The six trigonometric functions are the sine, cos, tan, csc, sec and cot. An equation consisting of one or more trigonometric functions is called a trigonometric equation, for instance cot^2(x) + 1 = csc^2(x) is a trigonometric equation. Some of Trigonometric Equations Examples are 2 cos^2(x) –sin(x) – 1 =0, 2sin(theta) – 1 = 0, cos(theta/2) = 1 + cos(theta), 6 sin^2(theta) – sin(theta) = 1, cot(2 theta) – tan(2 theta) = 0
Quadratic Trigonometric Equations
Trigonometric equations that involve trigonometric functions as products are called the quadratic trigonometric functions. In a quadratic equation with degree 2 written in the form of ax^2 + bx + c. So, a quadratic trigonometric equation will consist of trigonometric functions with degree 2. For instance, 2 sin^2(x) – sin(x) +1 = 0 is an example for a quadratic trigonometric equation.
Let us now learn the steps involved in simplifying trigonometric equations. We solve trigonometric equations as we solve any other algebraic equation except that we use the trigonometric identities in solving. They are also solved by the sign of the trigonometric value by determining the quadrant(s) for that particular trigonometric value. Let us simplify a trigonometric equation, 2 sin(pi/2 -alpha) =1. We shall use the identity sin (pi/2 – alpha) = cos(alpha) in solving the given equation. So, the trigonometric equation reduces to, cos(alpha)=1/2 . Now we need to check for what angle of cosine alpha gives the value ½. We know that cos(pi/3) equals ½. So, we write it as cos(+/-) pi/3 =1/2. The positive and negative sign depends on which quadrant the angle alpha lies and hence we can generalize the solution as alpha = (+/-)pi/3 + 2k pi where k is any integer. There is a significance for adding 2k pi to the solution, as per the periodic behavior of the function cosine. So, the solution set is written as {alpha = (+/-)pi/3 + 2k pi: k belongs to set of integers}
Sin(x) + cos(x) = 0
Simplifying the above equation we get, sin(x) = -cos(x). Here, the value of sine and cosine have to be equal with opposite sign. Let us write sine in terms of cosine using trigonometric identity, which gives us
cos(pi/2 – x+2 k pi) = cos(pi + x)
pi/2 – x + 2 k pi = pi + x
The general solution would be, x = -pi/4 + k pi where k belongs to set of integers
Now, in the interval [0, 2pi) let us consider k = 1 and k = 2, we get, x = (3/4) pi and x = (7/4) pi as the solution that is at 135 degrees and 315 degrees the value of sine and cosine are equal and opposite.
