Q. 46 The exact value of the expressio... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 46 (page 504)

The exact value of the expression.
$\tan 105 掳$

Short Answer

Expert verified

The exact value of $\tan 105 掳 = \frac{- \sqrt{3} \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}}}{\sqrt{2 + \sqrt{3} - \sqrt{3} \sqrt{2 - \sqrt{3}}}}$

Step by step solution

Step 1. Given information

The given expression is $\tan 105 掳$ $\tan 105^{鈭�}$

Step 2. Find the values of tan120∘ and tan15∘

Since, $105 掳$ is in the first quadrant so it will be positive.

We know that,

$\tan 15 掳 = \tan \frac{30 掳}{2} = = \sqrt{\frac{1 - \cos 30 掳}{1 + \cos 30 掳}} = \sqrt{\frac{1 - (\frac{\sqrt{3}}{2})}{1 + \frac{\sqrt{3}}{2}}} = \sqrt{\frac{2 - \sqrt{3}}{2 + \sqrt{3}}}$

Step 3. Find the value of sin105°

We know that,

$\tan (伪 - 尾) = \frac{\tan 伪 - \tan 尾}{1 + \tan 伪 \tan 尾}$

Substituting $伪 = 120 掳尾 = 15 掳$ , we get:

role="math" localid="1646589258220" $\tan (120 - 15 掳) = \tan 105 掳 \tan (120 掳 - 15 掳) = \frac{\tan 120 掳 - \tan 15 掳}{1 + \tan 120 掳 \tan 15 掳} = \frac{- \sqrt{3} - \sqrt{\frac{2 - \sqrt{3}}{2 + \sqrt{3}}}}{1 - \sqrt{3} \sqrt{\frac{2 - \sqrt{3}}{2 + \sqrt{3}}}} \tan 105 掳 = \frac{- \sqrt{3} \sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}}}{\sqrt{2 + \sqrt{3} - \sqrt{3} \sqrt{2 - \sqrt{3}}}}$

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91影视

The exact value of the expression.
$\tan 105 掳$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the values of tan120∘ and tan15∘

Step 3. Find the value of sin105°

One App. One Place for Learning.

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91影视

The exact value of the expression.tan105掳

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the values of tan120∘ and tan15∘

Step 3. Find the value of sin105°

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Statistics

Geometry

Study anywhere. Anytime. Across all devices.

The exact value of the expression.
$\tan 105 掳$