Trig EquationsTrig Equations

© Christine Crisp

AS Use of MathsAS Use of Maths

Trig Equations

0180 360

xy s in

0 180 360

xy cos

or

To solve trig equations you have to know what the sine and cosine curves look like

Due to the symmetrical appearance of the graphs when solving trig equations there will be more than one answer

Y=sinx Y=cosx

Trig Equations

0180 360

xy s in

0 180 360

xy cos

Y=sinx Y=cosx

Ex Sin 45 = 0.7And Sin 135 = 0.7

13545

Ex Cos 60 = 0.5And Cos 300 = 0.5

30060

0.50.7

Trig Equations

To solve trig equations use the forwards and backwards method

Solve the equation 50s in x

This means that if you find the sin of x then the answer is 0.5

This is pronounced inverse sin x and is on the same key as sin x but in yellow so use the 2nd F key

The opposite or inverse of sin x is sin–1x

Remember an inverse function is a function which has the opposite effect

The inverse (opposite) of x2 is x

Trig Equations

Solve the equation 50s in x

Forwardsx sin it = 0.5

Backwards0.5 sin-1it x

x = sin-10.5 = 30o

So the solution to the equation sinx = 0.5 is x = 30o

But unlike normal algebraic equations trig equations have many answers because the trig graph is periodic and repeats every 360o

Trig Equations

xy s in

BUT, by considering the graphs of and , we can see that there are many more solutions:

xy s in 50 y

e.g.1 Solve the equation .

50s in xSolution: The calculator gives us the solution x =

30

50 y

Every point of intersection of and gives a solution ! In the interval shown there are 10 solutions, but in total there are an infinite number.

xy s in 50 y

The calculator value is called the principal solution

30

principal solution

Trig Equations

We will adapt the question to:

Solution: The first answer comes from the calculator: Use the sin-1 key

Solve the equation for

50s in x 3600 x

Forwardsx sin it = 0.5

Backwards0.5 sin-1it x

x = sin-10.5 = 30o

This limits the number of solutions

Trig Equations

0180 360

xy s in

1

-1

50 yAdd the line 50 y

3600 xx andSketch between xy s in

There are 2 solutions.

The symmetry of the graph . . . 1 5 03 01 8 0 x

15030

. . . shows the 2nd solution is

It’s important to show the scale.

Tip: Check that the solution from the calculator looks

reasonable.

Trig Equations

Solution: The first answer from the calculator is

e.g. 2 Solve the equation in the interval

50cos x3600 x

Forwardsx cos it = -0.5

Backwards-0.5 cos-1it x

x = cos-1-0.5 = 120o

The opposite or inverse of cos x is cos–1x (inverse cos x)

Trig Equations

0 180 360

xy cos

1

-1

Solution: The first answer from the calculator is 120501 .cosx

50 y

Add the line 50 y

e.g. 2 Solve the equation in the interval

50cos x3600 x

3600 xx andSketch between xy cos

There are 2 solutions.

The symmetry of the graph . . . 2 4 01 2 03 6 0 x

240120

. . . shows the 2nd solution is

Trig Equations

0180 360

xy s in

0 180 360

xy cos

SUMMARY

• Find the principal solution from a calculator.

• Find the 2nd solution using symmetry

where c is a constant

To solve

cx s in 3600 xor cx c o s for

or

• Draw the line y = c.

• Sketch one complete cycle of the trig function. For example sketch from to .

3600

Trig Equations

Exercises

1. Solve the equations (a) and (b) for50cos x 3600 x2

3s in x

Forwardsx cos it = 0.5

Backwards0.5 cos-1it x

x = cos-10.5 = 60o

Trig Equations

0 180 360

xy cos

50 y1

-1

Exercises

30060

The 2nd solution is

60360 x300

1. Solve the equations (a) and (b) for50cos x 3600 x2

3s in x

Solution: (a) ( from calculator )60x

Trig Equations

(b) ,s in23x 3600 x

Exercises

Forwardsx sin it = 3

2 ,

Backwards sin-1it x 32

x = sin-

1

32

= 60o

Trig Equations

0180 360

xy s in

1

-1

Solution: ( from calculator )

60x

23y

12060

The 2nd solution is

60180 x120

(b) ,s in23x 3600 x

Exercises

Trig Equations

xy s in

y

180

1

-1

x360

180

e.g. 5 Solve the equation for50s in x 3600 x

50 y

30 330

Since the period of the graph is this solution . . .

360o360 30 330 . . . is

Solution:

1x sin 0.5 30

More Examples

Using forwards and back

Trig Equations

xy s in

y

180

1

-1

x360

180

Solution:

e.g. 5 Solve the equation for5.0s in x 3600 x

50 y

21030 330

21030180

Symmetry gives the 2nd value for .3600 x

1x sin 0.5 30

The values in the interval are and 3302103600 x

More Examples

Trig Equations

0 180 360

xy cos

1

-1

66 294

Solution: Principal

value

1x cos 0.4 66 e.g. 6 Solve for 40cos x 360180 x

2 9 46 63 6 0 xBy symmetry,

Method

40 y

Ans: 66 , 294

Using forwards and back

Trig Equations

SUMMARY

To solve or cx s in cx c o s

360• Once 2 adjacent solutions have been

found, add or subtract to find any others in the required interval.

• Find the principal value from the calculator. • Sketch the graph of the trig function showing at least one complete cycle and including the principal value.

• Find a 2nd solution using the graph.

Trig Equations

1. Solve the equations ( giving answers

correct to the nearest whole degree )

0 x 360 20s in x

(b) for650cos x 0 x 360

(a) for

Exercises

Trig Equations

xy s in

y

180

1

-1

x

20 y

360180

12

20s in x(a) for

192

Solution: Principal

value

12x

x 360 12 348 By symmetry,

Ans: 192 , 348

0 x 360

Exercises

Using forwards and back

348

Trig Equations

0 180 360

1

-1xy cos

Ans: 49 , 311

(b) for650cos x 0 x 360

Solution: Principal

value

1x cos 0.65 49

31149360 x

311

650 y

49

Exercises

Using forwards and back

Trig Equations

Solve the following

0 x 360

(b) Sinx = 0.49 for

0 x 360

(a) Sinx = 0.83 for

(d) Cosx = 0.65 for

(c) Cosx = 0.25 for 0 x 360

0 x 360

Answers

a) 56.2o, 123.9 b) 29.3o, 150.7

b) 75.5o, 284.5 c) 49.5o, 310.5

Trig Equations