Find the values of:

Question 1. (a) From the figure (1) given below, find the values of:

(i) sin θ

(ii) cos θ

(iii) tan θ

(iv) cot θ

(v) sec θ

(vi) cosec θ

(b) From the figure (2) given below, find the values of:

(i) sin

(ii) cos A

(iii) sin² A + cos² A

(iv) sec² A – tan² A.

 

Answer :

(a) From right angled triangle OMP,

By Pythagoras theorem,

OP² = OM² +MP²

⇒ MP² = OP² + OM²

⇒ MP² = (15)² – (12)²

⇒ MP² = 225 – 144

⇒ MP² = 81

⇒ MP² = 9²

⇒ MP = 9

(i) sin θ = MP/OP

= 9/15

= 3/5

(ii) cos θ = OM/OP

= 12/15

= 4/5

(iii) tan θ = MP/OP

= 9/12

= ¾

(iv) cot θ = OM/MP

= 12/9

= 4/3

(v) sec θ = OP/OM

= 15/12

= 5/4

(vi) cosec θ = OP/MP

= 15/9

= /3

(b) From right angled triangle ABC,

By Pythagoras theorem,

AB² = AC² + BC²

⇒ AB² = (12)² + (5)²

⇒ AB² = 144 + 25

⇒ AB² = 169

⇒ AB² = 13²

⇒ AB = 13

(i) sin A = BC/AB

= 5/13

(ii) cos A = AC/AB

= 12/13

(iii) sin² A + cos² A = (BC/AB)² + (AC/AB)²

= (5/13)² + (12/13)²

= (25/169) + (144/169)

= (25 + 144)/ 169

= 169/169

= 1

sin² A + cos² A = 1

(iv) sec² A – tan² A = (AB/AC)² – (BC/AC)²

= (13/12)² – (5/12)²

= (169/144) – (25/144)

= (169 – 25)/144

= 144/144

= 1

sec² A – tan² A = 1

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