Trigonometric Equations

An equation is a statement which says two expressions are equal. An equation can always be reduced in such a way that the expression on the right side becomes a constant. This helps us to evaluate the variable contained on the left side. In other words the variable can be solved and hence the idea of presenting an equation is to find the value of its variable. Sometimes, the variable may be a function of another independent variable and therefore, the solution involves knowledge of characteristics of such functions. If the functions are in trigonometric, the equations are called as trigonometric equations. Let us take a closer look on trigonometric functions.

How to Solve Trigonometric Equations

As mentioned earlier, the equations formed with trigonometric functions are called trigonometric equations. Hence, the initial solution is in the form of a trigonometric relation. At this point, we need to apply the trigonometric concepts and figure out the actual value of the variable for which the given equation is true. In trigonometric equations also we might land with extraneous solutions or no solutions. Let us explain this part with an example. The equation sin x + 3 = 4 is a trigonometric equation because of the presence of the term ‘sin x’. By simplifying we get sin x = 1. As per the properties of sine ratio of angles, the equation can be true for x = any co-terminal angle of ∏/2. On the other hand, if the equation was sin x + 3 = 5, there is no solution to the equation because on simplification we get sin x = 2, which is impossible. Thus one must be good at the fundamental trigonometric concepts to solve trigonometric equations. In the following sections let us see the different types of trigonometric equations.

Simplifying Trigonometric Equations

In many trigonometric equations, we may find that the involvement of more than one trigonometric ratios. Here again, the knowledge of relations between different trigonometric ratios help to a great extent. The best part in trigonometry is, any trigonometric ratio can be expressed in terms of another trigonometric ratio we desire. Thus, it is always possible to reduce any trigonometric equation to an equation with single trigonometric ratio. For example, the equation sec2 x + tan2 x = a, can be simplified as,
(1 + tan2 x) + tan2 x = a
1 + 2tan2 x = a
 tan2 x = [(a – 1)/2], which can now be easily evaluated.
Sometimes, numeric values may be required to be converted into trigonometric ratios of known angles to help simplification as sum or difference of two angles. Thus, the use of trigonometric identities and concepts are of great help in simplifying trigonometric equations.  


To begin with trigonometric equations, let us consider the simplest form of a trigonometric equation
sin x + cos x = 0.
The first tendency for any one is to divide throughout the equation by cos x, with an assumption cos x ≠ 0. (It is a valid assumption because, if cos x = 0, we get sin x is also equal to 0, which is impossible). That is,
(sin x/cos x) + 1 = 0 
tan x + 1 = 0,
tan x = -1.
The solution is, x = (2n∏ + 3∏/4) and x = (2n∏ - ∏/4) , where’ ‘n is any integer.


Now let us generalize the example that we had discussed in the previous section. Let us introduce different non zero coefficients ‘a’ and ‘b’ for the two terms. The equation is now in the form,
a*sin x + b* cos x = 0
a*sin x = -b*cos x
sin x = (-b/a)*cos x
tan x = -b/a.
The solution depends on the signs and magnitudes of the coefficients ‘a’ and ‘b’. If they are of the same sign, the solution is, x = an angle in second quadrant and x = an angle in the fourth quadrant. If the signs of ‘a’ and ‘b’ are opposite, then the solution is x = an angle in the first quadrant and x = an angle in the third quadrant. The reference angle of the solution is ‘arc tan (b/a). 

Quadratic Trigonometric Equations

We have already mentioned that trigonometric ratios are inter related and hence it is possible to restrict the equations with a single trigonometric ratio. However, in most cases the simplification turns into square of a trigonometric ratio. Thus, the equation becomes a quadratic equation in a single trigonometric equation. Hence, we need to follow the steps that are used in solution to quadratic equations. That is either by the method of factorization or by use of quadratic formula. It will be very convenient to substitute a variable for the trigonometric ratio and solve the quadratic. After it is done, we can reverse the substitution and figure out the final answer. It is important that the solutions to the quadratic trigonometric equations are checked to score out any extraneous solution.
Let us consider the same equation sin x + cos x = 0 and try to solve in a different method.
sin x + cos x = 0
sin x = - cos x
squaring both sides, sin2  x = cos2 x
We know that,  cos2 x  = 1 –  sin2  x  and hence,
sin2  x = 1 – sin2  x
or, 2  sin2  x  = 1
or,  sin2  x  = 1/2 
or, sin x = ±(√2)/2. Thus the solutions could be, x = (2n∏ + ∏/4),  x = (2n∏ - ∏/4), x = (2n∏ + 3∏/4) and x = (2n∏ - 3∏/4). But the solutions x = (2n∏ + ∏/4) and x = (2n∏ - 3∏/4) do not satisfy the given equation.
Hence the actual solutions are only x = (2n∏ + 3∏/4) and x = (2n∏ - ∏/4) , where’ ‘n is any integer. This agrees with the solutions we figured out earlier.
In some cases, the interval of the actual angle may be given. In such cases, it is sufficient to find only the particular solutions within the given interval.
We will illustrate different types of quadratic trigonometric equations in the worked out examples.

Trigonometric Equations Examples

Example 1: solve for all possible values of ‘x’ from the equation 1 + csc x = 3,  ∏/2  < x < ∏
1 + csc x = 3
csc x = 3 – 1 = 2
Therefore, sin x = 1/2
We arrived at the value of sin x is 1/2 and it is positive. We know the sine ratio of angles that lie in first and second quadrants are positive. Also we know that 1/2 is the value of the sine ratio for the reference angle ∏. Since the direction says that the angle is in second quadrant, we need to use the reference angle for the second quadrant. Hence, the particular solution is,                                                                      
x = ∏ – (∏/6) = 5∏/6 

Example 2: solve for all possible values of ‘x’ from the equation sin x + √3cos x = 2

The problem can be solved by expressing either sin x or cos x in terms of the other and simplifying. But the method is cumbersome. Instead, we can use a simpler method by remembering the sine and cosine values for reference angle of ∏/3 or ∏/6. Let us show how the reference angle of ∏/3 is used in this problem.
sin x + √3cos x = 2. Dividing the equation throughout by2,
(1/2)*sin x + (√3/2)cos x = 1
Expressing (1/2) and (√3/2) as the values of cos ∏/3 and sin ∏/3 respectively,
cos ∏/3*sin x + sin ∏/3*cos x = 1
The left side is in the form of sum formula of sine. That is,
sin (x + ∏/3) = 1.
Now the right side can be expressed as sin (2n∏ + ∏/2), where ‘n’ is any integer.
Therefore,  sin (x + ∏/3) = sin (2n∏ + ∏/2)
or, (x + ∏/3) = (2n∏ + ∏/2)
or, x = (2n∏ + ∏/2 + ∏/3) 
or, x = (2n∏ + ∏/6)
The same answer can be obtained by expressing (1/2) and (√3/2) as the values of sin ∏/6 and cos ∏/6 respectively and using difference formula of cosine.

Example 3: solve for all possible values of ‘x’ from the equation cos x + sec x = 2
cos x + sec x = 2
or, cos x + (1/cos x) = 2
or, cos2 x + 1 = 2cos x
or,  cos2 x – 2cos x + 1 = 0
The equation is a quadratic equation in ‘cos x’. It also appears to be an equation with identical roots. 
By factoring, completing the square or by using the quadratic formula, we get,
(cos x – 1)2 = 0
or, cos x – 1 = 0
or, cos x = 1
The general solution for the above relation is,
x = 2n∏, where ‘n’ is any integer. 

Example 4: solve for all possible values (in approximate degrees) of ‘x’ from the equation sec2 x + 1 = 3tanx, 0 < x < 2∏ 

sec2 x + 1 = 3tanx
(1 + tan2 x) + 1 = 3 tan x
tan2 x – 3tan x + 2 = 0
(tan x – 2)(tan x – 1) = 0
So, tan x = 2 and tan x = 1
The reference angles for tan x = 2 and for tan x = 1 are 63.4 deg (approximate) and 45 deg. Since the values obtained are positive, the angles of solution lie in first and third quadrants.
That is, the final solutions in degrees are, x = 45, x = 180 + 45 = 225, x ≈ 63.4 and x ≈ 180 + 63.4 ≈ 243.4

Example 5: solve for all possible values of ‘x’ from the equation cot x + csc x = 1
cot x + csc x = 1
or, cot x – 1 = csc x
Squaring both sides, or, (cot x – 1)2 = csc2 x
or, cot2 x – 2cot x + 1 =  csc2 x
or, cot2 x + 1 – 2cot x =  csc2 x
or,   csc2 x  – 2 cotx =   csc2 x     (since cot^2 x + 1 = csc^2 x)
or, -2cot x = 0,
or, cot x = 0.
This leads to a general solution x = any odd multiple of ∏/2. But for the given equation is not valid for all odd multiples of ∏/2. For example, if x = 3∏/2, cot (3∏/2) + csc (3∏/2) = -1 and not 1.
Therefore, the general solution can only be x = (2n∏ + ∏/2), where ‘n’ is any integer.

Example 6: solve for all possible values of ‘x’ from the equation 2sin2 x + 5 cos x = -1
2sin2 x + 5 cos x = -1
or, 2 – 2cos2 x + 5 cos x = -1
or, -2 cos2 x  + 5 cos x + 3 = 0
or, 2 cos2 x  – 5cos x – 3 = 0
The equation is a quadratic equation in ‘cos x’. By factoring, 
2 cos2 x  – 6cos x + cos x – 3 = 0
or, 2cos x(cos x – 3) + 1(cos x – 3) = 0
or, (cos x – 3)(2cos x + 1) = 0
Hence, cos x = 3 and cos x = -1/2.
Obviously cos x = 3 is an extraneous solution and hence the only solution is cos x = -1/2.
The sign is negative which means, the angle is in second and third quadrants. The principal angle for which cosine ratio is 1/2, is ∏/3. Therefore, the general solution is,
x = 2n∏ + (∏ - ∏/3) = ( 2n∏ + 2∏/3)  and  x = 2n∏ + (∏ + ∏/3) = ( 2n∏ + 4∏/3), where ‘n’ is any integer.