METHODS OF SOLVING TRIANGLESTo “solve” a triangle means to determine oneor more unknown values (such as the length ofa side or the size of an angle) from given knownvalues. Here are some of the methods used.Acute Angle of RightTriangle by TangentPythagorean TheoremWhen you know the lengths of two sides ofa right triangle, or its hypotenuse and one side,you can determine the length of the remainingside, or the length of the hypotenuse, by applyingthe Pythagorean theorem. The Pythagoreantheorem states that the square of the length of thehypotenuse of any right triangle equals the sumof the squares of the lengths of the other twosides.Figure 1-23 shows a right triangle with acuteangles A and B and right angle C. Sides oppositeA and B are designated as a and b; the hypotenuse(opposite C) is designated as c. Side a measures3.00 ft, side b measures 4.00 ft, and the hypot-enuse measures 5.00 ft. Any triangle with sidesand hypotenuse in the ratio of 3:4:5 is a righttriangle.If C^{2 }= a^{z }+ b^{2}, it follows that c = i=.The formulas for solving for either side, given theother side and the hypotenuse; or for thehypotenuse, given the two sides, areFigure 1-23.-A right triangle.One of the angles in a right triangle alwaysmeasures 900. Because the sum of the three anglesin any triangle is always 180°, it follows that eachof the other two angles in a right triangle mustbe an acute (less than 90°) angle. Also, if youknow the size of one of the acute angles, you candetermine the size of the other from the formulasA = (90° – B) and B = (90° – A).In any right triangle in which you know thelengths of the sides, you can determine the sizeof either of the acute angles by applying thetangent of the angle. Take angle A in figure 1-23,for example. You know thatReference to a table of natural tangents shows thatan angle with tangent 0.75 measures to the nearestminute, 36°52'.Side of Right Triangle by TangentIf you know the length of one of the sides ofa right triangle and the size of one of the acuteangles, you can determine the length of the otherside by applying the tangent. Suppose that for thetriangle shown in figure 1-23 you know that angleA measures 36°52' and that side b measures 4.00ft. You want to determine the length of side a.SinceSide of Right Triangle by CotangentSuppose that for the triangle shown in figure1-23, you know that angle B measures 53°08´ andthat side a measures 3.00 ft. You want to1-20