Given csc⁡(t)=[−127] and [(−π2)&lt;t&lt;(π2)] . Find sin⁡ t, cos⁡ t and tan⁡ t. Give exact answers without decimals.

To determine the value of sin⁡ t, cos⁡ t and tan⁡ t. Given: csc⁡(t)=[−127]and[(−π2)&lt;t&lt;(π2)] csc⁡(t)=[−127] Using trigonometric identity, sin⁡t=1csc⁡t sin⁡t=1−127 sin⁡t=−712 cos2t=1−sin2t cos⁡t=±1−(−712)2 cos⁡t=±1−49144 cos⁡t=±95144 Since, −π2≤t≤π2 ⇒cos⁡t=9512 tan⁡t=sin⁡tcos⁡t tan⁡t=(−712)(9512) tan⁡t=−795 Thus, sin⁡t=−712, cos⁡t=9512, tan⁡t=−795

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Marvin Mccormick

Answered question

2021-02-06

Given $\csc (t) = [\frac{- 12}{7}] and [(- \frac{π}{2}) < t < (\frac{π}{2})]$ .
Find $\sin t, \cos t and \tan t .$ Give exact answers without decimals.

Answer & Explanation

Malena

Skilled2021-02-07Added 83 answers

To determine the value of $\sin t, \cos t and \tan t .$
Given:
$\csc (t) = [\frac{- 12}{7}] and [(- \frac{π}{2}) < t < (\frac{π}{2})]$
$\csc (t) = [\frac{- 12}{7}]$
Using trigonometric identity, $\sin t = \frac{1}{\csc t}$
$\sin t = \frac{1}{\frac{- 12}{7}}$
$\sin t = - \frac{7}{12}$
$\cos^{2} t = 1 - \sin^{2} t$
$\cos t = \pm \sqrt{1 - {(- \frac{7}{12})}^{2}}$
$\cos t = \pm \sqrt{1 - \frac{49}{144}}$
$\cos t = \pm \sqrt{\frac{95}{144}}$
Since, $- \frac{π}{2} \leq t \leq \frac{π}{2}$
$\Rightarrow \cos t = \frac{\sqrt{95}}{12}$
$\tan t = \frac{\sin t}{\cos t}$
$\tan t = \frac{(- \frac{7}{12})}{(\frac{\sqrt{95}}{12})}$
$\tan t = - \frac{7}{\sqrt{95}}$
Thus,
$\sin t = - \frac{7}{12},$
$\cos t = \frac{\sqrt{95}}{12},$
$\tan t = - \frac{7}{\sqrt{95}}$

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