Trigonometry Ratios of Angle A in terms of secA

Trigonometry ratios are the ratios of the sides of a right triangle with respect to its acute angles. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). In this question, we are required to write all the trigonometry ratios of angle A in terms of secA.

Sine (sin) of angle A in terms of secA
Sine is defined as the ratio of the opposite side to the hypotenuse. Using the Pythagorean theorem, we can find the adjacent side in terms of the hypotenuse and secant. Therefore,

sinA = opposite/hypotenuse = (1/secA)/√(secA^2 - 1)

Cosine (cos) of angle A in terms of secA
Cosine is defined as the ratio of the adjacent side to the hypotenuse. Using the Pythagorean theorem, we can find the opposite side in terms of the hypotenuse and secant. Therefore,

cosA = adjacent/hypotenuse = √(secA^2 - 1)/secA

Tangent (tan) of angle A in terms of secA
Tangent is defined as the ratio of the opposite side to the adjacent side. Using the definitions of sine and cosine, we can simplify the expression for tangent. Therefore,

tanA = opposite/adjacent = (1/secA)/√(secA^2 - 1) * secA/√(secA^2 - 1) = 1/√(secA^2 - 1)

Cosecant (csc) of angle A in terms of secA
Cosecant is defined as the reciprocal of sine. Therefore,

cscA = 1/sinA = √(secA^2 - 1)/(1/secA)

Secant (sec) of angle A in terms of secA
Secant is defined as the reciprocal of cosine. Therefore,

secA = 1/cosA = secA/√(secA^2 - 1)

Cotangent (cot) of angle A in terms of secA
Cotangent is defined as the reciprocal of tangent. Therefore,

cotA = 1/tanA = √(secA^2 - 1)