Math  /  Trigonometry

QuestionQuestion If cosy=56\cos y=\frac{5}{6}, then what is the positive value of tan12y\tan \frac{1}{2} y, in simplest radical form with a rational denominator?

Studdy Solution
Use the half-angle identity for tangent:
tan12y=±1cosy1+cosy\tan \frac{1}{2} y = \pm \sqrt{\frac{1 - \cos y}{1 + \cos y}}
Substitute cosy=56\cos y = \frac{5}{6}:
tan12y=±1561+56\tan \frac{1}{2} y = \pm \sqrt{\frac{1 - \frac{5}{6}}{1 + \frac{5}{6}}}
Simplify the expression:
tan12y=±16116\tan \frac{1}{2} y = \pm \sqrt{\frac{\frac{1}{6}}{\frac{11}{6}}}
tan12y=±111\tan \frac{1}{2} y = \pm \sqrt{\frac{1}{11}}
tan12y=±111\tan \frac{1}{2} y = \pm \frac{1}{\sqrt{11}}
Rationalize the denominator:
tan12y=±111×1111\tan \frac{1}{2} y = \pm \frac{1}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}}
tan12y=±1111\tan \frac{1}{2} y = \pm \frac{\sqrt{11}}{11}
Since we are looking for the positive value:
tan12y=1111\tan \frac{1}{2} y = \frac{\sqrt{11}}{11}
The positive value of tan12y\tan \frac{1}{2} y is:
1111\boxed{\frac{\sqrt{11}}{11}}

View Full Solution - Free
Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord