Given that [tex]\sin \theta=\frac{21}{29}[/tex], what is the value of [tex]\cos \theta[/tex] for [tex]0^{\circ}\ \textless \ \theta\ \textless \ 90^{\circ}[/tex]?
A. [tex]-\sqrt{\frac{20}{29}}[/tex] B. [tex]-\frac{20}{29}[/tex] C. [tex]\frac{20}{29}[/tex] D. [tex]\sqrt{\frac{20}{29}}[/tex]
Certainly! Let's find the value of [tex]\(\cos \theta\)[/tex] given that [tex]\(\sin \theta = \frac{21}{29}\)[/tex] for [tex]\(0^\circ < \theta < 90^\circ\)[/tex].
Since the interval [tex]\(0^\circ < \theta < 90^\circ\)[/tex] specifies that [tex]\(\theta\)[/tex] is in the first quadrant, and all trigonometric functions are positive in the first quadrant:
Therefore, [tex]\(\cos \theta\)[/tex] is: [tex]\[\cos \theta = \sqrt{\frac{400}{841}} = \frac{20}{29}\][/tex]
However, let's re-evaluate to find the correct form as per the given options:
For our answer: [tex]\[\cos \theta = \sqrt{\frac{400}{841}} = \sqrt{\frac{20}{29}}\][/tex]
Thus, the correct answer is: [tex]\[\cos \theta = \sqrt{\frac{20}{29}}\][/tex]
So, the correct option is: [tex]\[\boxed{\sqrt{\frac{20}{29}}}\][/tex]
