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Higher Maths

www.maths4scotland.co.uk. Higher Maths. Strategies. Compound Angles. Click to start. Maths4Scotland Higher. The following questions are on. Compound Angles. Non-calculator questions will be indicated.

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Higher Maths

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

  2. 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

  3. 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

  4. p 4 8 Hint 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. Draw triangle Pythagoras Write down values for cos p and sin p Expand sin (2p) Expand cos (2p) Use m = tan (2p) Previous Next Quit Quit

  5. Hint 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 Simplify Previous Next Quit Quit

  6. Hint Maths4Scotland Higher Using triangle PQR, as shown, find the exact value of cos 2x Use Pythagoras Write down values for cosxandsinx Expand cos 2x Substitute values Simplify Previous Next Quit Quit

  7. 10 8 6 12 5 13 Hint 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). Mark up triangles Use Pythagoras Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Simplify Previous Next Quit Quit

  8. A and B are acute angles such that and . Find the exact value of a) b) c) 13 5 B A 3 5 12 4 Hint Maths4Scotland Higher 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) Substitute Previous Next Quit Quit

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

  10. 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 Hint Maths4Scotland Higher Use Pythagoras Write down sin x, cos x, sin y, cos y. Expand cos (x + y) Substitute Simplify Previous Next Quit Quit

  11. x 3 3 h 2 4 Hint 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 Use Pythagoras Draw in 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) Substitute Simplify Previous Next Quit Quit Table of exact values

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

  13. Hint 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 Previous Next Quit Quit Table of exact values

  14. 5 q 4 Hint Maths4Scotland Higher If find the exact value of a) b) 3 Opposite side = 3 Draw triangle Use Pythagoras Write down values for cosqandsinq Expand sin 2q Expand sin 4q (4q= 2q + 2q) Expand cos 2q Find sin 4q Previous Next Quit Quit

  15. 13 5 3 12 P Q 5 4 Hint Maths4Scotland Higher For acute angles P and Q Show that the exact value of 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 Simplify Previous Next Quit Quit

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

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

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

  19. S A T C Hint Maths4Scotland Higher 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 Rearrange Check range Find acute 2x Deduce 2x Previous Next Quit Quit Table of exact values

  20. S A T C Hint 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 Form equation Determine quadrants Replace sin 2x Rearrange Common factor Hence Previous Next Quit Quit Table of exact values

  21. S A T C Hint 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 Simplifies to 1st expression Rearrange: 2nd expression acute x Simplify 1st expr. Use exact values Determine quadrants Similarly for 2nd expr. Form Eqn. Previous Next Quit Quit Table of exact values

  22. S A T C Hint 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 Coords, P Determine quadrants for sin x Previous Next Quit Quit Table of exact values

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

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

  25. Hint 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 Try to solve: Replace cos 2q with Rearrange No solution Divide by 2 Hence there are no real solutions for q Factorise Equal roots for cos q Deduction Previous Next Quit Quit

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

  27. S A T C Hint Maths4Scotland Higher Find the exact solutions of 4sin2x = 1, 0  x  2 Rearrange Take square roots Find acute x + and – from the square root requires all 4 quadrants Determine quadrants for sin x Previous Next Quit Quit Table of exact values

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

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

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

  31. Maths4Scotland Higher Table of exact values Return

  32. Hint Maths4Scotland Higher Previous Next Quit Quit Table of exact values

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