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Math  /  Trigonometry

QuestionQuestion If cos⁡y=56\cos y=\frac{5}{6}, then what is the positive value of tan⁡12y\tan \frac{1}{2} y, in simplest radical form with a rational denominator?

Studdy Solution

STEP 1

1. We are given cos⁡y=56\cos y = \frac{5}{6}.
2. We need to find the positive value of tan⁡12y\tan \frac{1}{2} y.
3. We will use trigonometric identities to find tan⁡12y\tan \frac{1}{2} y.

STEP 2

1. Use the identity for sin⁡2y+cos⁡2y=1\sin^2 y + \cos^2 y = 1 to find sin⁡y\sin y.
2. Use the half-angle identity for tangent to find tan⁡12y\tan \frac{1}{2} y.
3. Simplify the expression to get the answer in simplest radical form with a rational denominator.

STEP 3

Use the Pythagorean identity sin⁡2y+cos⁡2y=1\sin^2 y + \cos^2 y = 1 to find sin⁡y\sin y.
Given cos⁡y=56\cos y = \frac{5}{6}, we have:
sin⁡2y+(56)2=1\sin^2 y + \left(\frac{5}{6}\right)^2 = 1
sin⁡2y+2536=1\sin^2 y + \frac{25}{36} = 1
Subtract 2536\frac{25}{36} from both sides:
sin⁡2y=1−2536\sin^2 y = 1 - \frac{25}{36}
Convert 1 to 3636\frac{36}{36} to have a common denominator:
sin⁡2y=3636−2536\sin^2 y = \frac{36}{36} - \frac{25}{36}
sin⁡2y=1136\sin^2 y = \frac{11}{36}
Take the square root of both sides to find sin⁡y\sin y:
sin⁡y=±1136\sin y = \pm \sqrt{\frac{11}{36}}
sin⁡y=±116\sin y = \pm \frac{\sqrt{11}}{6}
Assume sin⁡y\sin y is positive, then:
sin⁡y=116\sin y = \frac{\sqrt{11}}{6}

STEP 4

Use the half-angle identity for tangent:
tan⁡12y=±1−cos⁡y1+cos⁡y\tan \frac{1}{2} y = \pm \sqrt{\frac{1 - \cos y}{1 + \cos y}}
Substitute cos⁡y=56\cos y = \frac{5}{6}:
tan⁡12y=±1−561+56\tan \frac{1}{2} y = \pm \sqrt{\frac{1 - \frac{5}{6}}{1 + \frac{5}{6}}}
Simplify the expression:
tan⁡12y=±16116\tan \frac{1}{2} y = \pm \sqrt{\frac{\frac{1}{6}}{\frac{11}{6}}}
tan⁡12y=±111\tan \frac{1}{2} y = \pm \sqrt{\frac{1}{11}}
tan⁡12y=±111\tan \frac{1}{2} y = \pm \frac{1}{\sqrt{11}}
Rationalize the denominator:
tan⁡12y=±111×1111\tan \frac{1}{2} y = \pm \frac{1}{\sqrt{11}} \times \frac{\sqrt{11}}{\sqrt{11}}
tan⁡12y=±1111\tan \frac{1}{2} y = \pm \frac{\sqrt{11}}{11}
Since we are looking for the positive value:
tan⁡12y=1111\tan \frac{1}{2} y = \frac{\sqrt{11}}{11}
The positive value of tan⁡12y\tan \frac{1}{2} y is:
1111\boxed{\frac{\sqrt{11}}{11}}

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