Answer:
[tex]\cos\left(\theta\right)=\frac{\sqrt{15}}{4}[/tex]
Step-by-step explanation:
Given that [tex]\sin\left(-\theta\right)=-\frac{1}{4}[/tex] and [tex]\tan\left(\theta\right)=\frac{\sqrt{15}}{15}[/tex]
Using those values we need to find value of [tex]\cos\left(\theta\right)[/tex]
So let's begin with equation of [tex]\tan\left(\theta\right)=\frac{\sqrt{15}}{15}[/tex] and simplify it to get value of [tex]\cos\left(\theta\right)[/tex]
[tex]\tan\left(\theta\right)=\frac{\sqrt{15}}{15}[/tex]
[tex]\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}=\frac{\sqrt{15}}{15}[/tex]
[tex]\sin\left(\theta\right)=\frac{\sqrt{15}}{15}\cos\left(\theta\right)[/tex]
[tex]\sin\left(\theta\right)\cdot\frac{15}{\sqrt{15}}=\cos\left(\theta\right)[/tex]
[tex]\cos\left(\theta\right)=\sin\left(\theta\right)\cdot\frac{15}{\sqrt{15}}[/tex]
[tex]\cos\left(\theta\right)=-\sin\left(-\theta\right)\cdot\frac{15}{\sqrt{15}}[/tex] {since [tex]\sin\left(-\theta\right)=-\sin\left(\theta\right)[/tex] }
[tex]\cos\left(\theta\right)=-\sin\left(-\theta\right)\cdot\sqrt{15}[/tex]
[tex]\cos\left(\theta\right)=-\left(-\frac{1}{4}\right)\cdot\sqrt{15}[/tex]
[tex]\cos\left(\theta\right)=\frac{\sqrt{15}}{4}[/tex]
Answer:
[tex]\cos \theta = \frac{\sqrt{15}}{4}[/tex]
Step-by-step explanation:
The sine function is an odd function, that is:
[tex]\sin (-\theta) = - \sin \theta[/tex]
Then,
[tex]\sin \theta = \frac{1}{4}[/tex]
The cosine function is computed by the following relation:
[tex]\cos \theta = \frac{\sin \theta}{\tan \theta}[/tex]
[tex]\cos \theta = \frac{\frac{1}{4} }{\frac{\sqrt{15}}{15} }[/tex]
[tex]\cos \theta = \frac{\frac{1}{4} }{\frac{1}{\sqrt{15}} }[/tex]
[tex]\cos \theta = \frac{\sqrt{15}}{4}[/tex]
