KnowledgeBoat Logo
|

Mathematics

Write all the other trigonometric ratios of ∠A in terms of sec A.

Trigonometric Identities

6 Likes

Answer

We know that,

⇒ cos A = 1sec A\dfrac{1}{\text{sec A}} ………(1)

By formula,

⇒ sin2 A + cos2 A = 1

⇒ sin2 A = 1 - cos2 A

⇒ sin2 A = 1 - 1sec2A\dfrac{1}{\text{sec}^2 A}

⇒ sin2 A = sec2A−1sec2A\dfrac{\text{sec}^2 A - 1}{\text{sec}^2 A}

⇒ sin A = sec2A−1sec A\dfrac{\sqrt{\text{sec}^2 A - 1}}{\text{sec A}}

By formula,

⇒ sec2 A - tan2 A = 1

⇒ tan2 A = sec2 A - 1

⇒ tan A = sec2A−1\sqrt{\text{sec}^2 A - 1}

By formula,

⇒ cot A = 1tan A=1sec2A−1\dfrac{1}{\text{tan A}} = \dfrac{1}{\sqrt{\text{sec}^2 A - 1}}

By formula,

⇒ cosec A = 1sin A=1sec2A−1sec A=sec Asec2A−1\dfrac{1}{\text{sin A}} = \dfrac{1}{\dfrac{\sqrt{\text{sec}^2 A - 1}}{\text{sec A}}} = \dfrac{\text{sec A}}{\sqrt{\text{sec}^2 A - 1}}.

Hence,

sin A=sec2A−1sec A,cos A=1sec A,tan A=sec2A−1, cot A=1sec2A−1,cosec A=sec Asec2A−1\text{sin A} = \dfrac{\sqrt{\text{sec}^2 A - 1}}{\text{sec A}}, \text{cos A} = \dfrac{1}{\text{sec A}}, \text{tan A} = \sqrt{\text{sec}^2 A - 1}, \text{ cot A} = \dfrac{1}{\sqrt{\text{sec}^2 A - 1}}, \text{cosec A} = \dfrac{\text{sec A}}{\sqrt{\text{sec}^2 A - 1}}

Answered By

4 Likes

Related Questions