1

Problem

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

2

Step

The given values are

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

3

Formula

To find the exact value of tan(x + y) use the angle addition formula for tangent is \$tan(x + y) = (tan(x) + tan(y)) / (1- tan(x) tan(y))\$.

4

Hint

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

5

Formula

Then, use the Pythagorean theorem to find the adjacent side: \$("adjacent")^2 = ("hypotenuse")^2 - ("opposite")^2\$

6

Step

Plug the values in the formula:

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

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

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

\$"adjacent" = 4k\$

7

Step

So, for angle x, tan⁡(x):

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

\$(3k)/(4k)\$

\$3/4\$

8

Hint

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

9

Step

Using Pythagorean theorem, we can find opposite side in triangles:

\$(25m)^2 - (7m)^2\$

\$625 m^2 - 49 m^2 \$

\$576 m^2\$

opposite = 24m

10

Step

So, for angle y, tan⁡(y):

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

\$(24m)/(7m)\$

\$(24)/7\$

11

Step

Apply the values using the tangent angle addition formula now:

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

12

Step

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\$

13

Step

Make it simpler:

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

14

Solution

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

15

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

16

Choice-A

Incorrect due to calculation mismatch with correct procedure provided

Wrong \$ - (171)/(44) \$

17

Choice-B

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

Wrong \$ - (113)/(28) \$

18

Choice-C

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

Wrong \$ - (111)/(28) \$

19

Choice-D

Accurate because the calculation aligns with the correct procedure steps

Correct \$ - (117)/(44) \$

20

Answer

Option

D

21

Sumup

Please summarize choices