Trigonometric Equation Calculator

Instructions: Use calculator to solve trigonometric equations you provide, showing all the steps. Please type in the trigonometric equation you want to carry out in the box below.

Enter the trigonometric equation you need to solve (Ex: sin(x) = cos(x), etc.)

This calculator will allow you to solve trig equations, showing all the steps of the way. All you need to do is to provide a valid trigonometric equation, with an unknown (x). It could be something simple as 'sin(x) = 1/2', or something more complex like 'sin^2(x) = cos(x) + tan(x)'.

Once you are done typing your equation, just go ahead and click on "Solve" to get all the details of the processes of finding the solutions, if solutions can be found.

Trigonometric properties and rules almost always allow to reduce most trig equations into simpler ones, so this type of equation is one type that often times lead to solutions, but it can be extremely cumbersome at times.

What is a trigonometric Equation?

A trigonometric equation, in the simplest possible terms, is a math equation where the unknown x is inside of a trigonometric expression.

For example, the following expression is a trigonometric eqn:

$\displaystyle \sin(x) = 1$

Why? Simply because x appears inside of trig expression sine. Or for example:

$\displaystyle \tan(x) = x$

Now, these two are trig eqns, but the difference between the two is that for the first one, x appears ONLY inside of sine, whereas in the second one x appears inside of a trig function (tangent), but it also appears outside. This will usually make it difficult (or impossible) to solve the equation.

How to solve trigonometric equations

• Step 1: Make sure you are dealing with a trigonometric equation. Non-trigonometric equations will likely require a different approach
• Step 2: Make sure that the unknown x is inside of the trigonometric expression, but x does not appear outside a trig expression. If that is the case, it is likely you won't be able to solve the equation with elementary methods
• Step 3: Conduct an appropriate substitution, by first expressing all the trig functions present in the equation into one type (typically sine), and then use a substitution involving sine
• Step 4: With a little luck and if you did the correct substitution, you have reduced the original trigonometric equation into a polynomial equation to solve.

One of the key trig rules you need to use the ability to express all trigonometric functions in terms of any fixed trigonometric function. For example, we can write cosine in terms of sine:

$\displaystyle \cos(x) = \pm \sqrt{1 - \sin^2 x}$

Trigonometric Substitutions

Using trigonometric identities and substitutions is your way to go in this case. For example, suppose you want to solve this:

$\displaystyle \sin x = \cos x$

So we know this is a trig equation, and we know we can write cosine in terms of sine, so we do this:

$\displaystyle \sin x = \pm \sqrt{1 - \sin^2 x}$

Now what? Well, we can use the substitution: $$u = \sin x$$, so the equation above becomes:

$\displaystyle u = \pm \sqrt{1 - u}$

which is a rational equation, which by using simple algebraic manipulation means that we need to solve a polynomial equation in order to solve the original trig equation.

Application of trigonometry

• Step 1: All things mechanical: In manufacturing mechanical parts, circles and trigonometry play a crucial role
• Step 2: Analysis of periodic functions: Many phenomena are tightly related to periodicity, the point at which trigonometry comes to play
• Step 3: Advanced math: Mathematicians love their Fourier Series and Transform, which play a crucial role in spectral analysis

Circles and all their symmetry are so really important in real life, and trigonometry is the language by which we can quantify circles and its relationships. Solving trigonometric equations is at the center of math.

Why would you solve trigonometric equations

Trigonometric equations carry lots of value in practice especially in Engineering. Notable properties such as the period and frequency open a full spectrum of applications.

Circular structures play a crucial role in everything mechanical that we use today. Circles are synonymous of trigonometry, and trigonometric equations are at the center of it.

Example: Solving simple trig equations

Solve: $$\sin(x) = \frac{1}{2}$$

Solution:

We need to solve the following given trigonometric equation equation:

$\sin\left(x\right)=\frac{1}{2}$

The following is obtained:

$$\displaystyle \sin\left(x\right)=\frac{1}{2}$$
We apply the inverse trigonometric function $$\arcsin(\cdot)$$, so we get that
$$\displaystyle \,\,$$
$$\displaystyle \arcsin\left(\sin\left(x\right)\right)=\arcsin\left(\frac{1}{2}\right)$$
so then we get
$$\displaystyle \,\,$$
$$\displaystyle x =\arcsin\left(\frac{1}{2}\right)=\frac{1}{6}\pi{}$$

By direct application of the properties of the inverse trigonometric function $$\arcsin(\cdot)$$, as well as the properties of the trigonometric function $$\sin\left(x\right)$$, we obtain that

$x_1=\frac{5}{6}\pi{}+2\pi{}K_1 \text{ , for } K_1 \text{ an arbitrary integer constant}$ $= ... \, -\frac{7}{6}\pi{}, \, \,\, \frac{5}{6}\pi{}, \,\, \, \frac{17}{6}\pi{}, \, \, \, \frac{29}{6}\pi{} \, ...$
$x_2=\frac{1}{6}\pi{}+2\pi{}K_2 \text{ , for } K_2 \text{ an arbitrary integer constant}$ $= ... \, -\frac{11}{6}\pi{}, \, \,\, \frac{1}{6}\pi{}, \,\, \, \frac{13}{6}\pi{}, \, \, \, \frac{25}{6}\pi{} \, ...$

Therefore, solving for $$x$$ for the given equation leads to the solutions $$x=\frac{5}{6}\pi{}+2\pi{}K_1,\,\,x=\frac{1}{6}\pi{}+2\pi{}K_2$$, for $$K_1, K_2$$ arbitrary integer constants.

More equation calculators

Our trigonometric equation with steps will come in handy when dealing with equations with specific structure. If you are unsure of the type of equation you are dealing with, you can use our general equation solver, which will figure out the structure of the given equation, and will find a suitable approach.

The main difficulty with solving equations that are not linear equation or polynomial equation is that there is not a specific route to follow, nor there is any guarantee you will find solutions.

Usually, the strategy consists of simplifying expressions as much as possible, and after doing that, it is usually nowhere land, where you need to try whatever feels suitable.

Naturally, the idea is to try to reduce the equation to a simpler equation, by using some kind of substitution and a multi-step process, where you first find solutions of an auxiliary solution, which gives you CANDIDATES to the original equation. You would like to solve a linear equation, or even a quadratic equation, but perhaps the reduction you get will be a bit less generous.