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

The given values are

sin(α) = \$ 3/5 \$, cos(β) = \$ 4/5\$

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

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) ) \$

Plug the values in the hint

\$ cos(α) = \sqrt ( 1 - (3/5)^2 ) \$

\$ sin(β) = \sqrt ( 1 - (4/5)^2 ) \$

After simplification

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

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

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 \$

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

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

\$ tan(α+β) = ( 6/4 )/( 1 - 9/16 ) \$

After simplification

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

\$ tan(α+β) = ( 3/2 )/( 7/16 ) \$

To simplify, multiply the numerator and denominator by 16:

\$ tan(α+β) = 24/7 \$

Therefore, \$ tan(α+β) = 24/7 \$.

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\$

\$ 18/13 \$

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

\$ 7/12 \$

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\$​

\$ 13/9 \$

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

\$ 24/7 \$

Option

D

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