Question

Evaluate the expression under the given conditions. tan(2θ);cos(θ)=

3(frac)5

​
,θ in Quadrant I 1.13 Points] Find sin(2x),cos(2x), and tan(2x) from the given information. sin(x)=−

8(frac)17
​
,x in Quadrant III sin(2x)= cos(2x)= tan(2x)=

Answer 1

To find sin(2x), cos(2x), and tan(2x) when sin(x) = -8/17 and x is in Quadrant III, we use the double angle formulas, taking into account that both sine and cosine are negative in this quadrant.

Step-by-step explanation:

The subject of the question is to evaluate the expressions for sin(2x), cos(2x), and tan(2x) given that sin(x) = -8/17 and x is in Quadrant III. To find these values, we use the double angle formulas for sine, cosine, and tangent:

• sin(2x) = 2sin(x)cos(x)
• cos(2x) = cos^2(x) - sin^2(x) = 2cos^2(x) - 1 = 1 - 2sin^2(x)
• tan(2x) = sin(2x) / cos(2x)

Since sin(x) = -8/17 in Quadrant III and cosine is also negative in this quadrant, we can find cos(x) using the Pythagorean identity: cos(x) = -√(1 - sin^2(x)). Plugging these into the double angle formulas will yield the values for sin(2x), cos(2x), and tan(2x).

Answer 2

`sin(2x)=240/289`

`cos(2x)=161/289`

`tan(2x)=3/2`

For the first problem, its given that
`\(\cos(\theta) = (3)/(5)\)` and
`\(\theta\)` is in Quadrant I. To find
`\(\tan(2\theta)\)`, we'll use the double-angle identity for tangent:

`\[\tan(2\theta) = (2\tan(\theta))/(1 - \tan^2(\theta))\]`

Given that
`\(\cos(\theta) = (3)/(5)\)` and
`\(\theta\)` is in Quadrant I,
`\(\sin(\theta)\)` can be found using the Pythagorean identity:
`\(\sin^2(\theta) + \cos^2(\theta) = 1\)`.

`\[\sin(\theta) = √(1 - \cos^2(\theta)) = \sqrt{1 - \left((3)/(5)\right)^2} = (4)/(5)\]`

Now,
`\(\tan(\theta) = (\sin(\theta))/(\cos(\theta)) = ((4)/(5))/((3)/(5)) = (4)/(3)\)`.

So, using the double-angle formula for tangent:

`\[\tan(2\theta) = (2\tan(\theta))/(1 - \tan^2(\theta)) = (2 \cdot (4)/(3))/(1 - \left((4)/(3)\right)^2) = (8/3)/(1 - 16/9) = (8/3)/(-7/9) = -(24)/(7)\]`

For the second problem, its given
`\(\sin(x) = -(8)/(17)\)` and
`\(x\)` is in Quadrant III. To find
`\(\sin(2x)\)`,
`\(\cos(2x)\)`, and
`\(\tan(2x)\)`, we'll use the double-angle identities.

Given that
`\(\sin(x) = -(8)/(17)\)` and
`\(x\)` is in Quadrant III, we can use the Pythagorean identity to find
`\(\cos(x)\)`:

`\[\cos(x) = -√(1 - \sin^2(x)) = -\sqrt{1 - \left(-(8)/(17)\right)^2} = -\sqrt{1 - (64)/(289)} = -\sqrt{(225)/(289)} = -(15)/(17)\]`

Now, using the double-angle identities:

`\[\sin(2x) = 2\sin(x)\cos(x) = 2 \cdot \left(-(8)/(17)\right) \cdot \left(-(15)/(17)\right) = (240)/(289)\]`

`\[\cos(2x) = \cos^2(x) - \sin^2(x) = \left(-(15)/(17)\right)^2 - \left(-(8)/(17)\right)^2 = (225)/(289) - (64)/(289) = (161)/(289)\]`

`\[\tan(2x) = (\sin(2x))/(\cos(2x)) = ((240)/(289))/((161)/(289)) = (240)/(161) = (120)/(80) = (3)/(2)\]`

The complete question is here:

Evaluate the expression under the given conditions.

`$\tan (2 \theta) ; \cos (\theta)=(3)/(5), \theta$` in Quadrant I

Find
`$\sin (2 x), \cos (2 x)$` and
`$\tan (2 x)$` from the given information.

`$\sin (x)=-(8)/(17), x$` in Quadrant III

`$\sin (2 x)=$`

`$\cos (2 x)=$`

`$\tan (2 x)=$`