How to Find the Value of (tan(arcsin(frac{24}{25})))

When dealing with trigonometric functions and their inverses, one common question is to find the value of (tan(arcsin(frac{24}{25}))). This article will carefully walk through the steps to solve this problem using both a right triangle approach and the Pythagorean identity.

Using Trigonometric Identities

To start, we let (theta arcsin(frac{24}{25})). This implies that (sin(theta) frac{24}{25}).

Using the Pythagorean identity, we know that (sin^2(theta) cos^2(theta) 1). Substitute (sin(theta) frac{24}{25}) into the identity:

[left(frac{24}{25}right)^2 cos^2(theta) 1]

Simplify the equation:

[frac{576}{625} cos^2(theta) 1]

Solve for (cos^2(theta)):

[cos^2(theta) 1 - frac{576}{625} frac{625 - 576}{625} frac{49}{625}]

Taking the positive root, since (theta) is in the range ([-frac{pi}{2}, frac{pi}{2}]) where cosine is positive:

[cos(theta) sqrt{frac{49}{625}} frac{7}{25}]

Using a Right Triangle Approach

Consider a right triangle with angle (B arcsinleft(frac{24}{25}right)). Since (sin B frac{24}{25}), the side opposite to (B) is 24, and the hypotenuse is 25.

Using the Pythagorean theorem to find the length of the adjacent side:

[a^2 24^2 25^2]

Calculate the length of the adjacent side:

[a sqrt{25^2 - 24^2} sqrt{625 - 576} sqrt{49} 7]

Now, (tan B frac{text{opposite}}{text{adjacent}} frac{24}{7}).

Therefore, (tan(arcsin(frac{24}{25})) frac{24}{7}).

Generalizing the Approach

This method of drawing a triangle can be applied whenever you need to find a trig function composed with an inverse trig function, provided all the quantities are positive. Here is a general formula:

Given (tan(arcsin x) frac{x}{sqrt{1 - x^2}}).

For instance, in the case of (sin(theta) frac{24}{25}), you can directly substitute into the formula:

[tan(arcsin frac{24}{25}) frac{frac{24}{25}}{sqrt{1 - left(frac{24}{25}right)^2}} frac{frac{24}{25}}{sqrt{1 - frac{576}{625}}} frac{frac{24}{25}}{sqrt{frac{49}{625}}} frac{24}{25} cdot frac{25}{7} frac{24}{7}]

Conclusion

In conclusion, the value of (tan(arcsin(frac{24}{25}))) is (frac{24}{7}). This result can be derived using both the Pythagorean identity and the right triangle approach. Understanding these methods is crucial for solving a variety of trigonometric problems.