Question

Use the double-angle formulas to find the exact value of sin2u given that tanu=(3)/(5) and 0.

Answer 1

To find the exact value of sin2u with tanu = 3/5, we first calculate sinu and cosu using a right triangle, and then apply the double-angle formula sin2u = 2sinucosu to get the result sin2u = 15/17.

Step-by-step explanation:

To find the exact value of sin2u given that tanu =
`\((3)/(5)\)`and u is in the first quadrant, we must first recognize that we can relate tanu to sinu and cosu. Since tanu is the ratio of the opposite side to the adjacent side in a right-angled triangle, we can consider a triangle where the opposite side (O) is 3 and the adjacent side (A) is 5. By the Pythagorean theorem, the hypotenuse (H) will therefore be
`\(√(3^2 + 5^2) = √(34)\)`.

sinu is O/H, so sinu =
`\((3)/(√(34))\)` and cosu is A/H, so cosu =
`\((5)/(√(34))\)`. Now, by using the double-angle formula for sine, sin2u = 2sinucosu, we can compute the exact value: sin2u = 2
`\((5)/(√(34))\)`, which simplifies to sin2u =
`\((15)/(17)\)`.