Answer:
Given: [tex]\sin(x) = (4/5)[/tex].
Assuming that [tex]0 < x < 90^{\circ}[/tex], [tex]\cos(x) = (3/5)[/tex] while [tex]\tan(x) = (4/3)[/tex].
Step-by-step explanation:
By the Pythagorean identity [tex]\sin^{2}(x) + \cos^{2}(x) = 1[/tex].
Assuming that [tex]0 < x < 90^{\circ}[/tex], [tex]0 < \cos(x) < 1[/tex].
Rearrange the Pythagorean identity to find an expression for [tex]\cos(x)[/tex].
[tex]\cos^{2}(x) = 1 - \sin^{2}(x)[/tex].
Given that [tex]0 < \cos(x) < 1[/tex]:
[tex]\begin{aligned} &\cos(x) \\ &= \sqrt{1 - \sin^{2}(x)} \\ &= \sqrt{1 - \left(\frac{4}{5}\right)^{2}} \\ &= \sqrt{1 - \frac{16}{25}} \\ &= \frac{3}{5}\end{aligned}[/tex].
Hence, [tex]\tan(x)[/tex] would be:
[tex]\begin{aligned}& \tan(x) \\ &= \frac{\sin(x)}{\cos(x)} \\ &= \frac{(4/5)}{(3/5)} \\ &= \frac{4}{3}\end{aligned}[/tex].
