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Trigonometric Equations Trigonometric Identities : Double Angles : Trigonometric Graphs : Equations

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The Trigonometric Identities

In this essay we will combine the Trigonometric Function into equations that can be solved. We begin by reminding ourselves of the trigonometric relations:

In addition, there are relations called double angles:

Because Sin²X + Cos²X = 1, this last relation can also be written as:

Graphs and Values of Sine and Cosine

Sines are periodic. They oscilate between 1 and -1 over 360^o (2 p Radians) begining and ending at 0.

Below is the graph of Y = Cos X. This is similar but in a different phase.

Cosines also oscilate between 1 and -1 over 360^o (2 p Radians) begining and ending with 1.

The table below summarises the information for both Sines and Cosines between 0^o and 360^o (0 to 2 p Radians). This information will be used when solving trigonometric equations.

Solving Trigonometric Equations

Using the table, it is easy to see that X has two values in the required range. They are:

Re-arranging the equation (to get Cos X on one side and the numbers on the other side) gives:

Using the table, we can see that X has two values in the required range. They are:

Using the identity to replace Tan X gives:

The Cosines cancel out to give:

This gives two values of X:

Re-arrange the equation:

Therefore 2X = 120^o and 240^o which gives:

Using the double angle identity, replace Cos 2X by (1 - 2 Sin²X):

which re-arranges to a quadratic equation in Sin²X:

This can be solved by factorising:

This equation gives 0 if either 2 Sin X - 1 = 0 or Sin X + 1 = 0. In other words:

The first equation gives two vales (X = 30^o, X = 150^o), the second equation gives one value (270^o). Thus the solution of the original equation is:

Using the double angle identity, the Sin 2X can be replaced by 2 Sin X Cos X:

2 Cos X is common to both terms so this can be re-written:

This equation gives 0 if either 2 Cos X = 0 or Cos X + Sin X = 0. In other words:

The first equation gives two vales (X = 90^o, X = 270^o). The second equation also gives two values (135^o and 315^o - check these figures in the table). Thus the solution of the original equation is:

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KryssTal Related Pages

Introduction To Algebra

Trigonometry

Graphs

Look At Logarithms

Spherical Trigonometry