If in addition the values of the cosine of the angle are known the lengths of pairs of sides (b and c), which form this angle, to calculate the value of the unknown side (a) you can use the theorem of cosines. She argues that the squared length of the right side is equal to the sum of the squares of the lengths of the other two, if it is reduced by twice the product of the lengths of these sides on the well-known of the terms of the cosine of the angle between them: a2 = b2 + c2 - 2*a*b*cos(α).
Since the value of the angle α you do not know and calculate it is not necessary, indicate this in terms of the variable (the cosine of) any letter (e.g., f) and substitute into the formula: a2 = b2 + c2 - 2*a*b*f. Get rid of the extent in the left part of the expression to get a General view of the final calculation formula of the desired length of side: a = √(b2+c2-2*a*b*f).
To find the length of a side (a), provided that in addition to the value of cosine (f = cos(α)) lying opposite this side of the angle, given the value of another angle (β) and length lying on the opposite side (b), we can use the theorem of sines. According to her the ratio of desired length to the sine of the opposite angle is equal to the ratio of the length of the known side to the sine of the angle, which is also given in: a/sin(a) = b/sin(β).
The sum of the squares of the sine and cosine of the same angle equals one - use this identity to Express the sine in the left side of the equation using the given in terms of cosine: a/√(1-f2) = b/sin(β). Find a formula to calculate the length of the right side in General, moving the denominator from the left side of the identity in the right: a = √(1-f2)*b/sin(β).
In a right triangle to calculate the values of the parties is sufficient to complement the cosine of an acute angle (f = cos(α)) a single parameter - the length of any of the parties. To find the length of side (b) adjacent to the vertex, the cosine of which is known, multiply this value by the length of the hypotenuse (c): b = f*c. If you need to calculate the length of hypotenuse and length of side is known, transform this formula accordingly: c = b/f.