Sin = 5/13, and cos b = 3/5, evaluate cos (a-b).

cos (a - b) is 56/65

Step-by-step explanation:

Step 1:

Given sin a = 5/13, find cos a.

sin a = opposite side/hypotenuse = 5/13

The adjacent side can be found using Pythagoras Theorem.

Hypotenuse² = Opposite Side² + Adjacent Side²

⇒ Adjacent side² = Hypotenuse² - Opposite Side²

= 13² - 5² = 169 - 25 = 144

∴ Adjacent Side = 12

⇒ cos a = adjacent side/hypotenuse = 12/13

Step 2:

Given cos b = 3/5, find sin b.

cos b = adjacent side/hypotenuse = 3/5

The opposite side can be found using Pythagoras Theorem.

Hypotenuse² = Opposite Side² + Adjacent Side²

⇒ Opposite side² = Hypotenuse² - Adjacent Side²

= 5² - 3² = 25 - 9 = 16

∴ Opposite Side = 4

⇒ sin b = opposite side/hypotenuse = 4/5

Step 3:

Find cos (a - b).

cos (a - b) = cos a cos b + sin a sin b

= 12/13 * 3/5 + 5/13 * 4/5

= 36/65 + 20/65 = 56/65