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Maths 4 Scotland 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

Maths 4 Scotland 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

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 xaxis. 4 of the Draw Pythagoras b) Write down the exact value gradient of OB. p triangle 8 Write down values for cos p and sin p Expand sin (2 p) Expand cos (2 p) Use m = tan (2 p) Previous Hint Quit Next

Maths 4 Scotland 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 Hint Previous Quit Next

Maths 4 Scotland Higher 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 Simplify Hint Previous Quit Next

Maths 4 Scotland 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 Mark up =q triangles Find the exact value of sin (p + q). Use Pythagoras 10 8 6 12 13 5 Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Simplify Hint Previous Quit Next

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

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 Hypotenuse is 5 Write down sin x and cos x Expand sin (x + 30) Substitute Simplify Hint Previous Quit Table of exact values Quit Next

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°. Show that the exact value of Use Pythagoras 5 Write down sin x, cos x, sin y, cos y. Expand cos (x + y) Substitute 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 triangle Use Draw in Pythagoras perpendicular Use fact that sin x = sin ( ½ x + ½ x) Write down sin ½ x and cos ½ x 3 h 2 4 Expand sin ( ½ x + ½ x) Substitute Simplify Previous Hint Quit Table of exact values Quit Next

Maths 4 Scotland Higher Given that find the exact value of Draw triangle a 3 Use Pythagoras Write down values for cos a and sin a Expand sin 2 a Substitute values Simplify Previous Hint Quit Next

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

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

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

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Maths 4 Scotland Higher for 0 ≤ x ≤ correct to 2 Solve the equation decimal places Replace cos 2 x with Substitute Determine quadrants S A Simplify T Factorise C Hence Discard Find acute x Previous Hint Quit Next

Maths 4 Scotland 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. Equatio n Solve simultaneously Rearrange Determine quadrants S A Check range T C Find acute 2 x Deduce 2 x Previous Hint Quit Table of exact values Quit Next

Maths 4 Scotland Higher Functions f and g are defined on suitable domains by f(x) = sin (x) and g(x) = 2 x a) Find expressions for: Determine i) f(g(x)) ii) g(f(x)) x b) Solve 2 f(g(x)) = g(f(x)) for 0 x 360° 1 st expression 2 nd S A expression Form Determin equation e Replace sin quadrant T C 2 x s Rearrange Common factor Hence Previous Hint Quit Table of exact values Quit Next

Maths 4 Scotland Higher Functions set of real numbers a) Find expressions for b) i) are defined on a suitable i) f(h(x)) ii) Show that g(h(x)) ii) Find a similar expression for g(h(x)) Simplifies to 1 st iii) Hence solve the equation expression 2 nd expression Simplify 1 st expr. Use exact values Similarly for 2 nd expr. Rearrang e: acute x Determin e quadrant s S A T C Hint Form Eqn. Previous Quit Table of exact values Quit Next

Maths 4 Scotland a) b) Higher Solve the equation sin 2 x - cos x = 0 in the interval 0 x 180° The diagram shows parts of two trigonometric graphs, y = sin 2 x and y = cos x. Use your solutions in (a) to write down the co-ordinates of the point P. Replace sin 2 x Common factor Hence Solutions for where graphs cross By inspection (P) Find y value Determine x S Determine quadrants for sin x Previous T A Coords, P Hint C Quit Table of exact values Quit Next

Maths 4 Scotland Higher Solve the equation for 0 ≤ x ≤ 360° Replace cos 2 x with Substitute Determine quadrants Simplify S A T C Factorise Hence Find acute x Solutions are: x= 60°, 132°, 228° and 300° Previous Quit Table of exact values Quit Hint Next

Maths 4 Scotland Higher Solve the equation for 0 ≤ x ≤ 2 Rearrange Find acute x Note range Determine quadrants S A T C Solutions are: Hint Previous Quit Table of exact values Quit Next

Maths 4 Scotland Higher a) Write the equation cos 2 q + 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 2 q with Rearrange Try to solve: No solution Divide by 2 Hence there are no real solutions for Factorise Deductio n Equal roots for cos q Hint Previous Quit Next

Maths 4 Scotland Higher Solve algebraically, the equation sin 2 x + sin x = 0, 0 x 360 Determine quadrants for cos x S A Replace sin 2 x Common factor Hence T C Determine x x = 0°, Previous Quit Table of exact values 120°, 240°, 360° Quit Hint Next

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

Maths 4 Scotland Higher Solve the equation for 0 ≤ x ≤ 360° Replace cos 2 x with Substitute Determine quadrants Simplify S A T C Factorise Hence Find acute x Solutions are: x= 60°, 180° and 300° Previous Quit Table of exact values Quit Hint Next

Maths 4 Scotland Higher Solve algebraically, the equation 360° for 0 ≤ x ≤ Replace cos 2 x with Substitute Determine quadrants Simplify S A T C Factorise Hence Find acute x Discard above Solutions are: x= 60° and 300° Previous Quit Table of exact values Quit Hint Next

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Maths 4 Scotland Higher Table of exact values 30° 45° sin cos tan Return 1 60°