Hypotenuse call the party of a rectangular triangle lying opposite to a right angle. It is the greatest party of a rectangular triangle. It is possible to calculate it on Pythagorean theorem or by means of formulas of trigonometrical functions.
Instruction
1. Legs call the parties of a rectangular triangle adjacent to a right angle. In the drawing the legs are designated as AB and BC. Let lengths of both legs be set. Let's designate them as |AB| and |BC|. To find length of a hypotenuse |AC|, we will use Pythagorean theorem. According to this theorem the sum of squares of legs is equal to a hypotenuse square, i.e. in designations of our drawing |AB| ^2 + to |BC| ^2 = to |AC| ^2. From a formula we receive that length of a hypotenuse of AC is as |AC| = √ (|AB| ^2 + |BC| ^2).
2. Let's review an example. Let lengths of legs |AB| = 13 be set, = 21. On Pythagorean theorem we receive |BC| that |AC| ^2 = 13^2 + 21^2 = 169 + 441 = 610. To receive hypotenuse length, it is necessary to take a square root from the sum of squares of legs, i.e. from number 610: |AC| = √610. Having used the table of squares of integers, we find out that number 610 is not a full square of any integer. To receive final value the answer |AC| = √610. If the square of a hypotenuse was equal, for example, 675, then √675 = √ (3 * 25 * 9) = 5 * 3 * √3 = 15 * √3. In case similar reduction is possible, carry out the return check - square result and compare to a reference value.
3. Let one of legs and a corner, adjacent to it, be known to us. For definiteness let it will be the leg |AB| and a corner α. Then we can use a formula for trigonometrical function a cosine – the cosine of the angle is equal to the relation of an adjacent leg to a hypotenuse. I.e. in our designations cos α = |AB| / |AC|. From here we receive length of a hypotenuse |AC| = |AB| / cos α.Если to us the leg |BC| and a corner α are known, we will use a formula for calculation of a sine of the angle – the sine of the angle is equal to the relation of an opposite leg to a hypotenuse: α = |BC| / |AC|. We receive sin that length of a hypotenuse is as |AC| = |BC| / cos α.
4. For descriptive reasons we will review an example. Let length of a leg |AB| = 15. And a corner α = 60 ° be given. We receive |AC| = 15 / cos 60 ° = 15/0.5 = 30. Let's consider how it is possible to check the result by means of Pythagorean theorem. For this purpose we need to count length of the second leg |BC|. Having used a formula for tg tangent of angle α = |BC| / |AC|, we receive |BC| = |AB| * tg α = 15 * tg 60 ° = 15 * √3. Further we apply Pythagorean theorem, we receive 15^2 + (15 * √3) ^2 = 30^2 => 225 + 675 = 900. Check is executed.