Solving Trigonometric Equations

A trigonometric equation is an equation which has a trigonometric expression involving a variable, such as sinx, cosx, tanx, cotx, secx, cscx. There are the six trigonometric functions I mentioned in a previous introductory article on trigonometry. Some trigonometric equations are true for all values of the variable, such as 1 + tan²x = sec²x, 1 + cot²x = csc²x and sin²x + cos²x = 1. But many trigonometric equations are defined for only some values of the variable.

Fox example, suppose the equation is sinx = √2/2. We know from the unit circle, or we can solving using a calculator, that 45 degrees is a solution for x because sin 45 = √2/2. This is not the only value that satisfies the equation. Sine is positive in the first and second quadrant, which can be seen on the sine curve. So the answer is 45 degrees and the 45 degree angle in the second quadrant, which is 135 degrees. If answering in radians, it's π/4 and 3π/4.

Let's try an equation involving a single trigonometric function such as 3cosx - 2 = 5cosx - 1. We solve such equations the same as solving a basic algebra equation involving a single variable. We isolate the function to one side of the equals sign and solve for the variable. In this case, we want to get cosx on the left side of the equation and everything else to the right side of the equals sign. Therefore, we can subtract 5cosx from both sides. That gives us -2cosx - 2 = -1. Notice we have -2 on the left side of the equation. But we all we want is cosx on that side, so add 2 to both sides. Doing that gives us -2cosx = 1. From here we simply divide both sides by -2 to get cosx = -1/2. From our knowledge of the unit circle we know it's a 60 degree angle, but since the value is negative, it's not in the first quadrant. Cosine is negative in the second and third quadrants. So the answers are 120 and 240 degrees. In radians, the answer is 2π/3 and 4π/3. Note that we convert degrees to radians by multiplying each degree measure by π/180.

Suppose the equation is in quadratic form, meaning in the form au² + bu + c = 0. where u is a trigonometric function and a, b and c are real numbers and a cannot equal zero. We generally can solve this kind of equation by factoring, which I explained in a previous article.

Let's try the equation 2sin²x + sinx - 1 = 0. We'll attempt to solve this by factoring. Remember we use two sets of parentheses, each with two terms separated by an + or - sign. The first term in each must multiply to 2sin2x, the last terms must multiply to -1 and when multiplied out must give you sinx in the middle. Putting it together we get (2sinx - 1)(sinx + 1) = 0. We know this is correct because multiplying this out we get 2sin²x + 2sinx - sinx - 1, which is 2sinx²x + sinx - 1, the original expression. We set each factor equal to zero and solve. Therefore we have 2sinx - 1 = 0 and sinx + 1 = 0. Solving for x in the same manner we did in the previous example, we get x = 30 and 150 degrees and x = 270 degrees. In radians, the answers are π/6, 5π/6 and 3π/2.

You may also encounter a problem which involved multiple angles. An example of such an equation is cot3x = 1. To solve this equation, we are still looking for the values of x from 0 to 360 degrees (one rotation around the unit circle). We know cot is 1 at 45 degrees and 315 degrees, in radians this is π/4 and 7π/4. Therefore 3x =π/4 and 7π/4, we will add 2π to each to get 9π/4 and 15π/4 and again to complete three rotations around the unit circle. This gives us 17π/4 and 23π/4. If the problem was cot4x we'd need four rotations and so on. Now 3x is equal to all of those values. So in order to solve for x, we divide each angle to 3 to get π/12, 7π/12, 3π/4, 5π/3, 17π/12 and 23π/12.

There are more types of trigonometric equations, such as those using identities to solve, which I'll deal with in another article. Hope this guide will be useful for those that need to learn how to solve basic trigonometric equations.