Processing Math: Done
To print higher-resolution math symbols, click the
Hi-Res Fonts for Printing button on the jsMath control panel.

jsMath

4.4 Trigonometric equations

From Förberedande kurs i matematik 1

Revision as of 07:58, 20 August 2008 by Tek (Talk | contribs)
Jump to: navigation, search
       Theory          Exercises      

Contents:

  • The basic equations of trigonometry
  • Simple trigonometric equations

Learning outcomes:

After this section, you will have learned how to:

  • Solve the basic equations of trigonometry
  • Solve trigonometric equations that can be reduced to basic equations.

Basic equations

Trigonometric equations can be very complicated, but there are also many types of trigonometric equations which can be solved using relatively simple methods. Here, we shall start by looking at the most basic trigonometric equations, of the type sinx=a, cosx=a and tanx=a.

These equations usually have an infinite number of solutions, unless the circumstances limit the number of possible solutions (for example, if one is looking for an acute angle).

Example 1

Solve the equation sinx=21.


Our task is to determine all the angles that have a sine with the value 21. The unit circle helps us in this. Note that here the angle is designated as x.

4.4 - Figur - Två enhetscirklar med vinklar π/6 resp. 5π/6

In the figure, we have shown the two directions that give us points which have a y-coordinate 21 on the unit circle, i.e. angles with a sine value 21. The first is the standard angle 30=6 and by symmetry the other angle makes 30 with the negative x-axis. This means that the angle is 18030=150 or in radians 6=56. These are the only solutions to the equation sinx=21 between 0 and 2.

However, we can add an arbitrary number of revolutions to these two angles and still get the same value for the sine . Thus all angles with a value of the sine 21 are

xx=6+2n=65+2n

where n is an arbitrary integer. This is called the general solution to the equation.

The solutions can also be obtained in the figure below where the graph of y=sinx intersects the line y=21.

4.4 - Figur - Kurvorna y = sin x och y = ½

Example 2

Solve the equation cosx=21.


We once again study the unit circle.

4.4 - Figur - Två enhetscirklar med vinklar π/3 resp. -π/3

We know that cosine is 21 for the angle 3. The only other direction in the unit circle, which produces the same value for the cosine is the angle 3. Adding an integral number of revolutions to these angles we get the general solution

x=3+n2,

where n is an arbitrary integer.

Example 3

Solve the equation tanx=3 .


A solution to the equation is the standard angle x=3.

If we study the unit circle then we see that tangent of an angle is equal to the slope of the straight line through the origin making an angle x with the positive x-axis .

4.4 - Figur - Två enhetscirklar med vinklar π/3 resp. π+π/3

Therefore, we see that the solutions to tanx=3  repeat themselves every half revolution 3, 3+, 3++ and so on. The general solution can be obtained by using the solution 3 and adding or subtracting multiples of ,

x=3+n,

where n s an arbitrary integer.


Somewhat more complicated equations

Trigonometric equations can vary in many ways, and it is impossible to give a full catalogue of all possible equations. But let us study some examples where we can use our knowledge of solving basic equations.

Some trigonometric equations can be simplified by being rewritten with the help of trigonometric relationships. This, for example, could lead to a quadratic equation, as in the example below where one uses cos2x=2cos2x1.

Example 4

Solve the equation cos2x4cosx+3=0.


Rewrite by using the formula cos2x=2cos2x1 giving

(2cos2x1)4cosx+3=0,

which can be simplified to the equation (after division by 2)

cos2x2cosx+1=0.

The left-hand side can factorised by using the squaring rule to give

(cosx1)2=0.

This equation can only be satisfied if cosx=1. The basic equation cosx=1 can be solved in the normal way and the complete solution is

x=2n(n arbitrary integer).

Example 5

Solve the equation 21sinx+1cos2x=0.


According to the Pythagorean identity sin2x+cos2x=1, i.e. 1cos2x=sin2x, the equation can be written as

21sinx+sin2x=0.

Factorising out sinx one gets

sinx21+sinx=0. 

From this factorised form of the equation, we see that the solutions either have to satisfy sinx=0 or sinx=21, which are two basic equations of the type sinx=a and can be solved as in example 1. The solutions turn out to be

xxx=n=6+2n=76+2n(n  arbitrary integer).

Example 6

Solve the equation sin2x=4cosx.


By rewriting the equation using the formula for double-angles one gets

2sinxcosx4cosx=0.

We divide both sides with 2 and factorise out cosx, which gives

cosx(sinx2)=0.

As the product of factors on the left-hand side can only be zero if one of the factors is zero, we have reduced the original equation into two basic equations

  • cosx=0,
  • sinx=2.

But sinx can never be greater than 1, so the equation sinx=2 has no solutions. That leaves just cosx=0, and using the unit circle gives the general solution x=2+n.

Example 7

Solve the equation 4sin2x4cosx=1.


Using the Pythagorean identity one can replace sin2x by 1cos2x. Then we will have

4(1cos2x)4cosx44cos2x4cosx4cos2x4cosx+41cos2x+cosx43=1,=1,=0,=0.

This is a quadratic equation in cosx, which has the solutions

cosx=23andcosx=21.

Since the value of cosx is between 1 and 1 the equation cosx=23 has no solutions. That leaves only the basic equation

cosx=21,

that may be solved as in example 2.


Exercises

Study advice

Basic and final tests

After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.


Remember:

It is a good idea to learn the most common trigonometric formulas (identities) and practice simplifying and manipulating trigonometric expressions.


It is important to be familiar with the basic equations, such as sinx=a, cosx=a or tanx=a (where a is a real number). It is also important to know that these equations typically have infinitely many solutions.


Useful web sites

Experiment with the graph y = a sin b (x-c)