Question

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

Answer 1

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
`\( \cos \theta \).`