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Mathematics
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Oct 10th, 2014

NOTE:       Formulas for right triangles

                                 Sin A = opp/hyp

                                 Cos A =  adj/hyp

                                 Tan A = opp/adj

                      Where 'A' is the angle, 'opp' is the side opposite to angle A, 'adj' is the side adjacent to angle 'A', and 'hyp' is the hypotenuse (the longest side of the right triangle).

                       Pythagorean Theorem:               a^2 + b^2 = c^2

             Where 'a' and 'b' represent the legs of the triangle (opposite and adjacent) and 'c' is the hyptenuse. 



To solve these problems, focus on what is labeled on the sides.  If two sides are labeled, we match with the proper trigonometric formula to solve.  But if all three labels are labeled, with an unknown side being a variable, we can use the Pythagorean Theorem.


1.  a)         A = 62 degrees,           opp = x             adj = 9.8 meters

                 The opposite and adjacent sides are labeled, so we will use the Tan formula

                                 tan A = opp/adj           <Tangent formula>

                                 tan 62 = x/9.8              <Substitute and solve for the variable>

                      1.880726465 = x/9.8             <If allowed, we can use a calculator to evaluate tan 62>

                  Common denominator is 9.8

                  1.880726465*9.8 = (x/9.8) * 9.8   <Multiply both sides by common denominator>

                         18.43111936 = x

                                            x = 18.43111936

                                            x = 18.4             <As instructed, round your answer to 1 decimal place>

                 You can also include the side's units.

                SOLUTION:   18.4 m


     b)       A = 48 degrees                opp = 14.3 cm           hyp = x

                Opposite side and hypotenuse are labeled.  We will use the Sin formula

                                       Sin A = opp/hyp                  <Sine formula>

                                       Sin 48 = 14.3/x                   <Substitute>

                              0.7431448255 = 14.3/x               <Evaluate sin with calculator>

                            Common denominator is x

                           0.7431448255 * x = (14.3/x) * x         <Multiply both sides by common denominator>

                           0.7431448255x = 14.3

                           0.7431448255x = 14.3                    <Divide both sides by the 0.7431448255>

                            /0.7431448255      /0.7431448255

                             x = 19.24254803

                             x = 19.2                                        <Round to 1 decimal place>

                 SOLUTION:   x = 19.2 cm                


        c)        A = 60 degrees             adj = 3.5 m            hyp = x

                  Adjacent side and hypotenuse are labeled.  We will use Cos formula.

                            Cos A = adj/hyp

                            Cos 60 = 3.5/x

                            0.5 = 3.5/x

                            Common denominator is x

                           0.5*x = (3.5/x) * x

                           0.5x = 3.5

                           0.5x = 3.5                 <Dividing both sides by 0.5>

                             /0.5     /0.5

                              x = 7

                              x = 7.0                <If your result is an integer and we have to round to one decimal place,

                                                             we can put a .0 behind the integer>

                        SOLUTION:   x = 7.0 m


          d)     In this part, all three sides are labeled, so we would use the Pythagorean Theorem

                    Legs are labeled with  'x' and 4.3 mm, while the hypotenuse is labeled with 9.3 mm

                                a^2 + b^2 = c^2                        <Pythagorean Theorem>

                        (x)^2 + (4.3)^3 = (9.3)^2                   <Substitute into the variables>

                            x^2 + 18.49 = 86.49                    <Square all parts>

               x^2 + 18.49 - 18.49 = 86.49 - 18.49         <Subtract both sides by 18.49>

                                        x^2 = 68

                                sqrt(x^2) = sqrt(68)                  <Square root both sides, with a calculator if allowed>

                                      |x| = 8.246211251

                     x = 8.246211251   or   x = -8.246211251           <Absolute value rule>

                  We are solving for a side of a right triangle, so we want our x-value to be positive.

                                     x = 8.246211251

                                     x = 8.2                              <Round to one decimal place>

              SOLUTION:    x = 8.2 mm




2.  a)       NOTE:   Instead of using theta, I will use A

                opp = 7.9 cm                 adj = 12.3 cm

                Opposite and adjacent sides are labeled, so we will use the Tan formula

                                        tan A = opp/adj

                                        tan A = 7.9/12.3                <Substitute values>

                                        tan A = 0.6422764228        <Divide on the right side>

                              tan^-1(tan A) = tan^-1(0.6422764228)    <Take the inverse tan of both sides with a calculator

                                                                                                     if allowed>

                                        A = 32.71167684

                                        A = 33                           <As instructed round to the nearest degree (whole number)>

             SOLUTION:   33 degrees

Oct 10th, 2014

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