# Sin Calculator

**
Instructions:
**
Use this sin calculator to compute any operation involving sin. If it is an numeric expression with sine, the calculator will simplify
it, and if it is sin function, it will graph it. Please type the sin expression you want to work with.

## About this sin calculator

This *sin calculator* will two the following for you: you can provide a numeric expression such as sin(pi/4), in which case the calculator
will simplify it, and will give an approximate numeric value if necessary. Also, if you provide a sin function like
sin(3x+1), the calculator will graph it.

Then, the process is simple: once you have provided the *sin expression* you want to calculate, you just click on the "Calculate" button that is below the form to get the steps
of the solution.

Sine, along with cosine are two corner stones of trigonometry. You will see sine and cosine all around when solving triangles, for example, but also in fields like Physics.

## How to use a sin calculator?

The main idea of a sin calculator is to evaluate sin expressions that you provide. The are some notable angles, usually multiples or fractions of \(\pi\) that are simple, integer or fraction results when calculating their sin, so it is a good idea to use a sin expression calculator to help you with that.

It is not easy to remember all the sin calculations for ALL notable angles, and you will end up working with a triangle, trying to get the answer manually, and a calculator will come in handy to double check what you obtain manually.

Also, you can instead feed the calculator with a sin function, like sin(pi x), and rather than evaluating a few points, this calculator will give you the corresponding graph

## What are the steps for using a sin calculator?

- Step 1: Identify the sin expression you want to calculate
- Step 2: Type the expression in the corresponding box. You don't need to pre simplify, the calculator will do it for you
- Step 3: The calculator will check whether it is an expression that can be evaluated, in which case if will reduce to its simplest terms
- Step 4: If sin is still in the expression because it could not be simplified any further, like sin(3/4), the calculator will give you an approximated numeric value
- Step 5: If a sin function is provided instead, a graph will be provided

We cannot emphasize enough the importance of correctly calculating operations involving sine, as those will appear literally everywhere.

## sin and cos formula

Sine and cosine are two very close cousins, if not sisters. There is tight relationship between them, expressed in the following formula:

\[\displaystyle \sin\left(\frac{\pi}{2} - x \right) = \cos(x) \]Also, another formula that initimately links sine and cosine is:

\[\displaystyle \sin^2(x) + \cos^2(x) = 1 \]## Why is sin so important?

Sines are important because, along with cosines are at the center and core of the construction of a circle. And then circles harbor many other constructions, like triangles and so on.

Sine and cosine end up entangled in every single geometric construction, consequently.

### Example: Sin calculator

Calculate the following sin expression: \(\sin\left(\frac{\pi}{3}\right)\)

Solution: The following trigonometric expression has been provided to be calculated:

\[ \sin\left(\frac{\pi}{3}\right)\]By inspecting the given trigonometric expression, we can find one notable angle, which is \(\sin\left(\frac{\pi{}}{3}\right)\).

▹ For the angle \(\frac{\pi{}}{3}\) we graphically get:

The trigonometric expression given can be simplified as:

**Conclusion:** We conclude that \(\displaystyle \sin\left(\frac{\pi}{3}\right) = \frac{1}{2}\sqrt{3} \approx 0.866\).

### Example: More sine calculations

Calculate the following: \( \sin\left(\frac{5}{4}\right) \)

Solution: The following trigonometric expression has been provided to be calculated:

\[ \sin\left(\frac{5}{4}\right)\]but this given trigonometric expression cannot be further simplified.

**Conclusion:** The passed function cannot be simplified, and we get that approximately \(\displaystyle \sin\left(\frac{5}{4}\right) \approx 0.949\).

### Example: Sin function

Calculate \( \sin(3x + 1) \).

Solution: We need to work with the following trigonometric function

\[f(x) = \sin\left(3x+1\right)\]Based on the argument of the trigonometric function that was passed, the frequency and the period are computed as follows:

\[ \begin{array}{ccl} \text{Period} & = & \displaystyle\frac{2\pi}{3} \\\\ \\\\ & \approx & 2.0944 \end{array}\]and also

\[ \begin{array}{ccl} \text{Frequency} & = & \displaystyle\frac{3}{2\pi} \\\\ \\\\ & \approx & 0.4775 \end{array}\]Based on the provided trigonometric function, \(f(x) = \sin\left(3x+1\right)\), we obtain that:

» The amplitude in this case is \(A = 1\).

» The phase shift is equal to \(\displaystyle\frac{-1}{3} = -0.3333\).

» The vertical shift is equal to \( 0\).

Summarizing, the following has been found for the given trigonometric function

- Period = \(2.0944\)
- Frequency = \(0.4775\)
- Amplitude = \(1\)
- Phase Shift = \(-0.3333\)
- Vertical Shift = \(\displaystyle 0\)

The following is the corresponding graph

## More trigonometric calculators

Trigonometry blend all these concepts together, including circles and triangles, and both sin and cos are at the very core of it.

Dealing with trigonometric expressions is another crucial skill that is very important for you to acquire.