FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT

Question 1 :

Find the values of other five trigonometric functions for the following:

sin θ = -2/3, θ lies in the IV quadrant.

Solution :

sin θ  =  Opposite side/hypotenuse side

sin θ  =  -2/3

Opposite side  =  2, Hypotenuse side  =  3

(Hypotenuse side)2  =  (Opposite side)2 + (Adjacent side)2

Adjacent side  =  √(32 - 22)

  =  √(9 - 4)

  =  √5

Adjacent side  =  √5

Note : Since θ  lies in 4th quadrant, all trigonometric ratios other than cos and sec will have negative sign.

cos θ  =  Adjacent side/hypotenuse side  √5/3

tan θ  =  Opposite side/Adjacent side  =  -2/√5

cosec θ  =  Hypotenuse side/Opposite side  =  -3/2

sec θ  =  Hypotenuse side/Adjacent side  3/√5

cot θ  =  Adjacent side/Opposite side  =  -√5/2

Question 2 :

Find the values of other five trigonometric functions for the following:

tan θ = -2, θ lies in the II quadrant.

Solution :

tan θ  =  Opposite side/Adjacent side

tan θ  =  -2/1

Opposite side  =  2, Adjacent side  =  1

(Hypotenuse side)2  =  (Opposite side)2 + (Adjacent side)2

Hypotenuse side  =  √(22 + 12)

  =  √(4 + 1)

  =  √5

Hypotenuse side  =  √5

Note : Since θ  lies in 2nd quadrant, all trigonometric ratios other than sin and cosec will have negative sign.

sin θ  =  Opposite side/hypotenuse side   2/√5

cos θ  =  Adjacent side/hypotenuse side   -1/√5

cosec θ  =  Hypotenuse side/Opposite side  =  √5/2

sec θ  =  Hypotenuse side/Adjacent side  =  -√5/1

cot θ  =  Adjacent side/Opposite side  -1/2

Question 3 :

Find the values of other five trigonometric functions for the following:

sec θ = 13/5, θ lies in the IV quadrant

Solution :

sec θ  =  Hypotenuse side/Hypotenuse side 

sec θ = 13/5

Adjacent side  =  5, Hypotenuse side  =  13

(Hypotenuse side)2  =  (Opposite side)2 + (Adjacent side)2

Opposite side  =  √(132 - 52)

  =  √(169 - 25)

  =  √144

Opposite side  =  12

Note : Since θ  lies in 4th quadrant, all trigonometric ratios other than tan and cot will have negative sign.

sin θ  =  Opposite side/hypotenuse side   -12/13

cos θ  =  Adjacent side/hypotenuse side   5/13

tan θ  =  Opposite side/Adjacent side   12/5

cosec θ  =  Hypotenuse side/Opposite side  =  13/12

cot θ  =  Adjacent side/Opposite side  5/12

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