Find the exact value of tan (cos^-1 (-5/13))

-¹²/₅

Step-by-step explanation:

Let cos a = - ⁵/₁₃

Then tan a = sin a/cos a

= [√ (1 - cos²a) ]/cos a

= {√[1 - (-⁵/₁₃) ²]} / (-⁵/₁₃)

= - [√ (1 - ²⁵/₁₆₉) ] * ¹³/₅

= - [ (√ (¹⁴⁴/₁₆₉) ] * ¹³/₅

= - ¹²/₁₃ * ¹³/₅

= - ¹²/₅

tan (arccos (-⁵/₁₃)) = - ¹²/₅

The diagram shows the 2nd quadrant of a circle with radius 13, cos a = - ⁵/₁₃, and tan a = ¹²/₍₋₅₎ = - ¹²/₅.