The sine and cosine are direct trigonometrical functions for which there are several definitions - through a circle in the Cartesian system of coordinates, through solutions of the differential equation, through acute angles in a rectangular triangle. Each of such definitions allows to remove dependence between these two functions. The, perhaps, easiest way is given below to express a cosine through a sine - through their definitions for acute angles of a rectangular triangle.
Instruction
1. Express a sine of an acute angle of a rectangular triangle through lengths of the parties of this figure. According to definition, the sine of the angle (α) has to be equal to the relation of length of the party (a) lying opposite to it - a leg - to length of the party (c), opposite to a right angle - hypotenuses: sin(α) = a/c.
2. Find a similar formula for a cosine of the same corner. By definition this size has to be expressed by the relation of length of the party (b) adjoining this corner (second leg), length of the party (c) lying opposite to a right angle: cos (a) = a/c.
3. Rewrite the equality following from Pythagorean theorem so that the ratios between legs and a hypotenuse removed on two previous steps were involved in it. For this purpose at first divide both parts of the initial equation of this theorem (a² + b² = with²) into a hypotenuse square (a²/c² + b²/c² = 1), and then rewrite the received equality in such look: (a/c)² + (b/c)² = 1.
4. Replace in the received expression of a ratio of lengths of legs and hypotenuses with trigonometrical functions, proceeding from formulas of the first and second step: sin² (a) + cos² (a) = 1. Express a cosine from the received equality: cos(a) = √ (1 - sin² (a)). On it the task can be considered solved in a general view.
5. If except the common decision it is necessary to receive numerical result, use, for example, the calculator which is built in the Windows operating system. Find the reference to its start in the Subsection "Standard" of the section "All Programs" of the OS main menu. This reference is formulated laconically - "Calculator". To have an opportunity to calculate trigonometrical functions by means of this program include its "engineering" interface - press a combination of the Alt keys + 2.
6. Enter the value of a sine of the angle given in conditions and click the interface button with designation x² - so you will square a reference value. Then type on keyboard *-1, press Enter, enter +1 and press Enter once again - in such a way you subtract a sine square from unit. Click on a key with the radical's badge to take a square root and to receive final result.