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

3. 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.

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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 )

13. 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

14. 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

15. 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 

16. 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

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

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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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°

25. 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

26. 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

27. 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°

28. 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

29. 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°

30. 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

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

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