Q. 86 Find the exact value of:tan60掳+... [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 6: Q. 86 (page 381)

Find the exact value of: $\tan 60 掳 + \tan 150 掳$

Short Answer

Expert verified

The value of $\tan 60 掳 + \tan 150 掳 = \frac{2 \sqrt{3}}{3}$

Step by step solution

Step 1. Given information

The given value $= \tan 60 掳 + \tan 150 掳$

Step 2. Find the exact value of the given function by using unit circle

The unit circle

Step 3. Finding the value of tan60°+tan150°

Recall that the unit circle gives

(x, y) = (\cos 伪, \sin 伪)

for all the common angles:
Consider

\tan 60 掳

Let

伪 = 60 掳

From the above unit circle,
The angle

伪 = 60 掳

intersects the unit circle at

(\frac{1}{2}, \frac{\sqrt{3}}{2})

as shown in the above figure.
The pair of coordinates corresponding to

60 掳

(\frac{1}{2}, \frac{\sqrt{3}}{2})

,
Here the first coordinate gives

\cos 60 掳

and second coordinate gives

\sin 60 掳

.
Therefore,

\cos 60 掳 = \frac{1}{2}, \sin 60 掳 = \frac{\sqrt{3}}{2}

since $\tan 胃 = \frac{\sin 胃}{\cos 胃}$

Replace $胃$ by $60 掳$ then

$\tan 60 掳 = \frac{\sin 60 掳}{\cos 60 掳} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} [\sin c e \cos 60 掳 = \frac{1}{2}, \sin 60 掳 = \frac{\sqrt{3}}{2}] = \frac{\sqrt{3}}{1} = \sqrt{3}$

Therefore,the value of $\tan 60 掳 i s \sqrt{3}$

Consider $\tan 150 掳$
Let $伪 = 150 掳$
From the above unit circle,
The angle $伪 = 150 掳$ intersects the unit circle at $(- \frac{\sqrt{3}}{2}, \frac{1}{2})$ as shown in the above figure.
The pair of coordinates corresponding to $150 掳$ is $(- \frac{\sqrt{3}}{2}, \frac{1}{2})$ ,
Here the first coordinate gives $\cos 150 掳$ and second coordinate gives $\sin 150 掳$ .
Therefore,

$\cos 150 掳 = - \frac{\sqrt{3}}{2}, \sin 150 掳 = \frac{1}{2}$

since $\tan 胃 = \frac{\sin 胃}{\cos 胃}$

Replace $胃$ by $150 掳$ then

$\tan 150 掳 = \frac{\sin 150 掳}{\cos 150 掳} = \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}} [\sin c e \cos 150 掳 = - \frac{\sqrt{3}}{2}, \sin 150 掳 = \frac{1}{2}] = \frac{- 1}{\sqrt{3}}$

Therefore,the value of $\tan 150 掳 i s \frac{- 1}{\sqrt{3}}$

Finally,add these two exact functional values to obtain the required solution

Therefore,

$\tan 60 掳 + \tan 150 掳 = \sqrt{3} + \frac{- 1}{\sqrt{3}} = \sqrt{3} + (\frac{- 1}{\sqrt{3}}) = \frac{3 \sqrt{3} - \sqrt{3}}{3} = \frac{2 \sqrt{3}}{3}$

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91影视

Find the exact value of: $\tan 60 掳 + \tan 150 掳$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the exact value of the given function by using unit circle

Step 3. Finding the value of tan60°+tan150°

One App. One Place for Learning.

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91影视

Find the exact value of:tan60掳+tan150掳

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the exact value of the given function by using unit circle

Step 3. Finding the value of tan60°+tan150°

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Decision Maths

Statistics

Logic and Functions

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Find the exact value of: $\tan 60 掳 + \tan 150 掳$