Question

Evaluate the expression under the given conditions. sin(2theta);sin(theta) = 1/7, theta in Quadrant II

Answer 1

To evaluate the expression sin(2theta), given sin(theta) = 1/7 and theta in Quadrant II, use the double-angle identity sin(2theta) = 2sin(theta)cos(theta) and substitute the given value of sin(theta) into the equation.

Step-by-step explanation:

To evaluate the expression sin(2theta) given that sin(theta) = 1/7 and theta is in Quadrant II:

1. Since sin(theta) = 1/7, we can assign a value of 1/7 to the opposite side and 1 to the hypotenuse in a right triangle in Quadrant II.
2. Using the Pythagorean theorem, we can find the adjacent side of the triangle by taking the square root of (1 - (1/7)^2).
3. Now, we can evaluate sin(2theta) by applying the double-angle identity, which states that sin(2theta) = 2sin(theta)cos(theta).
4. Substituting the values we have, sin(2theta) = 2 * (1/7) * sqrt(48/49).
5. Simplifying further, sin(2theta) = 2/7 * sqrt(48/49).

Answer 2

The value of
`\( \sin(2\theta) \) when \( \sin(\theta) = (1)/(7) \) and \( \theta \) is in Quadrant II is \( -(48)/(49) \).`

Step-by-step explanation:

In Quadrant II, the sine function is positive. Given that
`\( \sin(\theta) = (1)/(7) \),` we can use this information to find the cosine of
`\( \theta \)` using the Pythagorean identity
`\( \sin^2(\theta) + \cos^2(\theta) = 1 \). Squaring \( (1)/(7) \)` and subtracting from 1, we get
`\( \cos^2(\theta) = (48)/(49) \). Taking the square root, we find \( \cos(\theta) = (√(48))/(7) = (4√(3))/(7) \).`

Now, to find
`\( \sin(2\theta) \),` we use the double angle identity
`\( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).` Substituting the values, we get
`\( \sin(2\theta) = 2 * (1)/(7) * (4√(3))/(7) = (8√(3))/(49) \).`

However, since
`\( \theta \)` is in Quadrant II,
`\( \sin(2\theta) \)` is negative. Therefore, the final answer is
`\( -(8√(3))/(49) \)` or equivalently
`\( -(48)/(49) \).` This represents the value of
`\( \sin(2\theta) \)` under the given conditions in Quadrant II.