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Vector components are added to determine the magnitude and direction of the resultant. Calculations using trigonometric functions are the most accurate method for making this determination. EO 1.3ADD vectors using the following methods: c. Analytical The graphic and components addition methods of obtaining the resultant of several vectors described in the previous chapters can be hard to use and time consuming. In addition, accuracy is a function of the scale used in making the diagram and how carefully the vectors are drawn. The analytical method can be simpler and far more accurate than these previous methods. Review of Mathematical Functions In earlier mathematics lessons, the Pythagorean Theorem was used to relate the lengths of the sides of right triangles such as in Figure 22. The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. This expression may be written as given in Equation 24.
Figure 22 Right Triangle Also, recall the three trigonometric functions reviewed in an earlier chapter and shown in Figure 23. The cosine will be used to solve for F_{x} The sine will be used to solve for F_{ y}. Tangent will normally be used to solve for , although sine and cosine may also be used. On a rectangular coordinate system, the sine values of are positive (+) in quadrants I and II and negative () in quadrants III and IV. The cosine values of are positive (+) in quadrants I and IV and negative () in quadrants II and III. Tangent values are positive (+) in quadrants I and III and negative () in quadrants II and IV.
Figure 23 Trigonometric Functions When mathematically solving for tan , calculators will specify angles in quadrants I and IV only. Actual angles may be in quadrants II and III. Each problem should be analyzed graphically to report a realistic solution. Quadrant II and III angles may be obtained by adding or subtracting 180° from the value calculated. 

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