Question

express in the product form 1+2 cos 20°​

Answer 1

Given:

Given expression is 1 + 2cos 20°​.

To find:

We need to find the product form of given expression.

Solution:

The given expression is 1 + 2cos 20°​.

We should find the product form.

To find the product form, first we should simplify the given expression.

$\Rightarrow 1 + 2cos 20^\circ$

$\Rightarrow 1 + 2(cos 20^\circ)$

The value $cos20^\circ=0.93969262078$.

So, substitute this value in above expression.

$\Rightarrow 1 + 2(0.93969262078)$

Multiplying the terms,

$\Rightarrow 1 + 1.87938524157$

Adding the terms,

$\Rightarrow 2.87938524157$

$\Rightarrow 2.9$ (Approximately)

If we multiply any number with 1, it results the same number.

$\Rightarrow 2.9\times1$

$\Rightarrow 1\times2.9$

Therefore, the product form of given expression is $1\times2.9$.

Answer 2

SOLUTION

TO EXPRESS

In the product form 1 + 2 cos 20°

FORMULA TO BE IMPLEMENTED

We are aware of the Trigonometric formula that

$\displaystyle \sf{ \cos \alpha + \cos \beta = 2 \cos \bigg( \frac{ \alpha + \beta }{2} \bigg)\cos \bigg( \frac{ \alpha - \beta }{2} \bigg)}$

EVALUATION

$\displaystyle \sf{ 1 + 2 \cos {20}^{ \circ} }$

$\displaystyle \sf{ = 2 \bigg( \frac{1}{2} + \cos {20}^{ \circ} \bigg) }$

$\displaystyle \sf{ = 2 \bigg( \cos {60}^{ \circ} + \cos {20}^{ \circ} \bigg) }$

$\displaystyle \sf{ =2 \times 2 \times \cos \bigg( \frac{ {60}^{ \circ} + {20}^{ \circ} }{2} \bigg)\cos \bigg( \frac{ {60}^{ \circ} - {20}^{ \circ} }{2} \bigg)}$

$\displaystyle \sf{ =2 \times 2 \times \cos \bigg( \frac{ {80}^{ \circ} }{2} \bigg)\cos \bigg( \frac{ {40}^{ \circ} }{2} \bigg)}$

$\displaystyle \sf{ = 4 \cos {40}^{ \circ} \cos {20}^{ \circ} }$

FINAL ANSWER

$\displaystyle \sf{ 1 + 2 \cos {20}^{ \circ} = 4 \cos {40}^{ \circ} \cos {20}^{ \circ}}$

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