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Higher Maths Strategies www.maths4scotland.co.uk Click to start Compound Angles

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Maths4Scotland Higher Compound Angles The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber.

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Maths4Scotland Higher Quit This presentation is split into two parts Using Compound angle formula for Exact values Solving equations Choose by clicking on the appropriate button

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Maths4Scotland Higher Hint PreviousNext Quit 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. b) Write down the exact value of the gradient of OB. Draw trianglePythagoras Write down values for cos p and sin p Expand sin (2p) Expand cos (2p) Use m = tan (2p) 8 4 p

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Maths4Scotland Higher Hint PreviousNext Quit 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 Simplify

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Maths4Scotland Higher Hint PreviousNext Quit Using triangle PQR, as shown, find the exact value of cos 2x Use Pythagoras Write down values for cos x and sin x Expand cos 2x Substitute values Simplify

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Maths4Scotland Higher Hint PreviousNext Quit 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 Find the exact value of sin ( p + q ). Use Pythagoras Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Simplify 6 8 5 12 10 13 Mark up triangles

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Maths4Scotland Higher Hint PreviousNext Quit Draw triangles Use Pythagoras Expand sin 2A A and B are acute angles such that and. Find the exact value of a) b) c) 4 3 A 12 5 B Hypotenuses are 5 and 13 respectively 5 13 Write down sin A, cos A, sin B, cos B Expand cos 2A Expand sin (2A + B) Substitute

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Maths4Scotland Higher Hint PreviousNext Quit Draw triangle Use Pythagoras Expand sin (x + 30) If x° is an acute angle such that show that the exact value of 3 4 x Hypotenuse is 5 5 Write down sin x and cos x Substitute Simplify Table of exact values

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Maths4Scotland Higher Hint PreviousNext Quit Use Pythagoras Expand cos (x + y) Write down sin x, cos x, sin y, cos y. Substitute Simplify 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 °. Show that the exact value of 5

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Maths4Scotland Higher Hint PreviousNext Quit Draw triangle Use Pythagoras The framework of a child’s swing has dimensions as shown in the diagram. Find the exact value of sin x° Write down sin ½ x and cos ½ x Substitute Simplify Table of exact values 3 3 4 x Draw in perpendicular 2 h Use fact that sin x = sin ( ½ x + ½ x ) Expand sin ( ½ x + ½ x )

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Maths4Scotland Higher Hint PreviousNext Quit Given that find the exact value of Write down values for cos a and sin a Expand sin 2a Substitute values Simplify 3 a Draw triangle Use Pythagoras

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Maths4Scotland Higher Hint PreviousNext Quit Find algebraically the exact value of Expand sin ( +120) Use table of exact values Combine and substitute Table of exact values Expand cos ( +150) Simplify

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Maths4Scotland Higher Hint PreviousNext Quit If find the exact value of a)b) Write down values for cos and sin Expand sin 2 Draw triangle Use Pythagoras 5 4 3 Opposite side = 3 Expand sin 4 (4 = 2 + 2 ) Expand cos 2 Find sin 4

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Maths4Scotland Higher Hint PreviousNext Quit Draw triangles Use Pythagoras Expand sin (P + Q) For acute angles P and Q Show that the exact value of 12 13 P 5 3 Q Adjacent sides are 5 and 4 respectively 5 4 Write down sin P, cos P, sin Q, cos Q Substitute Simplify

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Maths4Scotland Higher Previous Quit You have completed all 12 questions in this section Back to start

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Maths4Scotland Higher Quit Solving Equations Using Compound angle formula for Continue

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Maths4Scotland Higher Hint PreviousNext Quit Solve the equation for 0 ≤ x ≤ correct to 2 decimal places Replace cos 2x with Substitute Simplify Factorise Hence Discard Find acute x Determine quadrants AS CT

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Maths4Scotland Higher Hint PreviousNext Quit Table of exact values Solve simultaneously Rearrange Find acute 2 x Determine quadrants AS CT The diagram shows the graph of a cosine function from 0 to . a) State the equation of the graph. b) The line with equation y = - 3 intersects this graph at points A and B. Find the co-ordinates of B. Equation Check range Deduce 2x

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Functions f and g are defined on suitable domains by f(x) = sin (x) and g(x) = 2x a)Find expressions for: i) f(g(x)) ii) g(f(x)) b)Solve 2 f(g(x)) = g(f(x)) for 0 x 360° Maths4Scotland Higher Hint PreviousNext Quit Table of exact values 2 nd expression Form equation Rearrange Determine quadrants AS CT 1 st expression Common factor Replace sin 2x Hence Determine x

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Functions are defined on a suitable set of real numbers a)Find expressions for i) f(h(x)) ii) g(h(x)) b)i) Show that ii) Find a similar expression for g(h(x)) iii) Hence solve the equation Maths4Scotland Higher Hint PreviousNext Quit Table of exact values 2 nd expression Simplify 1 st expr. Similarly for 2 nd expr. Determine quadrants AS CT 1 st expression Use exact values Form Eqn. Simplifies to Rearrange: acute x

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a)Solve the equation sin 2x - cos x = 0 in the interval 0 x 180° b)The diagram shows parts of two trigonometric graphs, y = sin 2x and y = cos x. Use your solutions in (a) to write down the co-ordinates of the point P. Maths4Scotland Higher Hint PreviousNext Quit Table of exact values Determine quadrants for sin x AS CT Common factor Replace sin 2x Hence Determine x Solutions for where graphs cross By inspection (P) Find y value Coords, P

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Maths4Scotland Higher Hint PreviousNext Quit Solve the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants AS CT Table of exact values AS CT Solutions are: x= 60°, 132°, 228° and 300°

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Maths4Scotland Higher Hint PreviousNext Quit Solve the equation for 0 ≤ x ≤ 2 Rearrange Find acute x Determine quadrants AS CT Table of exact values Solutions are: Note range

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Maths4Scotland Higher Hint PreviousNext Quit a) Write the equation cos 2 + 8 cos + 9 = 0 in terms of cos and show that for cos it has equal roots. b) Show that there are no real roots for Rearrange Divide by 2 Deduction Factorise Replace cos 2 with Equal roots for cos Try to solve: Hence there are no real solutions for No solution

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Solve algebraically, the equation sin 2x + sin x = 0, 0 x 360 Maths4Scotland Higher Hint PreviousNext Quit Table of exact values Determine quadrants for cos x AS CT Common factor Replace sin 2x Hence Determine x x = 0°, 120°, 240°, 360°

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Find the exact solutions of 4sin 2 x = 1, 0 x 2 Maths4Scotland Higher Hint PreviousNext Quit Table of exact values Determine quadrants for s in x AS CT Take square roots Rearrange Find acute x + and – from the square root requires all 4 quadrants

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Maths4Scotland Higher Hint PreviousNext Quit Solve the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants AS CT Table of exact values Solutions are: x= 60°, 180° and 300°

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Maths4Scotland Higher Hint PreviousNext Quit Solve algebraically, the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants Table of exact values AS CT Solutions are: x= 60° and 300° Discard above

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Maths4Scotland Higher Previous Quit You have completed all 12 questions in this presentation Back to start

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Maths4Scotland Higher Return 30°45°60° sin cos tan1 Table of exact values

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