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deff fn [24]
2 years ago
13

If tan 0=5/4, calculate the value of cos 0

Mathematics
1 answer:
emmasim [6.3K]2 years ago
3 0
 
                     5
If  tan θ  =  —— , calculate the value of  cos θ:
                     4


Recall the definition of the tangent function:

                 sin θ
tan θ  =  ————
                cos θ
 
  5           sin θ
——  =  ————
  4          cos θ

Cross multiply:

5 · cos θ = 4 · sin θ

Square both sides:

(5 · cos θ)² = (4 · sin θ)²

5² · cos² θ = 4² · sin² θ

25 · cos² θ = 16 · sin² θ

But  sin² θ = 1 – cos² θ.  Substitute that for  sin² θ  into the equation above, then you get

25 · cos² θ = 16 · (1 – cos² θ)

25 · cos² θ = 16 – 16 · cos² θ

Isolate  cos² θ:

25 · cos² θ + 16 · cos² θ = 16

(25 + 16) · cos² θ = 16

41 · cos² θ = 16
 
                    16
cos² θ  =  ———
                    41

                      4²
cos² θ  =  ————
                  (√41)²

Take square root of both sides:
 
                       4
cos θ  =  ±  ———
                    √41

                       4                                    4
cos θ  =  –  ———     or     cos θ  =  ———          ✔
                     √41                               √41


The sign of  cos θ  depends on which quadrant  θ  lies. Since you first have a positive value for  tan θ, then that means  θ  lies either in the  1st  or the  3rd quadrant.

•  If  θ  is a  1st  quadrant angle, then

cos θ > 0
 
                   4
cos θ  =  ———          ✔
                 √41

•  If  θ  is a  3rd  quadrant angle, then

cos θ < 0
 
                       4
cos θ  =  –  ———          ✔
                     √41


I hope this helps. =)

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