Answer:
[tex]\displaystyle \cos(\theta) = \frac{\sqrt{7}}{4}[/tex].
[tex]\displaystyle \tan(\theta) = \frac{3\sqrt{7}}{7}[/tex].
Step-by-step explanation:
By the Pythagorean identity, for any given angle [tex]\theta[/tex], [tex]\sin^{2}(\theta) + \cos^{2}(\theta) = 1[/tex].
Given that [tex]\displaystyle \sin(\theta) = 3/4[/tex], solve this equation for [tex]\cos(\theta)[/tex].
[tex](\sin(\theta))^{2} + (\cos(\theta))^{2} = 1[/tex].
[tex]\displaystyle \left(\frac{3}{4}\right)^{2} + (\cos(\theta))^{2} = 1[/tex].
[tex]\begin{aligned} \cos(\theta) &= \sqrt{1 - \frac{9}{16}} \\ &= \sqrt{\frac{7}{16}} \\ &= \frac{\sqrt{7}}{4} \end{aligned}[/tex].
The tangent of an angle is equal to the ratio between the sine and cosine of that angle. In this question:
[tex]\begin{aligned} \tan(\theta) &= \frac{\sin(\theta)}{\cos(\theta)} \\ &= \frac{3/4}{\sqrt{7}/4} \\ &= \frac{3}{\sqrt{7}} \\ &= \frac{3\sqrt{7}}{7}\end{aligned}[/tex].
