Question

The value of \( \tan \left( \cos^{-1} \left( \frac{-24}{25} \right) \right) \) is equal to

• A. \( \frac{7}{24} \)
• B. \( \frac{-7}{24} \)
• C. \( \frac{-7}{25} \)
• D. \( \frac{-24}{7} \)
• E. \( \frac{24}{7} \)

Hint:

To evaluate \( \tan(\cos^{-1} x) \), use a right triangle and apply Pythagoras' theorem.

Answer 1

The Correct Option is B

Given:

We are asked to find the value of:

tan(cos^-1(-24/25))

Let θ = cos^-1(-24/25), so cos(θ) = -24/25.

We need to find tan(θ), which is given by:

tan(θ) = sin(θ) / cos(θ)

Using the Pythagorean identity:

sin²(θ) + cos²(θ) = 1

Substitute cos(θ) = -24/25:

sin²(θ) + (-24/25)² = 1

sin²(θ) + 576/625 = 1

sin²(θ) = 1 - 576/625 = 625/625 - 576/625 = 49/625

sin(θ) = 7/25

Now, calculate tan(θ):

tan(θ) = sin(θ) / cos(θ) = (7/25) / (-24/25) = 7 / -24 = -7/24

Thus, the value of tan(cos^-1(-24/25)) is -7/24.