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:
Sines are periodic. They oscilate between 1 and -1 over 360o (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 360o (2 p Radians) begining and ending with 1.
The table below summarises the information for both Sines and Cosines between 0o and 360o (0 to 2 p Radians). This information will be used when solving trigonometric equations.
(o) |
(Rad) |
||
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 = 120o and 240o 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 = 30o, X = 150o), the second equation gives one value (270o). 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 = 90o, X = 270o). The second equation also gives two values (135o and 315o - check these figures in the table). Thus the solution of the original equation is:
© 2000 Kryss Katsiavriades
Trigonometry
Right-angled triangles, Sines, Cosines, Tangents. Using trigonometric Functions, series and formulas.
Look At Logarithms
Index and base. Logarithms defined. Base 10 and base e. Uses of logarithms in calculations. Series for logarithms.
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