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

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

a) Find the exact values of: i) sin (2p) ii) cos (2p) Maths4Scotland 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 (2p) ii) cos (2p) The line OB is inclined at an angle 2p radians to the x-axis. b) Write down the exact value of the gradient of OB. 8 4 p Draw triangle Pythagoras Write down values for cos p and sin p Expand sin (2p) Expand cos (2p) Hint Use m = tan (2p) Previous Quit Quit Next

In triangle ABC show that the exact value of Maths4Scotland Higher 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 Hint Simplify Previous Quit Quit Next

cos x and sin x Maths4Scotland Higher 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 Hint Simplify Previous Quit Quit Next

On the co-ordinate diagram shown, A is the point (6, 8) and Maths4Scotland Higher 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). 10 13 6 8 5 12 Mark up triangles Use Pythagoras Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Hint Simplify Previous Quit Quit Next

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

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

The diagram shows two right angled triangles Maths4Scotland 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°. Show that the exact value of 5 Use Pythagoras Write down sin x, cos x, sin y, cos y. Expand cos (x + y) Substitute Hint Simplify Previous Quit Quit Next

The framework of a child’s swing has dimensions Maths4Scotland Higher The framework of a child’s swing has dimensions as shown in the diagram. Find the exact value of sin x° Draw triangle Draw in perpendicular Use Pythagoras 3 4 x 2 h 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) Substitute Hint Simplify Previous Quit Quit Next Table of exact values

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

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

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

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

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Using Compound angle formula for Maths4Scotland Higher Using Compound angle formula for Solving Equations Continue Quit Quit

Solve the equation for 0 ≤ x ≤  correct to 2 decimal places Maths4Scotland Higher Solve the equation for 0 ≤ x ≤  correct to 2 decimal places Replace cos 2x with Determine quadrants Substitute A S C T Simplify Factorise Hence Discard Hint Find acute x Previous Quit Quit Next

Maths4Scotland Higher Equation Determine quadrants 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 Determine quadrants Solve simultaneously A S C T Rearrange Check range Find acute 2x Hint Deduce 2x Previous Quit Quit Next Table of exact values

a) Find expressions for: i) f(g(x)) ii) g(f(x)) Maths4Scotland Higher 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° Determine x 1st expression 2nd expression A S C T Form equation Determine quadrants Replace sin 2x Rearrange Common factor Hint Hence Previous Quit Quit Next Table of exact values

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

a) Solve the equation sin 2x - cos x = 0 in the interval 0  x  180° Maths4Scotland Higher 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. Replace sin 2x Solutions for where graphs cross Common factor Hence By inspection (P) Determine x Find y value A S C T Coords, P Determine quadrants for sin x Hint Previous Quit Quit Next Table of exact values

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

Solutions are: Maths4Scotland Higher Solve the equation for 0 ≤ x ≤ 2 Rearrange Find acute x Note range A S C T Determine quadrants Solutions are: Hint Previous Quit Quit Next Table of exact values

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

x = 0°, 120°, 240°, 360° Maths4Scotland Higher Solve algebraically, the equation sin 2x + sin x = 0, 0  x  360 Replace sin 2x Determine quadrants for cos x Common factor A S C T Hence Determine x Hint x = 0°, 120°, 240°, 360° Previous Quit Quit Next Table of exact values

Find the exact solutions of 4sin2 x = 1, 0  x  2 Maths4Scotland Higher Find the exact solutions of 4sin2 x = 1, 0  x  2 Rearrange Take square roots Find acute x Determine quadrants for sin x + and – from the square root requires all 4 quadrants A S C T Hint Previous Quit Quit Next Table of exact values

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

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

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

30° 45° 60° sin cos tan 1 Table of exact values Maths4Scotland Higher Return