What Can I Do If sin(x) 4/5 and x is an Acute Angle? Solving for cos(x/2)

In this article, we will explore how to find the value of (cos(x/2)) given that (sin(x) frac{4}{5}) and (x) is a positive acute angle. This problem involves the application of trigonometric identities, particularly the half-angle identity, and the understanding of right-angled triangles.

Understanding the Problem

If (sin(x) frac{4}{5}) and (x) is an acute angle, we can determine the value of (cos(x)) using the Pythagorean identity. Since (sin^2(x) cos^2(x) 1), we have:

(cos^2(x) 1 - sin^2(x) 1 - left(frac{4}{5}right)^2 1 - frac{16}{25} frac{9}{25})

Therefore, (cos(x) frac{3}{5}). The positive acute angle corresponds to the larger angle in a 3-4-5 right triangle, as (3^2 4^2 5^2).

Using the Half-Angle Identity

To find (cos(x/2)), we use the half-angle identity:

(cos(x/2) sqrt{frac{1 cos(x)}{2}})

Substitute the value of (cos(x)) into the formula:

[cos(x/2) sqrt{frac{1 frac{3}{5}}{2}} sqrt{frac{frac{8}{5}}{2}} sqrt{frac{4}{5}} frac{2}{sqrt{5}} frac{2sqrt{5}}{5}]

Thus, the value of (cos(x/2)) is (frac{2sqrt{5}}{5}). We discard the negative value, as (x) is an acute angle, and all trigonometric ratios for acute angles are positive.

Alternative Methods

In a Right-Angled Triangle: In a 3-4-5 triangle, we have:

Sine: (sin(x) frac{4}{5})

Cosine: (cos(x) frac{3}{5})

Using the Double-Angle Identity for Cosine

The double-angle identity for cosine is given by:

(cos(2x) 1 - 2sin^2(x))

Substitute (sin(x) frac{4}{5}) into the identity:

[cos(2x) 1 - 2 left(frac{4}{5}right)^2 1 - 2 cdot frac{16}{25} 1 - frac{32}{25} frac{25}{25} - frac{32}{25} -frac{7}{25}]

However, this result is not necessary for finding (cos(x/2)). The main identity used here is the half-angle identity, which directly gives us the value of (cos(x/2)).

Conclusion

In conclusion, given that (sin(x) frac{4}{5}) and (x) is an acute angle, the value of (cos(x/2)) can be found using the half-angle identity:

[cos(x/2) sqrt{frac{1 cos(x)}{2}})

More accurately, we have:

[cos(x/2) frac{2sqrt{5}}{5})

This accurate approach avoids the confusion and simplifies the calculation by leveraging known trigonometric identities and the properties of right-angled triangles. Understanding and applying these identities is crucial in solving trigonometric problems efficiently.