Answer:
±80/289
Explanation:
If you want to learn how to calculate 2tanθ/(1 + tan²θ), you need to follow these steps. But be warned, this is not a piece of cake. It's more like a piece of pi.
First, you need to find the value of sinθ by dividing both sides of the equation by 17:
Next, you need to find the value of cosθ by using the Pythagorean identity: sin²θ + cos²θ = 1. You can do this by rearranging the equation and taking the square root of both sides:
- cos²θ = 1 - sin²θ
- cos²θ = 1 - (8/17)²
- cos²θ = 1 - 64/289
- cos²θ = 225/289
- cosθ = ±√(225/289)
- cosθ = ±15/17
Then, you need to find the value of tanθ by using the ratio identity: tanθ = sinθ/cosθ. You can do this by plugging in the values of sinθ and cosθ and simplifying:
- tanθ = sinθ/cosθ
- tanθ = (8/17) / (±15/17)
- tanθ = ±8/15
Finally, you need to find the value of 2tanθ/(1 + tan²θ) by plugging in the value of tanθ and simplifying:
- 2tanθ/(1 + tan²θ) = 2(±8/15) / (1 + (±8/15)²)
- 2tanθ/(1 + tan²theta) = ±16/15 / (1 + 64/225)
- 2tanθ/(1 + tan²theta) = ±16/15 / (225/225 + 64/225)
- 2tanθ/(1 + tan²theta) = ±16/15 / (289/225)
- 2tantheta/(1 + tan²theta) = ±(16/15) x (225/289)
- 2tantheta/(1 + tan²theta) = ±80/289
So, the value of 2tantheta/(1 + tan²theta) is ±80/289. Note that there are two possible values because cosθ and tanθ can be positive or negative depending on the quadrant of θ.
Congratulations! You have just solved a trigonometric equation. You deserve a round of applause. Or maybe just a round pi.