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, the sine of knowing

In order to obtain a formula linking

**the sine**and co**sine**of the angle, it is necessary to give or to recall some definitions. So,**the sine**of an angle is the ratio (quotient of the) opposite side of a right triangle to the hypotenuse. To**the sine**of an angle is the ratio adjacent side to the hypotenuse.Instruction

1

Draw a right triangle ABC, where angle ABC is a straight line (Fig.1). Consider the ratio of

sin CAB=BC/AC cos CAB=AB/AC.

**the sine**and co**sine**and angle CAB. According to the above definitionsin CAB=BC/AC cos CAB=AB/AC.

2

Remember the Pythagorean theorem - AB^2 + BC^2 = AC^2, where ^2 is the operation of squaring.

Divide left and right side of the equation by the square of the hypotenuse AC. Then the previous equation will look like this:

AB^2/AC^2 + BC^2 AC^2 = 1.

Divide left and right side of the equation by the square of the hypotenuse AC. Then the previous equation will look like this:

AB^2/AC^2 + BC^2 AC^2 = 1.

3

For convenience, we rewrite the equation obtained in step 2, as follows:

(AB/AC)^2 + (BC/AC)^2 = 1.

According to the definitions given in step 1, we get:

cos^2(CAB) + sin^2(CAB) = 1, i.e.

cos(CAB)=SQRT(1-sin^2(CAB)), where SQRT is the operation of taking the square root.

(AB/AC)^2 + (BC/AC)^2 = 1.

According to the definitions given in step 1, we get:

cos^2(CAB) + sin^2(CAB) = 1, i.e.

cos(CAB)=SQRT(1-sin^2(CAB)), where SQRT is the operation of taking the square root.

Useful advice

The magnitude of the sine and cosine of any angle cannot be greater than 1.

# Advice 3 : How to find the tangent using the cosine

Cosine, and sine, referred to as "direct" trigonometric functions. The tangent (along with the cotangent) is referred to the other pair, called "derivatives". There are several definitions of these functions that make possible finding the tangent of a specified angle for a known cosine value from the same value.

Instruction

1

Subtract from unity the quotient of the units on the squared value of the cosine of the specified angle, and from the result, extract the square root, this will be the value of the tangent of the angle, expressed via its cosine: tg(α)=√(1-1/(cos(α))2). In this case, note that in the formula for the cosine is in the denominator of the fraction. The impossibility of division by zero eliminates the use of this expression for angles equal to 90°, and differs from this value by multiples of 180° (270°, 450°, -90° etc.).

2

There is an alternative method of calculating the tangent at the known value of the cosine. It can be used, if not set a limit on the use of other trigonometric functions. To implement this method, first determine the angle for a known cosine value - this can be done using the inverse cosine. Then just calculate the tangent for the angle values. In General this algorithm can be written as: tg(α)=tg(arccos(cos(α))).

3

There are even more exotic option using the definition of cosine and tangent of acute angles using right triangle. The cosine in this definition corresponds to the ratio of the length adjacent to the considered corner of the leg to the length of the hypotenuse. Knowing the cosine ratio to find the corresponding lengths of these two sides. For example, if cos(α)=0.5, then the adjacent side can be taken equal to 10 cm and the hypotenuse is 20cm. The specific numbers here do not matter - the same and right decision you will get to any values having the same ratio. Then by the Pythagorean theorem determine the length of the missing side opposite leg. It will be equal to the square root of the difference between the lengths squared of the hypotenuse and the known leg: √(202-102)=√300. The tangent is, by definition, corresponds to the ratio of the lengths opposite and adjacent sides (√300/10) - calculate it and get the tangent value found using the classic definition of cosine.