angle a is acute and sin = u. find the other five trigonometric functions in the terms of u.

To find the other five trigonometric functions (cosine, tangent, cotangent, secant, and cosecant) in terms of u, we need to use the identity relationships between the trigonometric functions and utilize the given information.

Given:
Angle a is acute.
sin(a) = u

We start by using the Pythagorean identity for sine and cosine:

sin^2(a) + cos^2(a) = 1

Plugging in sin(a) = u into the equation, we get:

u^2 + cos^2(a) = 1

Rearranging the equation gives us:

cos^2(a) = 1 - u^2

Taking the square root of both sides:

cos(a) = ±√(1 - u^2)

We know that angle a is acute, so it lies in the first quadrant where cosine is positive. Therefore, we can take the positive square root:

cos(a) = √(1 - u^2)

Using cosine, we can now find the other trigonometric functions:

1. Tangent (tan(a)):
tan(a) = sin(a) / cos(a)
Since we know sin(a) = u and cos(a) = √(1 - u^2):
tan(a) = u / √(1 - u^2)

2. Cotangent (cot(a)):
cot(a) = 1 / tan(a)
cot(a) = √(1 - u^2) / u

3. Secant (sec(a)):
sec(a) = 1 / cos(a)
sec(a) = 1 / √(1 - u^2)

4. Cosecant (csc(a)):
csc(a) = 1 / sin(a)
csc(a) = 1 / u

Note: It's important to also consider the restrictions, such as the denominator cannot be zero or undefined values, depending on the given value of u.