Write expression for the other five trigonometric ratios for each acute angle:
A) cscĮ=a/b
B) secƒÀ= x+1/x-1, x�‚1
To find the expressions for the other five trigonometric ratios for each acute angle, we can use the given expressions for csc(A) and sec(A) and apply the reciprocal relationships between the trigonometric ratios.
A) csc(A) = a/b
The reciprocal of sine is cosecant:
cosec(A) = 1/sin(A) = 1/(a/b) = b/a
The reciprocal of cosine is secant:
sec(A) = 1/cos(A)
B) sec(A) = (x+1)/(x-1), for x ≠ 1
The reciprocal of secant is cosine:
cos(A) = 1/sec(A) = 1/((x+1)/(x-1)) = (x-1)/(x+1)
The other five trigonometric ratios for each acute angle are:
1) sine: sin(A) = √(1 - cos²(A)) = √(1 - ((x-1)/(x+1))²) = √((x+1)² - (x-1)²) / (x+1) = √(4x) / (x+1)
2) cosine: cos(A) = (x-1)/(x+1)
3) tangent: tan(A) = sin(A)/cos(A) = (√(4x) / (x+1)) / ((x-1)/(x+1)) = √(4x) / (x-1)
4) cosecant: csc(A) = 1/(sin(A)) = 1/((√(4x) / (x+1))) = (x+1) / √(4x)
5) cotangent: cot(A) = 1/(tan(A)) = 1/((√(4x) / (x-1))) = (x-1) / √(4x)
Note: These expressions are specific to the given equation for sec(A). If the equation changes, the expressions for the other five trigonometric ratios will be different.