Waves

Trigonometric Equations

Trigonometric Identities : Double Angles : Trigonometric Graphs : Equations

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The Trigonometric Identities

In Algebra we looked at solving simple equations like 5X - 2 = 0 and quadratic equations like X2 - 3X + 1 = 0. In Trigonometry we introduced the Trigonometric Functions (Sines, Cosines and Tangents).

In this essay we will combine the Trigonometric Function into equations that can be solved.

We begin by reminding ourselves of the trigonometric relations:

Sin X / Cos X = Tan X

Sin2X + Cos2X = 1

In addition, there are relations called double angles:

Sin 2X = 2 Sin X Cos X

Cos 2X = Cos2X - Sin2X

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

Cos 2X = 1 - 2 Sin2X

Cos 2X = 2 Cos2X - 1


Graphs and Values of Sine and Cosine

Before we can solve complicated trigonometric equations we must look at how sines and cosines vary. Below is the graph of Y = Sin X. X is measured in Radians.

Sine Wave

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

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

Cosine Wave

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

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

Angle
(o)
Angle
(Rad)
Sine Cosine
0
0
0
1
30
π / 6
1 / 2
√3 / 2
45
π / 4
1 / √2
1 / √2
60
π / 3
√3 / 2
1 / 2
90
π / 2
1
0
120
2π / 3
√3 / 2
-1 / 2
135
3π / 4
1 / √2
-1 / √2
150
5π / 6
1 / 2
-√3 / 2
180
π
0
-1
210
7π / 6
-1 / 2
-√3 / 2
225
5π / 4
-1 / √2
-1 / √2
240
4π / 3
-√3 / 2
-1 / 2
270
3π / 2
-1
0
300
5π / 3
-√3 / 2
1 / 2
315
7π / 4
-1 / √2
1 / √2
330
11π / 6
-1 / 2
√3 / 2
360
0
1


Solving Trigonometric Equations

Using the table or graphs above and some algebra, solve the following equations for values between 0o and 360o.

Sin X = 1 / 2

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

X = 30o and X = 150o.

Cos X + 1 / 2 = 0

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

Cos X = -1 / 2

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

X = 120o and X = 240o.

Cos X Tan X = 1 / √2

Using the identity to replace Tan X gives:

Cos X (Sin X / Cos X) = 1 / √2

The Cosines cancel out to give:

Sin X = 1 / √2

This gives two values of X:

X = 45o and X = 135o.

2 Cos 2X + 1 = 0

Re-arrange the equation:

Cos 2X = - 1 / 2

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

X = 60o and X = 120o.

Sin X - Cos 2X = 0

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

Sin X - (1 - 2 Sin2X) = 0

Sin X - 1 + 2 Sin2X = 0

which re-arranges to a quadratic equation in Sin2X:

2 Sin2X + Sin X - 1 = 0

This can be solved by factorising:

(2 Sin X - 1)(Sin X + 1) = 0

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

Sin X = 1 / 2 or Sin X = -1

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:

X = 30o, X = 150o and X = 270o.

2 Cos2X + Sin 2X = 0

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

2 Cos2X + 2 Sin X Cos X = 0

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

2 Cos X ( Cos X + Sin X) = 0

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

Cos X = 0 or Sin X = -Cos X

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:

X = 90o, X = 135o, X = 270o and 315o.

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