 13 13/ 12

Trig Identities www. mathsrevision. com Higher Unit 2 EF 1. 2 Given that cos θ = -2/ √ 5 where π< θ < Find sin θ and tan θ. sin 2 θ = 1 - cos 2 θ Hence sinθ = 5 sin θ = √(1/5) = 1/ √ 5 or - 1/ 3 π /2 sinθ < 0 = 1 – 4/ 5 1/ /2 Since θ is between π< θ < = 1 – (-2/ √ 5 )2 = 3 π √ 5 - 1/ tan θ = sinθ = cos θ - 1/ √ 5 = - 1/ √ 5 ÷ -2/ √ 5 X - √ 5 / Hence tan θ = 1/2 2

Higher Unit 2 EF 1. 2 www. mathsrevision. com Extra Practice HHM Ex 4 C

Trig Identities www. mathsrevision. com Higher Supplied on a formula sheet !! Unit 2 EF 1. 2 The following relationships are always true for two angles A and B. 1 a. 1 b. sin(A + B) = sin. Acos. B + cos. Asin. B sin(A - B) = sin. Acos. B - cos. Asin. B 2 a. cos(A + B) = cos. Acos. B – sin. Asin. B 2 b. cos(A - B) = cos. Acos. B + sin. Asin. B Quite tricky to prove but some of following examples should show that they do work!!

Trig Identities Higher Examples 1 Unit 2 EF 1. 2 www. mathsrevision. com (1) Expand cos(U – V). (use formula 2 b ) cos(U – V) = cos. Ucos. V + sin. Usin. V (2) Simplify sinf°cosg° - cosf°sing° (use formula 1 b ) sinf°cosg° - cosf°sing° = sin(f – g)° (3) Simplify cos 8 θ sinθ + sin 8 θ cos θ (use formula 1 a ) cos 8 θ sin θ + sin 8 θ cos θ = sin(8 θ + θ) = sin 9 θ

Trig Identities www. mathsrevision. com Higher Example 2 Unit 2 EF 1. 2 By taking A = 60° and B = 30°, prove the identity for cos(A – B). NB: cos(A – B) = cos. Acos. B + sin. Asin. B LHS = cos(60 – 30 )° = cos 30° = 3/ 2 RHS = cos 60°cos 30° + sin 60°sin 30° =(½ Hence LHS = RHS !! X = 3/ 4 = 3/ 2 3/ + 3/ X ½) ) + ( 2 2 3/ 4

Trig Identities www. mathsrevision. com Higher Example 3 Unit 2 EF 1. 2 Prove that sin 15° = ¼( 6 - 2) sin 15° = sin(45 – 30)° = sin 45°cos 30° - cos 45°sin 30° = (1/ 2 X 3/ 1/ ) ( 2 2 X ½) = ( 3/2 2 - = ( 3 - 1) 2 2 = ( 6 - 2) 4 = ¼( 6 - 2) 1/ 2 2) X 2 2

Trig Identities www. mathsrevision. com Higher Example 4 NAB type Question Unit 2 EF 1. 2 y 41 x 40 Show that 3 4 cos( - ) = 187/ 205 Triangle 2 Triangle 1 If missing side = x If missing side = y Then x 2 = 412 – 402 = 81 Then y 2 = 42 + 32 = 25 So So x=9 y=5 sin = 9/41 and cos = 40/41 sin = 3/5 and cos = 4/5

Trig Identities Higher Unit 2 EF 1. 2 www. mathsrevision. com sin = 9/41 and cos = 40/41 sin = 3/5 and cos = 4/5 cos( - ) = cos + sin = (40/41 X 4/5) + (9/41 X 3/5 ) = 160/ 205 = 187/ 205 + 27/ 205 Remember this is a NAB type Question

Higher Unit 2 EF 1. 2 www. mathsrevision. com Extra Practice HHM Ex 11 B, Ex 11 C, Ex 11 D

Paper 1 type questions Trig Identities www. mathsrevision. com Higher Example Unit 2 EF 1. 2 Simplify sin(θ - /3) + cos(θ + /6) + cos( /2 θ) sin(θ - /3) + cos(θ + /6) + cos( /2 - θ) = sin θ cos /3 – cos θ sin /3 + cos θ cos /6 – sin θ sin /6 + cos /2 cos θ + sin /2 sin θ = 1/2 sin θ – 3/2 cos θ + 3/ = 2 cos θ – 1/2 sin θ + 0 x cos θ + 1 X sin θ

Paper 1 type questions Trig Identities Higher Example Unit 2 EF 1. 2 www. mathsrevision. com Prove that (sin. A + cos. B)2 + (cos. A - sin. B)2 = 2(1 + sin(A - B)) LHS = (sin. A + cos. B)2 + (cos. A - sin. B)2 = sin 2 A + 2 sin. Acos. B + cos 2 A – 2 cos. Asin. B + sin 2 B = (sin 2 A + cos 2 A) + (sin 2 B + cos 2 B) + 2 sin. Acos. B - 2 cos. Asin. B = 1 + 2(sin. Acos. B - cos. Asin. B) = 2 + 2 sin(A – B) = 2(1 + sin(A – B)) = RHS

Extra Practice Unit 2 EF 1. 2 www. mathsrevision. com Higher HHM Ex 11 E

Double Angle Formulae www. mathsrevision. com Higher Unit 2 EF 1. 2

Double Angle formulae www. mathsrevision. com Higher Mixed Examples: Unit 2 EF 1. 2 Substitute form the tan (sin/cos) equation +ve because A is acute Similarly: 3 -4 -5 triangle ! A is greater than 45 degrees – hence 2 A is greater than 90 degrees.

Double Angle formulae www. mathsrevision. com Higher Unit 2 EF 1. 2

Double Angle formulae www. mathsrevision. com Higher Unit 2 EF 1. 2

Double Angle formulae www. mathsrevision. com Higher Unit 2 EF 1. 2

Extra Practice Unit 2 EF 1. 2 www. mathsrevision. com Higher HHM Ex 11 G & Ex 11 I

Maths 4 Scotland Higher Application of Addition and Double Angle Formulae Non-calculator questions will be indicated You will need a pencil, paper, ruler and rubber.

Maths 4 Scotland Higher A is the point (8, 4). The line OA is inclined at an angle p radians to the x-axis a) Find the exact values of: i) sin (2 p) ii) cos (2 p) The line OB is inclined at an angle 2 p radians to the x -axis. 4 b) Draw Write down the exact value of. Pythagora the gradient of OB. p triangle s 8 Write down values for cos p and sin p Expand sin (2 p) Expand cos (2 p) Use m = tan (2 p)

Maths 4 Scotland In triangle ABC show that the exact value of Use Pythagoras Write down values for sin a, cos a, sin b, cos b Expand sin (a + b) Substitute values Simplif y Higher

Maths 4 Scotland Using triangle PQR, as shown, find the exact value of cos 2 x Use Pythagoras Write down values for cos x and sin x Expand cos 2 x Substitute values Simplif y Higher

Maths 4 Scotland On the co-ordinate diagram shown, A is the point (6, 8) and B is the point (12, -5). Angle AOC = p and angle COB = q Mark up triangles Use Find the exact value of sin (p + q). Pythagoras Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Simplif y Higher 10 8 6 13 12 5

Maths 4 Scotland Higher A and B are acute angles such that and . Find the exact value of a) b) Draw triangles 5 A c) 4 3 13 B 12 Use Hypotenuses are 5 and 13 respectively Pythagoras Write down sin A, cos A, sin B, cos B Expand sin 2 A Expand cos 2 A Expand sin (2 A + B) Substitu te 5

Maths 4 Scotland Higher If x° is an acute angle such that 5 show that the exact value of 4 x 3 Draw triangle Use Pythagoras Write down sin x and cos x Expand sin (x + 30) Substitut e Simplify Hypotenuse is 5

Maths 4 Scotland Higher The diagram shows two right angled triangles ABD and BCD with AB = 7 cm, BC = 4 cm and CD = 3 cm. Angle DBC = x° and angle ABD is y°. 5 Show that the exact value of Use Pythagoras Write down sin x, cos x, sin y, cos y. Expand cos (x + y) Substitut e Simplify Hint Previous Quit Next

Maths 4 Scotland Higher The framework of a child’s swing has dimensions as shown in the diagram. Find the exact value of sin x° Draw in triangle perpendicular Use fact that sin x = sin ( ½ x + ½ x) Write down sin ½ x and cos ½ x Expand sin ( ½ x + ½ x) Substitut e Simplify Use Pythagoras 3 x h 2 4 3

Maths 4 Scotland Higher Given that find the exact value of Draw triangle Use Pythagoras Write down values for cos a and sin a Expand sin 2 a Substitute values Simplif y a 3

Maths 4 Scotland Find algebraically the exact value of Expand sin (q +120) Expand cos (q +150) Use table of exact values Combine and substitute Simplif y Higher

Maths 4 Scotland If exact value of a) Higher find the b) Draw Use triangle Pythagoras Write down values for cos q and sin q Expand sin 2 q Expand sin 4 q (4 q = 2 q + 2 q) Expand cos 2 q Find sin 4 q 5 3 q Opposite side = 3 4

Maths 4 Scotland Higher For acute angles P and Q Show that the exact value of Draw triangles Substitu te Simplify 12 P 5 3 Q 5 Use Adjacent sides are 5 and 4 respectively Pythagoras Write down sin P, cos P, sin Q, cos Q Expand sin (P + Q) 13 4

The Wave Function www. mathsrevision. com Higher Heart beat Unit 2 EF 1. 2 Many wave shapes, whether occurring as sound, light, water or electrical waves, can be described mathematically as a combination of sine and cosine waves. Spectrum Analysis Electrical

General shape for y = sinx + cosx The Wave Function 1. Like y = sin(x) shifted left Higher 2. Like. Unit y = cosx right 2 EFshifted 1. 2 www. mathsrevision. com 3. Vertical height different y = sin(x)+cos(x) y = sin(x) y = cos(x) Demo

The Wave Function www. mathsrevision. com Higher Unit 2 EF 1. 2 Whenever a function is formed by adding cosine and sine functions the result can be expressed as a related cosine or sine function. In general: With these constants the expressions on the right hand sides = those on the left hand side FOR ALL VALUES OF x

The Wave Function www. mathsrevision. com Higher Worked Example: Unit 2 EF 1. 2 Re-arrange The left and right hand sides must be equal for all values of x. So, the coefficients of cos x and sin x must be equal: A pair of simultaneous equations to be solved

The Wave Function www. mathsrevision. com Higher Unit 2 EF 1. 2 Find tan ratio Square and add note: sin(+) and cos(+)

The Wave Function www. mathsrevision. com Higher Unit 2 EF 1. 2 Note: sin(+) and cos(+) 90 o 180 Demo o S A T C 270 o 0 o

The Wave www. mathsrevision. com Higher Example Expand Function equate coefficients Unit 2 EF 1. 2 Square and add Find tan ratio note: sin(+) and cos(+) 90 o 180 o S A T C 270 o 0 o

The Wave Function Unit 2 EF 1. 2 www. mathsrevision. com Higher Finally: Demo

Extra Practice www. mathsrevision. com Higher Unit 2 EF 1. 2 HHM Ex 16 C , Ex 16 D , Ex 16 E

The Wave www. mathsrevision. com Higher Example Expand equate Function coefficients Unit 2 EF 1. 2 Square and add Find tan ratio noting sign ofo 90 sin(+) and cos(+) 180 Demo o S A T C 270 o 0 o

The Wave Function Unit 2 EF 1. 2 www. mathsrevision. com Higher Finally:

Extra Practice www. mathsrevision. com Higher Unit 2 EF 1. 2 HHM Ex 16 F

Are you on Target ! Unit 2 EF 1. 2 www. mathsrevision. com Higher • Update you log book • Make sure you complete and correct ALL of the Trigonometry questions in the past paper booklet.