Since \( x \) is in the first quadrant, \( \cos(x) = \frac{4}{5} \).

["Understanding Why ( \cos(x) = \frac{4}{5} ) When ( x ) Is in the First Quadrant", "When working with trigonometric functions, particularly in the context of right triangles or the unit circle, understanding the signs and values of sine, cosine, and tangent based on quadrant placement is crucial. One common scenario encountered in trigonometry is determining the exact value of cosine when an angle ( x ) lies in the first quadrant and ( \cos(x) = \frac{4}{5} ).", "Why the First Quadrant Matters", "The first quadrant spans angles ( 0^\circ < x < 90^\circ ) (or ( 0 < x < \frac{\pi}{2} ) radians), where all trigonometric functions—sine, cosine, and tangent—are positive. This positive value property directly affects how we interpret the cosine value of ( \frac{4}{5} ) when ( x ) is in this region.", "Interpreting ( \cos(x) = \frac{4}{5} ) Geometrically", "Consider a right triangle located in the first quadrant with angle ( x ). In the unit circle or standard triangle representation:", "- The cosine of an angle represents the ratio of the adjacent side over the hypotenuse:\n [\n \cos(x) = \frac{\ ext{adjacent side}}{\ ext{hypotenuse}} = \frac{4}{5}\n ]\n This tells us that if the hypotenuse is 5 units, the adjacent side (along the x-axis) measures 4 units.", "- Using the Pythagorean theorem, the opposite side (along the y-axis) is:\n [\n \ ext{opposite} = \sqrt{(\ ext{hypotenuse})^2 - (\ ext{adjacent})^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3\n ]\n Thus, ( \sin(x) = \frac{3}{5} ) and ( \ an(x) = \frac{3}{4} ).", "Why the Value is Positive", "Since ( x ) is in the first quadrant, all ratios are positive, confirming ( \cos(x) = \frac{4}{5} ) fits perfectly within standard trigonometric identities. This value not only maintains consistency with right triangle definitions but also aligns with the Pythagorean identity:\n[\n\cos^2(x) + \sin^2(x) = \left(\frac{4}{5}\right)^2 + \left(\frac{3}{5}\right)^2 = \frac{16}{25} + \frac{9}{25} = 1\n]\nThis identity holds true, reinforcing the correctness of the value.", "Applications of ( \cos(x) = \frac{4}{5} )", "This cosine value commonly appears in:", "- Physics problems involving right-angle motion or forces\n- Engineering calculations requiring precise angular components\n- Graphing trigonometric functions and solving triangles\n- Computer graphics and simulations where first-quadrant angles are standard", "Conclusion", "When ( x ) is in the first quadrant, ( \cos(x) = \frac{4}{5} ) reflects a consistent, positive value grounded in geometric definitions and validated by Pythagorean identities. Recognizing this relationship enhances understanding of trigonometric ratios, supports accurate problem-solving, and strengthens foundational knowledge for more complex mathematics.", "---", "Keywords for SEO: ( \cos(x) = \frac{4}{5} ), first quadrant trigonometry, cosine in right triangles, trigonometric identities, unit circle, Pythagorean theorem, trigonometric ratios, geometry applications, mathematical reasoning."]