The sine of a certain acute angle is 1 4 ; find the numerical value of the other trigonometric ratios of this angle.

Let θ be the given acute angle and sin θ = 1 4 . We know that, sin 2 θ + cos 2 θ = 1 ⇒ cos θ = 1 - sin 2 θ ⇒ cos θ = 1 - 1 4 2 So, cos θ = 15 16 = 15 4 ∵ tan θ = sin θ cos θ ∴ tan θ = 1 4 15 4 ⇒ tan θ = 1 15 We also know, cot θ = 1 tan θ ⇒ cot θ = 15 Similarly, sec θ = 1 cos θ = 4 15 And cosec θ = 1 sin θ = 4 .

If sin θ = 12 13 , where θ is an acute angle, find tan θ and versin θ .

Given, sin θ = 12 13 and θ is an acute angle. We know, sin 2 θ + cos 2 θ = 1 ⇒ cos 2 θ = 1 - sin 2 θ ⇒ cos θ = 1 - sin 2 θ ⇒ cos θ = 1 - 12 13 2 ⇒ cos θ = 25 169 ⇒ cos θ = 5 13 We know, tan θ = sin θ cos θ ⇒ tan θ = 12 13 5 13 ⇒ tan θ = 12 5 We also know, versin θ = 1 - cos θ ⇒ versin θ = 1 - 5 13 ⇒ versin θ = 8 13

If sin A = 11 61 , find tan A , cos A and sec A , where A is an acute angle.

Given, sin A = 11 61 and A is an acute angle. We know, sin 2 A + cos 2 A = 1 ⇒ cos 2 A = 1 - sin 2 A ⇒ cos A = 1 - sin 2 A ⇒ cos A = 1 - 11 61 2 ⇒ cos A = 3600 3721 ⇒ cos A = 60 61 We know, tan A = sin A cos A ⇒ tan A = 11 61 60 61 ⇒ tan A = 11 60 We also know, sec A = 1 cos A ⇒ sec A = 1 60 61 ⇒ sec A = 61 60

If cos θ = 4 5 , find sin θ and cot θ , where θ is an acute angle.

Given, cos θ = 4 5 and θ is an acute angle. We know, sin 2 θ + cos 2 θ = 1 ⇒ sin 2 θ = 1 - cos 2 θ ⇒ sin θ = 1 - cos 2 θ ⇒ sin θ = 1 - 4 5 2 ⇒ sin θ = 9 25 ⇒ sin θ = 3 5 We know, cot θ = cos θ sin θ ⇒ cot θ = 4 5 3 5 ⇒ cot θ = 4 3

If cos A = 9 41 , find tan A and cosec A , where A is an acute angle.

Given, cos A = 9 41 and A is an acute angle. We know, sin 2 A + cos 2 A = 1 ⇒ sin 2 A = 1 - cos 2 A ⇒ sin A = 1 - cos 2 A ⇒ sin A = 1 - 9 41 2 ⇒ sin A = 1600 1681 ⇒ sin A = 40 41 We also know, tan A = sin A cos A ⇒ tan A = 40 41 9 41 ⇒ tan A = 40 9 We know, cosec A = 1 sin A ⇒ cosec A = 1 40 41 ⇒ cosec A = 41 40

If tan θ = 1 7 , find the value of cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ

Given, tan θ = 1 7 We know, sec 2 θ - tan 2 θ = 1 ⇒ sec 2 θ = 1 + 1 7 2 ⇒ sec 2 θ = 8 7 . . . i We know, cot 2 θ = 1 tan 2 θ ⇒ cot 2 θ = 1 1 7 ⇒ cot 2 θ = 7 We also know, cosec 2 θ - cot 2 θ = 1 Now, cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ = 1 + cot 2 θ - sec 2 θ 1 + cot 2 θ + sec 2 θ ⇒ cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ = 1 + 7 - 8 7 1 + 7 + 8 7 ⇒ cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ = 48 7 64 7 ⇒ cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ = 48 64 ⇒ cosec 2 θ - sec 2 θ cosec 2 θ + sec 2 θ = 3 4 .