Question #3a2df

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Question #3a2df

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Answer 1 · 2016-12-04T17:28:30

Please see the explanation.

Explanation:

Given: $cos (x) = \frac{\sqrt{3}}{2}$ $cos(x) = sqrt(3)/2$

Use the identity $sin (x) = \pm \sqrt{1 - {cos}^{2} (x)}$ $sin(x) = +-sqrt(1 - cos^2(x))$ :

Because we are given that $tan (x)$ $tan(x)$ is a positive number, we shall drop the $\pm$ $+-$ , thereby, making the sine positive, only:

$sin (x) = \sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}$ $sin(x) = sqrt(1 - (sqrt(3)/2)^2)$

$sin (x) = \sqrt{\frac{4}{4} - \frac{3}{4}}$ $sin(x) = sqrt(4/4 - 3/4)$

$sin (x) = \frac{1}{2}$ $sin(x) = 1/2$

Verify that $tan (x) = \frac{\sqrt{3}}{3}$ $tan(x) = sqrt(3)/3$ :

$\frac{sin (x)}{cos (x)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} = tan (x)$ $sin(x)/cos(x) = (1/2)/(sqrt(3)/2) = 1/sqrt(3) = sqrt(3)/3 = tan(x)$

Verified.

Use the identity $cot (x) = \frac{1}{tan (x)}$ $cot(x) = 1/tan(x)$ :

= 1/(sqrt(3)/3)#

$cot (x) = \sqrt{3}$ $cot(x) = sqrt(3)$

Use the identity $csc (x) = \frac{1}{sin (x)}$ $csc(x) = 1/sin(x)$

$csc (x) = \frac{1}{\frac{1}{2}}$ $csc(x) = 1/(1/2)$

$csc (x) = 2$ $csc(x) = 2$

Use the identity $sec (x) = \frac{1}{cos (x)}$ $sec(x) = 1/cos(x)$

$sec (x) = \frac{1}{\frac{\sqrt{3}}{2}}$ $sec(x) = 1/(sqrt(3)/2)$

$sec (x) = \frac{2}{\sqrt{3}}$ $sec(x) = 2/sqrt(3)$

$sec (x) = \frac{2 \sqrt{3}}{3}$ $sec(x) = (2sqrt(3))/3$

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Question #3a2df

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