Free Trigonometry Calculator with steps

With the help of a trigonometry calculator, you will be able to solve two trigonometric problems. You can use it to find the values of sine, cosine, or tangent and cotangent in the first part of the calculator. In the second part of the trigonometry calculator, you can calculate the triangle sides or angle of right triangle trigonometry.

This article provides you with tips on how to use a Trigonometry Calculator, what cos is in trigonometry, how to do trigonometry. In addition, you will learn what are and how to use trigonometry formulas. If you have other problems, you can also use our Trigonometric Functions Calculator, Cotangent Calculator, Tangent Calculator, and Power Reducing Calculator. Also, learn about fractions with our Fractions Calculator.

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What is Trigonometry – Definition?

A part of math science that deals with the specific functions of angles, degrees, and their application is trigonometry. Trigonometry’s name comes from the Greek words trigon, meaning triangle, and metron, meaning measure. We can divide it into planar (angles and distances in the plane) and spherical (angles and distances in space).

Trigonometry refers to studying the properties of a right triangle, lengths, and angles. We learn the relationships between the lengths of the sides of a right triangle. However, it also deals with waves and oscillations.

Trigonometry is not just a study of plane figures. It has a branch called spherical trigonometry that studies triangles in three-dimensional spaces. Astronomers use spherical trigonometry, but we can also use it for navigation.

Apart from what we use in pure mathematics, we can also apply trigonometry in many branches of science. There are six trigonometric relationships. Three basic ones, called Sine (sin), Cosine (cos), and Tangent (tan), together with Secant (sec), Cosecant (csc), and Cotangent (cot).

Trigonometry – History

Trigonometry occurs in ancient Babylon, Egypt, India, and China. These people knew practical geometry and some concepts that were an introduction to trigonometry. In the Rhind Papyrus (1800 BC, Egypt), among the 84 problems in algebra, arithmetic, and geometry, there are also five problems in which seqt is one of them.

We can define seqt in terms of the cotangent, meaning that the Egyptians knew the numerical relations in a triangle. The founder of modern trigonometry is considered to be the Hipparchus of Greece (around 190-120 BC). He was the first to calculate the values ​​of trigonometric functions. In the 2nd century, Ptolemy contributed to the development of trigonometry in his work Almagest.

Ptolemy was an astronomer and had to use some basics of trigonometry. The last major contribution to the development of trigonometry was the invention of a new term – logarithms by the Scottish mathematician John Napier in 1614. Logarithm tables greatly facilitated numerical computation, including compiling tables of trigonometric functions.

Trigonometric Identities

We can define trigonometric identity in terms of the equality of two expressions. These expressions remain accurate even when we include any values instead of general numbers. In short, these identities connect trig functions when math expressions need to be simplified.

Basic trigonometric identities are:

• Connection of sine and cosine: sin²α+cos²α=1
• Sine and tan connection: sin²α=tg²α / 1+tg²α
• Cosine and tan connection: cos²α=1 / 1+tg²α

To learn more about trigonometric identities, see this related Sum and Difference Identities Calculator.

Trigonometry Table

This table will make it easier for you to remember and easily find trigonometric numbers of basic angles (in degrees).

First of all, write all the functions in columns. In the first column, you write the angles: 0°, 30°, 45°, 60°, 90°. Fill in the sine column using the expression √x / 2.

For an angle of 0°, set x = 0 and then the value √x / 2 = √0 / 2 = 0. In the same column for the angle 30° set that x = 1 and from that follows √x / 2 = √1 / 2 = 1/2.

Likewise, for an angle of 45°, set x = 2 and it follows √x / 2 = √2 / 2. For an angle of 60°, set x = 3 and when we replace it, we get √x / 2 = √3 / 2 and for sin 90°, set x = 4 and when we replace it, we get the following √x / 2 = √4 / 2 = 1.

Then we fill in the column that belongs to the cosine. Fill it using the reverse order because the rule sin x° = cos (90-x)° applies.

We fill in the column that starts with a tangent. It is easy because we know that tanα = sinα / cosα. Fill in the column belonging to the cotangent in the reverse order. Learn about cofunctions, with this related Cofunction Calculator.

Trigonometry vs. Geometry

Mathematics has three main branches, called Arithmetic, Algebra, and Geometry. Geometry is the study of the shapes, sizes, and properties of a space of a certain number of dimensions. The geometry we can categorize as planar geometry, solid geometry, and spherical geometry.

Geometry is used daily, almost everywhere, and by everyone. You can find geometry in physics, engineering, architecture, and many others. So, we ask ourselves, what is trigonometry in geometry?

Geometry is the main branch of math, while trigonometry is the branch of geometry. Geometry is the study of the elements of a figure. Trigonometry is a study of the properties of a right triangle.

Trigonometry Calculator – How to Use With Steps?

When you want to calculate the missing sides or the angles of a right triangle, the first step is to enter the values you know. It could not be a problem. You need to calculate the value you are missing and it is enough to know the following variables: one side and one angle, two sides, area, and one side.

It is essential to know that you need more familiar variables if you have the values of two triangles. When you have two triangles of the same shape, so the angles are the same, the sizes can be different. Thus, in such triangles, the cell lengths are not the same. Such a relationship is called triangle similarity. Only if the cells have the same length do the triangles coincide.

On the whole, using these steps you can solve every trigonometry problem.

Trigonometry Calculator – Example

At what angle did the sports plane fly until it reached an altitude of 2,500 m if we know that it crossed 5 km? At that height, a paratrooper jumped out of the plane. How far is the parachute landing site from the take-off point of the plane?

\sin\alpha =\frac{2.5}{5}=\frac{1}{2}=30

d=\sqrt{5^{2}-2.5^{2}}= \sqrt{\frac{75}{4}}=\frac{5\sqrt{3}}{2}\approx 4.33 \;km

Additionally, here is one more example. We can do the trigonometry in the right triangle step by step. For instance, a right triangle has a side length of 8 cm, and a hypotenuse length of 15 cm. We can find the other side length with our calculator:

• type given a side of a triangle,
• type given hypotenuse length.

Trigonometry calculator will calculate the next:

• the b side will be 12.689 cm length,
• angle α is 32.231 and angle β is 57.769.

Real-world Applications

To sum up, trigonometry is used in all mathematics, and we can apply it to all other sciences and find a result for our problem. Surveyors have used it for centuries, as have military and other engineers. We can use it in optics and statics and in physical chemistry in physics.

Trigonometry has its application in music, more precisely in computer music and modern architecture. Beautifully curved surfaces of stone and glass would be impossible without it.

The trigonometry of a right triangle can also use in aerial control. Air traffic control employees at small airports have to determine the height of the clouds in the evening. Triangle trigonometry has applications for finding unknown lengths and measuring angles in images, movies, and television. There is an ideal distance from which to look to create the best possible picture from the eye.

Some of the reasons people use angles in everyday life are position, direction, precision. Crossings on the street are usually performed as close as possible to an angle of 90°, possibly even higher, so that visibility when turning is as high as possible.

Angles are also often used in sports. Athletes who try to throw balls at certain distances, such as baseballs, basketball, soccer balls, and golf balls, strategically use angles. If they want to hit a short and high ball, they will search for an angle close to 90°. It is now clear what trigonometry is in the real world.

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