Given that sin(x) = \$3/5\$ and cos(y )= \$7/25\$, find the exact value of tan(x + y).

The given values are

sin(x) = \$3/5\$ and cos(y) = \$7/25\$

To find the exact value of tan(x + y) use the angle addition formula for tangent

\$tan(x + y) = (tan(x) + tan(y))/(1 - tan(x)tan(y))\$

Utilize \$sin⁡(x) = 3/5\$ by representing the opposite side as 3k and hypotenuse as 5k, where k is a constant.

Then, use the Pythagorean theorem to find the adjacent side

\$("adjacent")^2 = ("hypotenuse")^2−("opposite")^2\$

\$(5k)^2 - (3k)^2\$

\$25k^2 - 9k^2\$

\$("adjacent")^2 = 16k^2\$

\$"adjacent" = 4k\$

So, for angle x, tan⁡(x)

\$tan(x) = "opposite"/"adjacent"\$

\$tan(x) = (3k)/(4k)\$

\$tan(x) = 3/4\$

cos(y) = \$7/25\$, where cos(y) = \$"adjacent" / "hypotenuse"\$, let’s denote adjacent as 7m and hypotenuse as 25m. where m is a constant.

Using Pythagorean theorem, we can find opposite side in triangles

\$("opposite")^2 = ("hypotenuse")^2−("adjacent")^2\$

\$("opposite")^2 = (25m)^2 − (7m)^2\$

\$("opposite")^2 = 625m^2 - 49m^2\$

\$("opposite")^2 = 576m^2\$

opposite = 24m

So, for angle y, tan⁡(y)

\$tan(y) = "opposite"/"adjacent"\$

\$tan(y) = (24m)/(7m)\$

\$tan(y) = 24/7\$

Apply the values using the tangent angle addition formula now

\$tan(x + y) = (3/4 + (24)/7) / (1 - 3/4 times 24/7)\$

After simplification

\$tan(x + y) = ((21 + 96)/28) / ((1 - (72))/28)\$

\$tan(x + y) = (117/28)/((28 - 72)/28)\$

\$tan(x + y) = 117 / (28) times (- 28) / 44\$

Make it simpler

\$tan(x + y) = - 117/(44)\$

So, the exact value of tan⁡(x + y) is \$− 117/(44)\$.​

Incorrect due to calculation mismatch with correct procedure provided

\$ - 171/44 \$

Calculation doesn’t align with the correct steps; thus, it’s incorrect

\$ - 113/28 \$

This is wrong because the calculation provided doesn’t match the correct steps

\$ - 111/28 \$

Accurate because the calculation aligns with the correct procedure steps

\$ - 117/44 \$

Option

D

Can you summarize what you’ve understood in the above steps?