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

Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. Maths4Scotland Higher Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. Express y = 2 sin x + 5 cos x in the form k sin (x + a) where k > 0 and 0  a  360 b) Find the coordinates of the minimum turning point P. Expand ksin(x + a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Hint Minimum when: P has coords. Previous Quit Quit Next

Maths4Scotland Higher Write sin x - cos x in the form k sin (x - a) stating the values of k and a where k > 0 and 0  a  2 b) Sketch the graph of sin x - cos x for 0  a  2 showing clearly the graph’s maximum and minimum values and where it cuts the x-axis and the y-axis. Expand ksin(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Sketch Graph Hint Previous Quit Quit Next Table of exact values

Maths4Scotland Higher Express in the form Expand kcos(x + a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are +) Dividing: Put together: Hint Previous Quit Quit Next

Maths4Scotland Higher Find the maximum value of and the value of x for which it occurs in the interval 0  x  2. Express as Rcos(x - a): Equate coefficients: Square and add a is in 4th quadrant (sin is - and cos is +) Dividing: Put together: Hint Max value: when Previous Quit Quit Next Table of exact values

Express in the form Maths4Scotland Higher Expand ksin(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are both +) Dividing: Put together: Hint Previous Quit Quit Next

The diagram shows an incomplete graph of Maths4Scotland Higher The diagram shows an incomplete graph of Find the coordinates of the maximum stationary point. Max for sine occurs Max value of sine function: Sine takes values between 1 and -1 Max value of function: 3 Coordinates of max s.p. Hint Previous Quit Quit Next

a) Express f (x) in the form Maths4Scotland Higher a) Express f (x) in the form b) Hence solve algebraically Expand kcos(x - a): Equate coefficients: Square and add a is in 1st quadrant (sin and cos are both + ) Dividing: Put together: Solve equation. Hint Cosine +, so 1st & 4th quadrants Previous Quit Quit Next

Solve the simultaneous equations where k > 0 and 0  x  360 Maths4Scotland Higher Solve the simultaneous equations where k > 0 and 0  x  360 Use tan A = sin A / cos A Divide Find acute angle Determine quadrant(s) Sine and cosine are both + in original equations Solution must be in 1st quadrant Hint State solution Previous Quit Quit Next

Solve the equation in the interval 0  x  360. Maths4Scotland Higher Solve the equation in the interval 0  x  360. Use Rcos(x - a): Equate coefficients: Square and add a is in 2nd quadrant (sin + and cos - ) Dividing: Put together: Solve equation. Hint Cosine +, so 1st & 4th quadrants Previous Quit Quit Next

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