Trig graphs and equations

Write the formula for finding the area of a non right angled triangle

A =½ absinC

What are the three types of equation which can be solved?

Compound angle eg. Sin3x =1/2

Basic Quadratic-squared function sin2x = ½

Harder Quadratic-trinomial sin2x +5sinx +6=0

How many solutions should cos2x = 1/2 have?

4- one in each quadrant of the cast diagram (positive and negative solutions)

How many solutions should cos3x = 1/2 have between 0 and 360 degrees?

6 solutions- double the coefficient of x.

Be careful and check interval properly for number of solutions. To find the next two larger/smaller solutions add/ subtract 360 onto the original solutions respectively.

State the max and min value and period of y=cosx

Max = 1 min = -1 period 360 degrees

State the max and min value and period of y=sinx

Max = 1 min = -1 period 360 degrees

state the max and min value and the period of y=tanx

Max/min undefined

Period = 180 degrees

What do we use to find the period of any graph?

Period = 360

Where n is the number of cycles

Y = acosbx + c . What do the values of a, b and c mean?

A is amplitude

B is number of cycles between 0 and 360

C is vertical movement

Y = 2sin4x + 3 What is the amplitude of this graph?

a = 2

Y = 2sin4x + 3 how many cycles of the curve appear between 0 and 360 degrees

4 cycles – each with a period of 90 degrees

Y = 2sin4x + 3 What is the maximum and minimum value of this function?

a= 2 but moved up 3 so max = 5 and min=1

Y = -sinx, what has happened to the graph?

Reflected on the x axis (upside down as y values have been multiplied by -1 so a reflection on the x axis has occured)

If y = sin(x+30) what has happened to the sin graph?

Moved 30 degrees left

When solving an equation involving a phase angle you may need to check to see if there are solutions greater than 360 degrees which will be included since the graph has moved left. Add on 360 to original solutions for x + a then solve.

If y = sin(x-30), what has happened to the sin graph?

Moved 30 degrees right

When solving an equation involving a phase angle you may need to check for solutions which are less than 0 degrees before finding x so subtract 360 from the original solutions for x - a then solve.

Where does the sin graph cut the x axis?

At 180 and 360 degrees

Where does the cosine graph cut the x axis?

At 90 and 270 degrees

How would you solve a trig equation?

1. rearrange numbers so only the trig function is on the left hand side of the equation

2. use inverse function to find acute angle solution in quadrant 1 or use exact values in non calculator questions

3. Use cast diagram to identify both quadrant answers and apply the rule for these quadrants.

What is a cast diagram used to find?

The angles where trig graphs are positive and negative

When solving a trig equation, how would you find the value of a solution in the sine quadrant?

180 – x

When solving a trig equation, how would you find the value of a solution in the cosine quadrant?

360 – x

When solving a trig equation, how would you find the value of a solution in the tan quadrant?

180 + x

When solving a trig equation sinx = -1/2 Which quadrants would solutions be in ?

Sin is negative in the tan and cosine quadrants Q3 and Q4

When solving a trig equation cosx = -1/2 Which quadrants would solutions be in ?

Cos is negative in the sin and tan quadrants Q2 and Q3

When solving a trig equation sinx = 1/3 Which quadrants would solutions be in ?

Sine is positive in Q1 and Q2

When solving a trig equation tanx = 1/2 Which quadrants would solutions be in ?

Tan is positive in Q1 and Q3

When solving cosx = -1/5, the solutions are in quadrant 2 and 3, why is this and how would you find the solutions in these quadrants?

Because only sine is positive in Q2 and only tan is positive in Q3. In Q2 180-x, in Q3 180+x

What are the two trig identities?

• sin2x + cos2x = 1

• tanx= sinx/cosx

How would you know when to use sin2x + cos2x =1 as an identity?

When you have a squared term or 1 in your expression

How would you know when to use tanx = sinx/cosx?

When you have tan(x) in the expression or no squared terms

When using identities what two types of question can be asked?

To simplify or prove LHS=RHS

When you are asked to solve a trig equation with a trinomial what should you do?

1. Factorise

2. Solve each bracket=0

3. Use CAST diagram to check where solutions lie

4. Calculate solutions and ensure they lie within the given interval

What is the exact value of sin30?

1/2

What is the exact value of sin60?

√3/2

What is the exact value of sin45?

1/√2

What is the exact value of cos30?

√3/2

What is the exact value of cos60?

1/2

What is the exact value of cos45?

1/√2

What is the exact value of tan30?

1/√3

What is the exact value of tan60?

√3

What is the exact value of tan45?

1

How do you find the exact value of a negative angle?

Either work backwards anti clockwise on CAST diagram or add 360 onto the negative angle and use CAST as normal

How do you find the exact value of a related angle?

Work out what quadrant the angle lies in and whether sin/cos/tan is positive or negative in this quadrant.

Then in Q2 (180-x)=RA

Q3 (x -180) = RA

Q4 (360 – x) = RA

How do you convert between degrees and radians?

Degrees to radians ×π/180Radians to degrees x180/π