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Use the given information to determine the exact trigonometric value. sin0 = -1/5, pi<0<3pi/2, cos0

User Tohasanali
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1 Answer

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For
\( \sin \theta = -(1)/(5) \) and
\( \pi < \theta < (3\pi)/(2) \),
\( \cos \theta = -(2√(6))/(5) \), derived using the Pythagorean identity and considering the third quadrant.

Given:


\[ \sin \theta = -(1)/(5) \] and


\[ \pi < \theta < (3\pi)/(2) \]

Using the Pythagorean identity:


\[ \cos^2 \theta = 1 - \sin^2 \theta \]


\[ \cos^2 \theta = 1 - \left(-(1)/(5)\right)^2 \]


\[ \cos^2 \theta = 1 - (1)/(25) \]


\[ \cos^2 \theta = (24)/(25) \]

Taking the square root of both sides (as \( \theta \) lies in the third quadrant where cosine is negative):


\[ \cos \theta = -\sqrt{(24)/(25)} \]


\[ \cos \theta = -(√(24))/(√(25)) \]


\[ \cos \theta = -(2√(6))/(5) \]

Therefore, when
\( \sin \theta = -(1)/(5) \) and
\( \pi < \theta < (3\pi)/(2) \),
\( \cos \theta = -(2√(6))/(5) \).

complete the question

To determine
\( \cos \theta \) given
\( \sin \theta = -(1)/(5) \) and
\( \pi < \theta < (3\pi)/(2) \), the Pythagorean identity for trigonometric functions and
\( \sin \theta \)to solve for
\( \cos \theta \).

User Alf Moh
by
8.9k points
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