# Exact Value of tan 11¼°

How to find the exact value of tan 11¼° using the value of cos 45°?

Solution:

For all values of the angle A we know that, 2 sin$$^{2}$$ $$\frac{A}{2}$$ = 1 - cos A

Again, for all values of the angle A we know that, 2 sin $$\frac{A}{2}$$ cos $$\frac{A}{2}$$ = sin A

Now tan 11¼°

= $$\frac{sin 11¼°}{cos 11¼°}$$

= $$\frac{sin 11¼°}{cos 11¼°}$$ × $$\frac{2 sin 11¼°}{2 sin 11¼°}$$

= $$\frac{2 sin^{2} 11¼°}{2 sin 11¼° cos 11¼°}$$

= $$\frac{1 - cos 22½°}{sin 22½°}$$

= $$\frac{1 - \sqrt{\frac{1 + cos 45°}{2}}}{\sqrt{\frac{1 - cos 45°}{2}}}$$

= $$\frac{\sqrt{2} - \sqrt{1 + cos 45°}}{\sqrt{1 - cos 45°}}$$

= $$\frac{\sqrt{2} - \sqrt{1 + \frac{1}{\sqrt{2}}}}{\sqrt{1 - \frac{1}{\sqrt{2}}}}$$

= $$\frac{\sqrt{2} - \sqrt{\frac{\sqrt{2} + 1}{\sqrt{2}}}}{\sqrt{\frac{\sqrt{2} - 1}{\sqrt{2}}}}$$

= $$\frac{\sqrt{2\sqrt{2}} - \sqrt{\sqrt{2} + 1}}{\sqrt{\sqrt{2} - 1}}$$

= $$\frac{\sqrt{2\sqrt{2}} - \sqrt{\sqrt{2} + 1}}{\sqrt{\sqrt{2} - 1}}$$ × $$\frac{\sqrt{\sqrt{2} + 1}}{\sqrt{\sqrt{2} + 1}}$$

= $$\frac{\sqrt{2\sqrt{2}}\cdot \sqrt{\sqrt{2} + 1} - \sqrt{(\sqrt{2} + 1)^{2}}}{\sqrt{(\sqrt{2} + 1)(\sqrt{2} - 1)}}$$

= $$\frac{\sqrt{2\sqrt{2}{(\sqrt{2} + 1})}-(\sqrt{2} + 1)}{{\sqrt{2 - 1}}}$$

= $$\sqrt{4 + 2\sqrt{2}} - (\sqrt{2} + 1)$$

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

## Recent Articles

1. ### Successor and Predecessor | Successor of a Whole Number | Predecessor

May 24, 24 06:42 PM

The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

2. ### Counting Natural Numbers | Definition of Natural Numbers | Counting

May 24, 24 06:23 PM

Natural numbers are all the numbers from 1 onwards, i.e., 1, 2, 3, 4, 5, …... and are used for counting. We know since our childhood we are using numbers 1, 2, 3, 4, 5, 6, ………..

3. ### Whole Numbers | Definition of Whole Numbers | Smallest Whole Number

May 24, 24 06:22 PM

The whole numbers are the counting numbers including 0. We have seen that the numbers 1, 2, 3, 4, 5, 6……. etc. are natural numbers. These natural numbers along with the number zero

4. ### Math Questions Answers | Solved Math Questions and Answers | Free Math

May 24, 24 05:37 PM

In math questions answers each questions are solved with explanation. The questions are based from different topics. Care has been taken to solve the questions in such a way that students