Final answer:
To find trigonometric functions from the given pair (7/25, -24/25), which represents cosine and sine, we use trigonometric identities and the Pythagorean theorem to determine the tangent, secant, cosecant, and cotangent of the angle.
Step-by-step explanation:
The student's question is asking to find the trigonometric functions for a given ordered pair (7/25, -24/25), which represents the cosine and sine of an angle respectively. In the context of a right-angled triangle, cosine corresponds to the adjacent side over the hypotenuse, and sine corresponds to the opposite side over the hypotenuse.
We can deduce that the hypotenuse is 1, since the sine and cosine values given are components of a unit circle.
If θ is the angle in question, then we can state:
- θ's cosine value cos(θ) is 7/25
- θ's sine value sin(θ) is -24/25
Using the pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find the other trigonometric functions:
- The tangent of θ, tan(θ), is sin(θ)/cos(θ) which simplifies to (-24/25)/(7/25) = -24/7.
- The secant of θ, sec(θ), which is 1/cos(θ), is hence 25/7.
- The cosecant of θ, csc(θ), which is 1/sin(θ), is -25/24.
- And the cotangent of θ, cot(θ), which is 1/tan(θ), is -7/24.