# Trigonometric Functions Calculator

By Hanna Pamuła, PhD candidate

This trigonometric functions calculator can help in determining the values of six trig functions in no time. You'll find here not only the three basic functions - sine, cosine and tangent, but also their reciprocals: cosecant, secant and cotangent, respectively. Scroll down to read more about trigonometric ratios, to find sin cos tan charts and learn the mnemonic rule to remember functions definitions - sohcahtoa. Don't wait any longer, give this trigonometric functions calculator a try!

## Sine, cosine, tangent...

Sine, cosine and tangent are the trigonometric function of an angle. If the angle is acute, you can define these trig functions in the context of a right triangle. Starting with the triangle sides nomenclature: we can name the sides of a right triangle as • hypotenuse - probably you heard that term when learning about the Pythagorean theorem. So it's the longest side in the right triangle - the opposite one to the right angle (90°).
• adjacent side - the side forming the angle of interest (in our case, angle α), which is not a hypotenuse. This side is adjacent to both the angle of interest and the right angle.
• opposite side - the side opposite to the angle of interest (in this case, angle α).

Given the sides definitions, you can state that:

• `sin(α) = opposite / hypotenuse` so `sin(α) = a / c`
• `cos(α) = adjacent / hypotenuse` so `cos(α) = b / c`
• `tan(α) = opposite / adjacent` so `tan(α) = a / b` If you want to read more about those three famous functions, jump directly to our dedicated tools:

But if you don't care too much about the details, and just want to calculate all functions at once, use this tool as a complete sin cos tan calculator.

## ... And other trig functions: cosecant, secant and cotangent. Trigonometric ratios

In previous paragraph you learned about basic trigonometric ratios: But what if we would like to invert the ratios? Like `c / a` instead of `a / c`? Apparently, these reciprocal trigonometric ratios have their own names, and are even used sometimes! So, for the right triangle:

1. Cosecant is the reciprocal of the sine

It's the ratio of the hypotenuse to the opposite. The abbreviation of cosecant is csc, e.g. csc(30°) and it's range is csc(α)≥ 1 and csc(α) ≤ -1:

`csc(α) = 1 / sin(α) = c / a`

2. Secant is the reciprocal of the cosine

It's the ratio of the hypotenuse to the adjacent. The abbreviation of secant is sec e.g. sec(30°) and it's range is sec(α)≥ 1 and sec(α) ≤ -1:

`sec(α) = 1 / cos(α) = c / b`

1. Cotangent is the reciprocal of the tangent

It's the ratio of the adjacent to the opposite side. The abbreviation of cotangent is cot, e.g. cot(30°) and it's range is -∞ < cot(α) < ∞:

`cot(α) = 1 / tan(α) = b / a` If you think that's all trigonometric ratios, the six above - you're wrong. There are many other trig functions - like e.g. hacoversine, exsecant or versine - but, fortunately, they are archaic and obsolete, so not used nowadays. If you want to read more about those trigonometric functions used in dark pre-calculators ages, check out this post by Evelyn Lamb from Scientific American Blog.

## Sohcahtoa (soh cah toa) mnemonics

• SOH stands for Sine equals Opposite over Hypotenuse.

• CAH stands for Cosine equals Adjacent over Hypotenuse.

• TOA stands for Tangent equals Opposite over Adjacent.

Trig function Abbreviation Definition
Sine SOH sin(α) = opposite / hypotenuse
Cosine CAH cos(α) = adjacent / hypotenuse
Tangent TOA tan(α) = opposite / adjacent

It's one of the methods to remember the definition of the basic trigonometric functions, but many other techniques exist. For example, you can simply pick one sentence which appeals to you the most:

• Sailors Often Have Curly Auburn Hair Till Old Age ⛵
• Some Old Horses Can Always Hear Their Owners Approach 🐎
• Some Old Hen Caught Another Hen Taking One Away 🐔
• Studying Our Homework Can Always Help To Obtain Achievement 📖
• Some Old Hippy Caught Another Hippy Tripping On Acid ☮️

Or invent your own, they are always the easiest to remember!

The trigonometric functions can also be defined in terms of the rise, run, and slope of a line segment relative to horizontal.

## Charts of sin cos tan csc sec cot

Below you'll find the sin cos tan charts for <-2π, 2π> range, as well as graphs for their reciprocal functions:

Sine Cosecant  Cosine Secant  Tangent Cotangent  ## How to use trigonometric functions calculator - an example

Are you still wondering how to use this trigonometric functions calculator? Don't worry; it's a piece of cake! 🍰

1. Enter the angle. Choose between degrees and radians (in two versions rad or π rad). Let's take π/3 as an example, so choose π rad unit and type 1/3.
2. Here it is! Six trig functions calculated by the trigonometric functions calculator in the blink of an eye! And also an explanatory image, if the angle is acute. What more can you ask, right?

As you know six basic trigonometric functions, you're ready for more advanced topics, such as law of sines and law of cosines.

Users also like our other math tools:

• Sig fig calc which performs conversion of any number into a number with desired amount of significant figures.
• Exponent calculator helping in raising any number to any power, in line with scientific notation tool to convert from any decimal value to scientific notation.
• LCM and GCF calculators to find least common multiple and greatest common factor of a set of the numbers.

Maybe you'll like them too? 😉

Hanna Pamuła, PhD candidate

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