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Question

In each of the following, one of the six trigonometric ratios is given. Find the values of the other trigonometric ratios.

(i) sin A=23
(ii) cos A=45
(iii) tan θ = 11
(iv) sin θ=1115
(v) tan α=512
(vi) sin θ=32
(vii) cos θ=725
(viii) tan θ=815
(ix) cot θ=125
(x) sec θ=135
(xi) cosec θ=10
(xii) cos θ=1215


Solution

(i) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Perpendicular side = 2 and

Hypotenuse = 3

Therefore, by Pythagoras theorem,

Now we substitute the value of perpendicular side (BC) and hypotenuse (AC) and get the base side (AB)

Therefore,

Hence, Base =

Now,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(ii) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 4 and

Hypotenuse = 5

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and hypotenuse (AC) and get the perpendicular side (BC)

Hence, Perpendicular side = 3

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(iii) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 1 and

Perpendicular side = 5

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and the perpendicular side (BC) and get hypotenuse (AC)

Hence, Hypotenuse =

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(iv) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Perpendicular side = 11 and

Hypotenuse = 15

Therefore,

By Pythagoras theorem,

Now we substitute the value of perpendicular side (BC) and hypotenuse(AC) and get the base side (AB)

Hence, Base =

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(v) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 12 and

Perpendicular side = 5

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and the perpendicular side (BC) and get hypotenuse (AC)

Hence, Hypotenuse = 13

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(vi) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Perpendicular side = and

Hypotenuse = 2

Therefore,

By Pythagoras theorem,

Now we substitute the value of perpendicular side (BC) and hypotenuse(AC) and get the base side (AB)

Hence, Base = 1

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(vii) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 7 and

Hypotenuse = 25

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and hypotenuse (AC) and get the perpendicular side (BC)

Hence, Perpendicular side = 24

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(viii) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 15 and

Perpendicular side = 8

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and the perpendicular side (BC) and get hypotenuse (AC)

Hence, Hypotenuse = 17

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(ix) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 12 and

Perpendicular side = 5

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and the perpendicular side (BC) and get hypotenuse (AC)

Hence, Hypotenuse = 13

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(x) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 5 and

Hypotenuse = 13

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and hypotenuse (AC) and get the perpendicular side (BC)


Hence, Perpendicular side = 12

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

(xi) Given:

…… (1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Perpendicular side = 1 and

Hypotenuse =

Therefore,

By Pythagoras theorem,

Now we substitute the value of perpendicular side (BC) and hypotenuse (AC) and get the base side (AB)

Hence, Base side = 3

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

 

(xii) Given: ……(1)

By definition,

…... (2)

By Comparing (1) and (2)

We get,

Base = 12 and

Hypotenuse = 15

Therefore,

By Pythagoras theorem,

Now we substitute the value of base side (AB) and hypotenuse (AC) and get the perpendicular side (BC)

Hence, Perpendicular side = 9

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,

Now,

Therefore,


Mathematics
RD Sharma (2014)
Standard X

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