Find the missing measures. Write all answers in radical by Noodlezs

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									Warm – up
 Find the missing measures. Write all
 answers in radical form.
            y
                            60°
                30                     10
                 45    y
            x
    60
      45          z                         30°
                                    z
    x6
    y3 3
                                  z 5 3
    z3 2                         y 5
       Introduction to
        Trigonometry
This section presents the 3 basic trigonometric
ratios sine, cosine, and tangent. The concept of
similar triangles and the Pythagorean Theorem
can be used to develop the trigonometry of
right triangles.
 Engineers and scientists have
found it convenient to formalize
the relationships by naming the
       ratios of the sides.


   You will memorize these
       3 basic ratios.
The Trigonometric Functions


       SINE
     COSINE
   TANGENT
SINE
Pronounced like “sign”

COSINE
Pronounced like “co-sign”

TANGENT
Pronounced “tan-gent”
                                    B
        Opp to A
Sin(A)  With Respect to angle A,
           Hyp
         label the three sides


         Adj to A
Cos(A) 
          Hyp

                        A           C
         Opp to A
Tan(A) 
         Adj to A
 We need a way
to remember
all of these
ratios…
SOHCAHTOA   Sin
            Opp
            Hyp
            Cos
            Adj
            Hyp
            Tan
            Opp
            Adj
Finding sin, cos, and tan.
  (Just writing a ratio or decimal.)
Find the sine, the cosine, and the tangent of M.
Give a fraction and decimal answer (round to 4 places).

N                              opp    9
                       sin M 
                               hyp  10.8  .8333

9        10.8
                               adj  6
                       cos M 
                               hyp 10.8  .5556
P        6       M

                                opp   9
                        tan M                 1.5
                                adj   6
   A
                                Find the sine,
                 24.5           cosine, and the
 8.2
                                tangent of angle A
  C           23.1         B

                                 opp   23.1
                         sin A             .9429
Give a fraction                  hyp   24.5
and decimal
answer.
                                adj     8 .2
                        cos A               .3347
                                hyp     24.5
Round to 4 decimal
places
                                opp     23.1
                        tan A               2.8171
                                adj     8.2
      Finding a side.
(Figuring out which ratio to use and
    getting to use a trig button.)
    Ex: 1 Find x. Round to the nearest tenth.
Figure out which ratio to use.
                                         55
What we’re looking for…
What we know…
                                 20 m
                                   adj
                  opp
            tan 
                  adj                                x
      tan55 
                      x                              opp

                     20
 20 tan55   x                     20        tan         55   )

                                          x  28.6 m
  We can find the tangent of 55
  using a calculator
Ex: 2 Find the missing side.
Round to the nearest tenth.


           283 m
                             sin 24 
x                                          x
              24                        283
                         283 sin 24   x
                               x  115.1 m
Ex: 3 Find the missing side.
Round to the nearest tenth.


                              cos40 
                                         x
           40
  20 m           x                      20
                         20 cos40   x

                              x  15.3 m
Ex: 4 Find the missing side.
Round to the nearest tenth.
                                      tan72  80
                                                 x

              80 m                   x tan72   80
              Note: When the variable is     x     80
 72                in the denominator,
                                                  tan(72 )
    x               you end up dividing

   80          tan       72     )      =
                                           x  26 m
Sometimes the right triangle is hiding
ABC is an isosceles triangle as
marked. Find sin C.
                              Answer as a fraction.
           A




     13             13
          12
                                     opp 12
                             sin C     
 B                       C           hyp 13
           10   5
Ex. 5
        A person is 200 yards from a river. Rather than walk
        directly to the river, the person walks along a straight
        path to the river’s edge at a 60° angle. How far must
        the person walk to reach the river’s edge?
                                        cos 60°


200                                  x (cos 60°) = 200
        60°          x                        x


                                           X = 400 yards
Ex: 6
A surveyor is standing 50 metres from the
base of a large tree. The surveyor measures
the angle of elevation to the top of the tree
as 71.5°. How tall is the tree?
                                                  Opp
                                      tan 71.5° 
                                                  Adj
                                                   y
?                                     tan 71.5° 
                                                  50
                      71.5°      50 (tan 71.5°) = y
             50 m
                                        y  149.4 m
  For some applications of trig,
we need to know these meanings:
     angle of elevation and



      angle of depression.
Angle of Elevation
If an observer looks UPWARD toward
an object, the angle the line of sight
makes with the horizontal.




                 Angle of
                 elevation
Angle of Depression
If an observer looks DOWNWARD toward
an object, the angle the line of sight
makes with the horizontal.

                Angle of
                depression
      Finding an angle.
(Figuring out which ratio to use and getting to
use the 2nd button and one of the trig buttons.
       These are the inverse functions.)
Ex. 1: Find . Round to four decimal places.
                                   17.2
                           tan  
                                    9
17.2            2nd      tan 17.2               9       )


              
        9                 62.3789

       Make sure you are in degree mode (not radians).
Ex. 2: Find . Round to three decimal places.
          7                          7
                             cos 
                                     23
    23
                   2nd      cos 7                23       )




                            72.281
         Make sure you are in degree mode (not radians).
Ex. 3: Find . Round to three decimal places.
                                sin  
                                         200
200                                      400

                 2nd       sin    200         400 )


                                30

      Make sure you are in degree mode (not radians).
When we are trying to find a side
we use sin, cos, or tan.


When we need to find an angle
we use sin-1, cos-1, or tan-1.

								
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