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Math  /  Trigonometry

QuestionIf sin⁡(θ)=−57\sin (\theta)=-\frac{5}{7} and θ\theta is in the 4 th quadrant, find th If the value doesn't exist, type DNE. cos⁡(θ)=□csc⁡(θ)=□sec⁡(θ)=□tan⁡(θ)=□cot⁡(θ)=□\begin{array}{l} \cos (\theta)=\square \\ \csc (\theta)=\square \\ \sec (\theta)=\square \\ \tan (\theta)=\square \\ \cot (\theta)=\square \end{array}

Studdy Solution

STEP 1

1. θ\theta is in the 4th quadrant.
2. sin⁡(θ)=−57\sin(\theta) = -\frac{5}{7}.
3. The trigonometric identities and Pythagorean identity will be used.
4. The trigonometric functions cos⁡(θ)\cos(\theta), csc⁡(θ)\csc(\theta), sec⁡(θ)\sec(\theta), tan⁡(θ)\tan(\theta), and cot⁡(θ)\cot(\theta) will be evaluated.

STEP 2

1. Find cos⁡(θ)\cos(\theta) using the identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1.
2. Determine csc⁡(θ)\csc(\theta) using the definition csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}.
3. Determine sec⁡(θ)\sec(\theta) using the definition sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}.
4. Determine tan⁡(θ)\tan(\theta) using the definition tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.
5. Determine cot⁡(θ)\cot(\theta) using the definition cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}.

STEP 3

Use the identity sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 to find cos⁡(θ)\cos(\theta).
sin⁡2(θ)+cos⁡2(θ)=1 \sin^2(\theta) + \cos^2(\theta) = 1 (−57)2+cos⁡2(θ)=1 \left(-\frac{5}{7}\right)^2 + \cos^2(\theta) = 1 2549+cos⁡2(θ)=1 \frac{25}{49} + \cos^2(\theta) = 1 cos⁡2(θ)=1−2549 \cos^2(\theta) = 1 - \frac{25}{49} cos⁡2(θ)=4949−2549 \cos^2(\theta) = \frac{49}{49} - \frac{25}{49} cos⁡2(θ)=2449 \cos^2(\theta) = \frac{24}{49} cos⁡(θ)=±2449 \cos(\theta) = \pm \sqrt{\frac{24}{49}} cos⁡(θ)=±247 \cos(\theta) = \pm \frac{\sqrt{24}}{7}

STEP 4

Because θ\theta is in the 4th quadrant, cos⁡(θ)\cos(\theta) is positive.
cos⁡(θ)=247 \cos(\theta) = \frac{\sqrt{24}}{7} cos⁡(θ)=267 \cos(\theta) = \frac{2\sqrt{6}}{7}

STEP 5

Determine csc⁡(θ)\csc(\theta) using the definition csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}.
csc⁡(θ)=1sin⁡(θ) \csc(\theta) = \frac{1}{\sin(\theta)} csc⁡(θ)=1−57 \csc(\theta) = \frac{1}{-\frac{5}{7}} csc⁡(θ)=−75 \csc(\theta) = -\frac{7}{5}

STEP 6

Determine sec⁡(θ)\sec(\theta) using the definition sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}.
sec⁡(θ)=1cos⁡(θ) \sec(\theta) = \frac{1}{\cos(\theta)} sec⁡(θ)=1267 \sec(\theta) = \frac{1}{\frac{2\sqrt{6}}{7}} sec⁡(θ)=726 \sec(\theta) = \frac{7}{2\sqrt{6}} Rationalize the denominator: sec⁡(θ)=726⋅66 \sec(\theta) = \frac{7}{2\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} sec⁡(θ)=7612 \sec(\theta) = \frac{7\sqrt{6}}{12}

STEP 7

Determine tan⁡(θ)\tan(\theta) using the definition tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}.
tan⁡(θ)=sin⁡(θ)cos⁡(θ) \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} tan⁡(θ)=−57267 \tan(\theta) = \frac{-\frac{5}{7}}{\frac{2\sqrt{6}}{7}} tan⁡(θ)=−526 \tan(\theta) = \frac{-5}{2\sqrt{6}} Rationalize the denominator: tan⁡(θ)=−526⋅66 \tan(\theta) = \frac{-5}{2\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} tan⁡(θ)=−5612 \tan(\theta) = \frac{-5\sqrt{6}}{12}

STEP 8

Determine cot⁡(θ)\cot(\theta) using the definition cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}.
cot⁡(θ)=1tan⁡(θ) \cot(\theta) = \frac{1}{\tan(\theta)} cot⁡(θ)=1−5612 \cot(\theta) = \frac{1}{\frac{-5\sqrt{6}}{12}} cot⁡(θ)=12−56 \cot(\theta) = \frac{12}{-5\sqrt{6}} Rationalize the denominator: cot⁡(θ)=12−56⋅66 \cot(\theta) = \frac{12}{-5\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} cot⁡(θ)=126−30 \cot(\theta) = \frac{12\sqrt{6}}{-30} cot⁡(θ)=−265 \cot(\theta) = -\frac{2\sqrt{6}}{5}
Final values: cos⁡(θ)=267 \cos(\theta)=\frac{2\sqrt{6}}{7} csc⁡(θ)=−75 \csc(\theta)=-\frac{7}{5} sec⁡(θ)=7612 \sec(\theta)=\frac{7\sqrt{6}}{12} tan⁡(θ)=−5612 \tan(\theta)=-\frac{5\sqrt{6}}{12} cot⁡(θ)=−265 \cot(\theta)=-\frac{2\sqrt{6}}{5}

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