1

Problem

If sin(α) = \$ 3/5\$ and cos(β) = \$ 4/5\$, where α and β are acute angles, find tan(α+β).

2

Step

The given values are

sin(α) = \$ 3/5 \$, cos(β) = \$ 4/5\$

3

Formula

\$ tan(α+β) = (tan(α) + tan(β) )/( 1 - tan(α) + tan(β) ) \$.

4

Step

We can find cos(α) and sin(β) using the Pythagorean identity:

\$ cos(α) = \sqrt ( 1 - sin^2(α) ) \$

\$ sin(β) = \sqrt ( 1 - cos^2(β) ) \$

5

Step

Plug the values in the step:

\$ cos(α) = \sqrt ( 1 - (3/5)^2 ) \$

\$ sin(β) = \sqrt ( 1 - (4/5)^2 ) \$

6

Step

After simplification:

\$ cos(α) = \sqrt ( 1 - 9/(25) ) = 4/5 \$

\$ sin(β) = \sqrt ( 1 - (16)/(25) ) = 3/5 \$

7

Step

Now, find tan(α) and tan(β):

\$ tan(α) = sin(α) / cos(α) = (3/5) / (4/5) = 3/4 \$

\$ tan(β) = sin(β) / cos(β) = (3/5) / (4/5) = 3/4 \$

8

Step

Now, substitute these values into the formula for tan(α+β):

\$ tan(α+β) = (3/4 + 3/4 )/( 1 - (3/4 \times 3/4) ) \$

\$ tan(α+β) = ( 6/4 )/( 1 - 9/(16) ) \$

9

Step

After simplification

\$ tan(α+β) = ( 3/2 )/( (16 - 9) /(16) ) \$

\$ tan(α+β) = ( 3/2 )/( 7/16 ) \$

10

Step

To simplify, multiply the numerator and denominator by 16:

\$ tan(α+β) = (24)/7 \$

11

Solution

Therefore, \$ tan(α+β) = (24)/7\$.

12

Sumup

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

Choices

13

Choice-A

Option A is not correct because the calculation for tan⁡(α+β) using the given values and the tangent addition formula does not result in \$(18)/(13)\$

Wrong \$ (18)/(13) \$

14

Choice-B

Which is \$7/(12)\$​, is not correct because it does not match the derived value from the tangent addition formula

Wrong \$ 7/(12) \$

15

Choice-C

Option C states that tan(α+β) = \$(13)/9\$​, but our detailed calculations above clearly show that the correct value is \$(24)/7\$​, not \$(13)/9\$​

Wrong \$ (13)/9 \$

16

Choice-D

This is correct because it does match the correct result for tan(α+β) by using the formula

Correct \$ (24)/7 \$

17

Answer

Option

D

18

Sumup

Please summarize choices