Homework 5.6 Right Triangle Trig by happo7

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Homework 5.6 Right Triangle Trig
MAT 125: Precalculus
Fall 2009: Sections 6, 9, and 13
Mr. Jonckheere

Due Homework

CP 5.6.1. Fill in the blanks to complete each theorem

     a) If α is an accute angle of a right triangle, opp is the length of the side opposite α, adj is the length of
     the side adjacent to α, hyp is the length of the hypotenuse, and x = m(α) then

                            opp                  adj                    opp
              sin ( x ) =          cos ( x ) =            tan ( x ) =
                            hyp                  hyp                    adj

     b) If a right triangle has legs of length a, and b, and a hypotenuse of length c then c2 = a2 + b2.
     c) The sum of the primary measures of the angles in any triangle is π = 180°.
     d) If x is in [0, π/2] then

             arcsin(sin(x)) = x
             arcos(cos(x)) = x


CP 5.6.2. Let α be an accute angle in a right triangle, and x = m(α). Let the side opposite α have length 6,
and the side adjecent to α have length 3. Evaluate all six trig functions at x.

             h2 = 62 + 32 = 36 + 9 = 45
             h = 45 = 3 5
                          opp   6         2                       5
              sin ( x ) =     =   =                csc ( x ) =
                          hyp 3 5          5                     2
                          adj   3         1
              cos ( x ) =     =   =                sec ( x ) = 5
                          hyp 3 5          5
                          opp 6                                  3 1
              tan ( x ) =     = =2                 cot ( x ) =    =
                          adj 3                                  6 2
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CP 5.6.3. A right triangle has an angle with measure 68°, and a hypotenuse of length 10. Solve the triangle.

           x = 68°                                                                                       b
           c = 10

           y = 90° – 68° = 22°                                                                       y
                                                                                            c
           sin ( x ) =
                       a                                                                                 a
                       c
           a = c sin ( x )
             = 10 sin ( 68° )                                                              x
             ≈ 9.3                                                                 a
                                                                                                b

                      b
           cos ( x ) =
                      c
           b = c cos ( x )
             = 10 cos ( 68° )
             ≈ 3.7
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CP 5.6.4. A right triangle has legs of lengths 20 and 31. Solve the triangle.
                                                                                                          b
           a = 20
           b = 31
                                                                                                      y
            2      2       2
           c =a +b                                                                           c
                       2       2   2
           c = a + b = 20 + 31 = 1361
                                       2                                                                  a

                      a
            tan ( x ) =                                                                     x
                      b
                       a                                                          a
            x = arctan                                                                         b
                       b
                        20 
              = arctan  
                        31 
              ≈ 32.8°


            y = 90° − x
              ≈ 90° − 32.8°
                = 57.2°



CP 5.6.5. A 40 meter guy wire is attached to the top of a 35 meter atenna and to a point on the ground. What
is the primary measure (in degrees) of the angle made by the guy wire and the ground?

                      a
           sin ( x ) =
                      c
                       a           35 
            x = arcsin   = arcsin   = 61°
                       c           40 


CP 5.6.6. Two country roads meet at a right angle. A biker travelling on the North-South road and
approaching the intersection decides to take a shortcut across the field. When he reaches a large oak, he turns
40° and heads across the field for the East-West road. If he travelled a straight 0.25 miles across the field to
get to the East-West road, how far is the intersection from the oak? Round to the nearest 100th of a mile.

                     b
           cos( x) =
                     c
           b = c cos( x) = 0.25cos ( 40° ) ≈ 0.19 mi
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CP 5.6.7. The forest ranger at the top of Kendrick mountain is watching a forst fire spread is her direction. In
10 minutes the angle of depression of the leading edge of the fire changed from 11° to 13°. At what speed is
the fire spreading in the direction of the ranger. Assume the ranger is 3430 feet above the fire.

                             a
           tan ( x1 ) =
                          b1 + b2
                          a
           tan ( x2 ) =
                          b2
                                     b1 + b2 b2 b1
           cot ( x1 ) − cot ( x2 ) =          − =
                                        a      a a
           b1 = a ( cot ( x1 ) − cot ( x2 ) )
                 b1
           v=
                  t
                 a ( cot ( x1 ) − cot ( x2 ) )
             =
                            t
                 3430 ( cot (11° ) − cot (13° ) )
             =
                                 10
             ≈ 279 ft/min
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Extra Practice

CP 5.6.8. A triangle has a hypotenuse of length 100 and a leg of length 10. Solve the triangle.

           c = 100                                                                                       b
           a = 10

           c2 = a2 + b2                                                                              y
           b2 = c2 – a2                                                                     c
           b = c −a
                   2         2                                                                           a
                         2       2
             = 100 − 10
             = 9900                                                                        x
             = 90 11                                                                a
                                                                                                b

                     a
           sin ( x ) =
                     c
                      a
           x = arcsin  
                      c
                      10 
            = arcsin      
                      100 
            ≈ 5.7°

           y = 90° – x
             ≈ 90° – 5.7°
             = 84.3°
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CP 5.6.9. One angle of a right triangle measures 39° and the leg adjacent to it is 15 units long. Solve the
triangle.
                                                                                                         b
           x = 39°
           b = 15
                                                                                                     y
           y = 90° – x                                                                      c
               = 90° – 39°                                                                                a
               = 51°

                         b                                                                  x
           cos ( x ) =
                         c
                   b                                                                   a
           c=                                                                                   b
                 cos ( x )
                    15
             =
                 cos ( 39° )
             ≈ 19.3

                      a
           tan ( x ) =
                      b
           a = b tan ( x )
             = 15 tan ( 39° )
             ≈ 12.1


CP 5.6.10. A tree casts a shadow which is 6.4 feet long. The primary measue of
the angle of elevation from the shadow of the tree’s top to the actual tree’s top is
71°. How tall is the tree to the nearest 100th of a foot?

                      a
           tan ( x ) =
                      b
           a = b tan ( x ) = 6.4 tan ( 71° ) ≈ 18.59
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CP 5.6.11. A hot-air baloonist spots two mile-posts on the straight road over which he hovers. He measures
the angle of depression to mile post 184 to be 17° and the angle of depression to mile post 183 to be 21°.
How high is he?

                         a
           tan ( x1 ) =
                        1+ b
                        a
           tan ( x2 ) =
                        b
                                       1+ b b 1
           cot ( x1 ) − cot ( x2 ) =       − =
                                        a   a a
                            1
           a=
                 cot ( x1 ) − cot ( x2 )
                            1
             =
                 cot (17° ) − cot ( 21° )
             ≈ 1.5 mi

								
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