PLEASE I NEED YOUR HELP WITH THESE

Mathematics
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Oct 9th, 2014

2.   b)         opp = 1.8 m            hyp = 2.3 m

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

                         sin A = opp/hyp

                         sin A = 1.8/2.3                           <Substitute values>

                         sin A = 0.7826086957               <Divide with calculator>

              sin^-1(sin A) = sin^-1(0.7826086957)       <Take the inverse sin of both sides>

                               A = 51.50004959

                               A = 52                                 <Round to the nearest angle (whole number)

               SOLUTION:  52 degrees



     c)       adj = 5.5 mm          hyp = 7.75 mm

                Adjacent side and hypotenuse are labeled.  We will use cos.

                          cos A = 5.5/7.75

                          cos A = 0.796774194

                cos^-1(cos A) = cos^-1(0.796774194)

                               A = 44.791324466

                               A = 45

               SOLUTION:  45 degrees




3.  a)         Hint:  Side 'a' is opposite to angle A              Side 'a' is adjacent to angle B

                           Side 'b' is opposite to angle B             Side 'b' is adjacent to angle b

                           If angle C is the right angle, then side 'c' is the hypotenuse


              ANGLES:  We are given two angles, angle C = 90 degrees, and angle B = 25 degrees.  The missing angle is angle A.  Keep in mind that the sum of three angles is 180 degrees.  And since we already have two given angles, we can use this concept to find the remaining angle.

                                    Angle A + Angle B + Angle C = 180         <Sum of three angles in a triangle is 180>

                                   A  +  25  + 90  = 180                     <Substitute angles>

                                   A + 115 = 180                                <Add on the left side>

                                   A + 115 - 115 = 180 - 115               <Subtract both sides by 115>

                                   A = 65


             SIDES:  We are given one side, a = 16.  We need to find sides 'b' and 'c'.

                          Side 'a' is opposite to angle A

                          We can use one of the trigonometric formulas that contain the opposite side as a variable, like the sin formula.

                                        sin A = opp/hyp

                                        sin 65 =  16/c                      <Substitute, remember that 'c' is the hypotenuse>

                             0.906307787 = 16/c

                                denominator is c

                        0.906307787 * c  = (16/c)*c

                        0.906307787c = 16

                        0.906307787c = 16                            <Divide both sides by 0.906307787>

                        /0.906307787      /0.90637787

                                  c = 17.6540467

                                  c = 17.7                                 <Round side to 1 decimal place>


                    With two sides now known, we can use the Pythagorean Theorem to find the remaining side.

                              a^2 + b^2 = c^2

                           (16)^2 + b^2 = (17.7)^2

                           256 + b^2 = 313.29

                          256 + b^2 - 256 = 313.29 - 256            <Subtract both sides by 256>

                                   b^2 = 57.29

                         sqrt(b^2) = sqrt(57.29)                       <Square root both sides>

                                  |b| = 7.569015788 

                       b = 7.569015788   or   b = -7.569015788

                             Use positive value for the side only.

                                b = 7.569015788

                                b = 7.6                                          <Round side to 1 decimal place>


              SOLUTION:          Angle A = 65 degrees

                                            side b = 7.6 

                                            side c = 17.7



     b)       ANGLES:    Angle A is opposite of side 'a' and 'c' is the hypotenuse.  We can use sin.

                                      sin A = opp/hyp

                                      sin A = 5/13

                                      sin A = 0.3846153846

                             sin^-1(sin A) = sin^-1(0.3846153846)

                                          A = 22.61986495

                                          A = 23                                   <Rounded to the nearest angle (whole number)


                              Sum of three angles is 180

                                        Angle A + Angle B + Angle C = 180

                                            23 + B + 90 = 180

                                             B + 113 = 180

                                            B + 113 - 113 = 180 - 113                 <Subtract both sides by 113>

                                            B = 67


               SIDES:       Two sides are known, so we can use the Pythagorean Theorem

                                      a^2 + b^2 = c^2

                                      (5)^2 + b^2 = (13)^2

                                      25 + b^2 = 169

                                      25 + b^2 - 25 = 169 - 25

                                      b^2 = 144

                                sqrt(b^2) = sqrt(144)

                                     |b| = 12

                                   b = 12  or  b = -12

                                  Use the positive version

                                    b = 12


                SOLUTION:         Angle A = 23 degrees

                                             Angle B = 67 degrees

                                            Side b = 12



4.    Let 'x' represent the height of the tree

       If you draw a diagram, it should represent a right triangle.  The angle of elevation is opposite to the vertical side (which represents the height of the tree), and adjacent to the horizontal side (which represents the tree's shadow).  In other words:

                           A = 41 degrees           opp = x              adj = 13.4 m

         Opposite and adjacent sides are labeled.  We will use the tan formula.

                                 tan A = opp/adj

                                tan 41 = x/13.4

                                0.8692867378 = x/13.4

                               0.8692867378 * 13.4 = (x/13.4)*13.4

                               11.64844229 = x

                                x = 11.64844229

                                x = 11.6  m

                 SOLUTION:    The tree is about 11.6 meters long

     (I apologize, I wasn't able to post the diagram part.  But it should represent a right triangle.)

                          

Oct 10th, 2014

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