# Advice 1: How to find angle if you know the sine

The sine and cosine pair of the basic trigonometric functions, which indirectly Express the measure of an angle in degrees. All of these functions there are more than a dozen, and among them there are those that allow, for instance, to recover the sine value of an angle in degrees. For practical work with them, you can use a software calculator or network services.
Instruction
1
Use the inverse sine to calculate the angle in degrees, if you know the sine of this angle. If the angle denoted by the letter α, in General form, this can be written as: α = arcsin(sin(α)).
2
If you have the ability to use the computer for practical calculations it is easiest to use the built in calculator operating system. In the last two versions of Windows it can be run like this: press the Windows key, type the letters "ka" and press Enter. In earlier releases of this OS the link "Calculator" look at the subsection "Standard" under "All programs" in the main menu system.
3
After starting the application, switch to a mode that allows you to work with trigonometric functions. You can do this by selecting the string "Engineering" in the "View" menu of the calculator or by pressing Alt + 2.
4
Enter the value of the sine. By default, the calculator interface is no button to calculate the inverse sine. To be able to use this feature, you need to invert the values of the default button - click on the Inv button in the program window. In earlier versions, this button replaces the checkbox with the same name - put a checkmark.
5
Click on the button calculate sine - inverting functions symbol will change to sin⁻1. The calculator will calculate the angle and display its value.
6
Can be used in the calculations, and various online services, which is more than enough on the Internet. For example, go to the page http://planetcalc.com/326/, scroll it down a bit and in the Input field enter the value of the sine. To start the calculation procedure here is the orange button labeled Calculate - click it. The result of the calculation you will find in the first row of the table under this button. In addition to the arcsine and it displays the value of the arc cosine, arc tangent and arc cotangent of the entered value.

# Advice 2: How to find the cosine if you know the sine

The sine and cosine is the direct trigonometric functions, for which there are several definitions through a circle in a Cartesian coordinate system, using the solution of a differential equation, using acute angles in a right triangle. Each of these definitions allows us to derive a relationship between these two functions. The following is probably the most simple way to Express the cosine in terms of sine, through their definitions for acute angles of a right triangle.
Instruction
1
Express the sine of an acute angle of a right triangle using the lengths of the sides of this figure. According to the definition, the sine of the angle (α) should be equal to the ratio of the length of the sides (a), lying in front of him - the side to side length (c) opposite the right angle is the hypotenuse: sin(α) = a/c.
2
Find a similar formula for the cosine ofa same angle. By definition, this value should be expressed by the ratio of the length of the sides (b) adjacent to this corner (second side) to the length of the side (c), which lies opposite the right angle: cos(a) = a/c.
3
Rewrite the equation resulting from the Pythagorean theorem, so that it was involved in the relationship between the legs and the hypotenuse, derived from the previous two steps. To do this, first divide both sides of the original equation of this theorem (a2 + b2 = c2) on the square of the hypotenuse (a2/c2 + b2/c2 = 1), and then the obtained equality rewrite in this form: (a/c)2 + (b/c)2 = 1.
4
Replace in the resulting expression of the ratio of the lengths of the legs and hypotenuse trigonometric functions based on the formulas of the first and second step: sin2(a) + cos2(a) = 1. Express the cosine of the obtained equality: cos(a) = √(1 - sin2(a)). This problem can be considered solved in General.
5
If in addition to the common solutions you need to obtain a numerical result, use, for example, a calculator built into the Windows operating system. The link to his startup, locate the subsection "Standard" under "All programs" on the main menu OS. This link is articulated succinctly - "Calculator". To be able to calculate using this program trigonometric functions turn it on "engineering" the interface is press Alt + 2.
6
Enter this in terms of the value of the sine of the angle and click interface designation x2 - so you put up the original value in the square. Then enter on the keypad *-1, hit enter, then type +1 and hit Enter again - this way you subtract one square of sine. Click the button with the icon of the radical, to extract the square root and get the final result.

# Advice 3: What is the sine and cosine

The study of triangles being mathematicians for several millennia. The science of triangles - trigonometry - uses special values: sine and cosine.

## Right triangle

Initially, the sine and cosine arose from the need to calculate values in right triangles. It was observed that if the value of the degree measures of the angles in a right triangle do not change, the aspect ratio, no matter how these parties are neither changed in length, remains always the same.

And it was introduced the concepts of sine and cosine. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, cosine – adjacent to the hypotenuse.

## Cosines and sines

But the cosines and sines can be applied not only in right triangles. To find the value of the obtuse or acute angle, the sides of any triangle, it is enough to apply theorem law of cosines and law of sines.

The theorem of the cosines is quite simple: the squares of the sides of a triangle is equal to the sum of the squares of the other two sides minus twice the product of these sides into the cosine of the angle between them."

There are two interpretations of the theorem of sines: small and extended. According to small: "In a triangle the angles opposite the sides are proportional". This theorem often extend through the properties of a triangle circumscribed about the circle: In a triangle the angles opposite the sides are proportional and their ratio is equal to the diameter of the circumscribed circle".

## Derivatives

The derivative is a mathematical tool that indicates how quickly changes in the function of relative change of its argument. Derivatives are used in algebra, geometry, Economics and physics, the number of technical disciplines.

When solving problems you must know the table values of the derivatives of trigonometric functions: sine and cosine. The derivative of sine is cosine, and cosine - sine, but with the sign "minus".

## Application in mathematics

Very often the sines and cosines are used when solving right-angled triangles and problems associated with them.

The convenience of sines and cosines is reflected in the technique. Corners and sides were just judged by the theorems of cosines and sinuses by breaking up the complex shapes and objects with "simple" triangles. Engineers and architects, often dealing with the calculations of the aspect ratio and degree measures, spending a lot of time and effort to calculate the cosines and sines of angles is not tabular.

Then "help" came to the table Bradis containing thousands of values of sines, cosines, tangents and cotangent different angles. In Soviet times, some teachers were forced to teach their charges page tables Bradis by heart.
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