How to find unknown side

If in a right triangle the hypotenuse is known and one of the legs, the second leg find using the Pythagorean theorem. Since the sum of the squares of the legs a and b equals the square of c the hypotenuse (c2=a2+b2), performing simple conversion, will receive the equality to find unknown side. Designate the unknown side as b. To find it, find the difference of the squares of the hypotenuse and the known leg, and from the result select the square root b=√(c2-a2).

Example. The hypotenuse of a right triangle is 5 cm and one of the sides is 3 cm Find the value of the second leg. Substitute the values in the derived formula and get the b=√(52-32)=√(25-9) =√16=4 see

If in a right triangle length of the hypotenuse and one of the acute angles, use the properties of trigonometric functions in order to find the right leg. If you want to find the side adjacent to a known corner to find it, use one of the definitions of the cosine of the angle that says that he is equal to the ratio of adjacent sides of a to the hypotenuse c (cos(α)=a/c). Then to find the length of side, multiply the hypotenuse the cosine of the adjacent leg to the angle a=c∙cos(α).

Example. The hypotenuse of a right triangle is 6 cm and the acute angle 30º. Find the length of the side adjacent to this angle. This leg will be equal to a=c∙cos(α)=6∙cos(30º)=6∙√3/2≈5.2 cm

If you need to find the leg opposite the acute angle, use the same method of calculation, only the cosine of the angle in the formula change for its sine (a=c∙sin(α)). For example, using the previous problem, find the length of the side opposite the acute angle of 30º. Using the proposed formula, we get: a=c∙sin(α)= 6∙sin(30º)= 6∙1/2=3 cm.

If you know one of the other two sides and an acute angle, to calculate the length of another use the tangent of an angle which is equal to the ratio of opposite over adjacent. Then, if a is the leg adjacent to the acute corner, find it by dividing the opposite side b to the tangent of the angle a=b/tg(α). If the leg protivorechit a sharp corner, it is equal to the product of the known side b on the tangent of an acute angle a=b∙tg(α).