Higher Unit 3 Exam Type Questions What is a Wave Function Connection with Trig Identities Earlier Maximum and Minimum Values Solving Equations involving the Wave Function Exam Type Questions www.mathsrevision.com

The Wave Function Heart beat Many wave shapes, whether occurring as sound, light, water or electrical waves, can be described mathematically as a combination of sine and cosine waves. Spectrum Analysis Electrical

The Wave Function General shape for y = sinx + cosx Like y = sin(x) shifted left Like y = cosx shifted right Vertical height different The Wave Function y = sin(x)+cos(x) y = sin(x) y = cos(x)

The Wave Function Whenever a function is formed by adding cosine and sine functions the result can be expressed as a related cosine or sine function. In general: With these constants the expressions on the right hand sides = those on the left hand side FOR ALL VALUES OF x

The left and right hand sides must be equal for all values of x. The Wave Function Worked Example: Re-arrange The left and right hand sides must be equal for all values of x. So, the coefficients of cos x and sin x must be equal: A pair of simultaneous equations to be solved

Find tan ratio note: sin(+) and cos(+) The Wave Function Find tan ratio note: sin(+) and cos(+) Square and add

Note: sin(+) and cos(+) The Wave Function Note: sin(+) and cos(+) C A S T 0o 180o 270o 90o

The Wave Function Example 90o S A 180o 0o T C 270o Expand and equate coefficients The Wave Function Example Find tan ratio note: sin(+) and cos(+) C A S T 0o 180o 270o 90o Square and add

The Wave Function Finally:

The Wave Function Example 90o S A 180o 0o T C 270o Expand and equate coefficients The Wave Function Example Square and add Find tan ratio noting sign of sin(+) and cos(+) C A S T 0o 180o 270o 90o

The Wave Function Finally:

Maximum and Minimum Values Worked Example: b) Hence find: i) Its maximum value and the value of x at which this maximum occurs. ii) Its minimum value and the value of x at which this minimum occurs.

Maximum and Minimum Values Expand and equate coefficients Maximum and Minimum Values Square and add C A S T 0o 180o 270o 90o Find tan ratio note: sin(+) and cos(-)

Maximum and Minimum Values Maximum, we have:

Maximum and Minimum Values Minimum, we have:

Maximum and Minimum Values Example A synthesiser adds two sound waves together to make a new sound. The first wave is described by V = 75sin to and the second by V = 100cos to, where V is the amplitude in decibels and t is the time in milliseconds. Find the minimum value of the resultant wave and the value of t at which it occurs. For later, remember K = 25k

Maximum and Minimum Values Expand and equate coefficients Maximum and Minimum Values Find tan ratio note: sin(-) and cos(+) C A S T 0o 180o 270o 90o Square and add

Maximum and Minimum Values remember K = 25k =25 x 5 = 125 The minimum value of sin is -1 and it occurs where the angle is 270o Therefore, the minimum value of Vresult is -125 Adding or subtracting 360o leaves the sin unchanged

Maximum and Minimum Values Minimum, we have:

Solving Trig Equations True for ALL x means coefficients equal. Solving Trig Equations Worked Example: Step 1: Compare Coefficients: Square &Add

Solving Trig Equations Find tan ratio note: sin(+) and cos(+) Solving Trig Equations C A S T 0o 180o 270o 90o

Solving Trig Equations Step 2: Re-write the trig. equation using your result from step 1, then solve. C A S T 0o 180o 270o 90o

Solving Trig Equations Step 2:

Solving Trig Equations Expand and equate coefficients Solving Trig Equations Example Find tan ratio note: sin(-) and cos(-) C A S T 0o 180o 270o 90o Square and add

Solving Trig Equations 2x – 213.7 = 16.1o , (180-16.1o),(360+16.1o),(360+180-16.1o) 2x – 213.7 = 16.1o , 163.9o, 376.1o, 523.9o, …. 2x = 229.8o , 310.2o, 589.9o, 670.2o, …. x = 114.9o , 188.8o, 294.9o, 368.8o, ….

Solving Trig Equations (From a past paper) Example A builder has obtained a large supply of 4 metre rafters. He wishes to use them to build some holiday chalets. The planning department insists that the gable end of each chalet should be in the form of an isosceles triangle surmounting two squares, as shown in the diagram. If θo is the angle shown in the diagram and A is the area m2 of the gable end, show that 4 Find algebraically the value of θo for which the area of the gable end is 30m2.

Solving Trig Equations 4 s Solving Trig Equations (From a past paper) Part (a) Let the side of the square frames be s. Use the cosine rule in the isosceles triangle: This is the area of one of the squares. The formula for the area of a triangle is Total area = Triangle + 2 x square:

Solving Trig Equations (From a past paper) Part (b) Find tan ratio note: sin(+) and cos(+) C A S T 0o 180o 270o 90o Square and add Finally:

Solving Trig Equations (From a past paper) Part (c) From diagram θo < 90o ignore 2nd quad Find algebraically the value of θo for which the area is the 30m2 C A S T 0o 180o 270o 90o

Higher Maths The Wave Function Strategies Click to start www.maths4scotland.co.uk Higher Maths Strategies The Wave Function Click to start

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 k sin(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

Cosine +, so 1st & 4th quadrants 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

Cosine +, so 1st & 4th quadrants Maths4Scotland Higher Solve the equation in the interval 0  x  360. Use R cos(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

You have completed all 9 questions in this presentation Maths4Scotland Higher You have completed all 9 questions in this presentation Previous Quit Quit Back to start

Are you on Target ! Update you log book Make sure you complete and correct ALL of the Wave Function questions in the past paper booklet.