To find the exact value of the expression sin(x - π/3), we can use the following steps:
Step 1: Calculate the value of cos(Ï€/3) using the unit circle or trigonometric identities. We know that cos(Ï€/3) = 1/2.
Step 2: Use the trigonometric identity sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
Step 3: Substitute the values sin(x) = -5/13, tan(x) > 0, and cos(π/3) = 1/2 into the identity sin(x - π/3) = sin(x)cos(π/3) - cos(x)sin(π/3).
Substituting the values into the expression:
sin(x - π/3) = (-5/13)(1/2) - cos(x)(√3/2)
Step 4: We need to find the value of cos(x). Since tan(x) > 0 and sin(x) = -5/13, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos(x).
Squaring both sides of the equation sin(x) = -5/13 and using the identity, we get:
(-5/13)^2 + cos^2(x) = 1
25/169 + cos^2(x) = 1
cos^2(x) = 1 - 25/169
cos^2(x) = 144/169
cos(x) = ± √(144/169) = ± 12/13
However, we know that tan(x) > 0, so cos(x) must be positive.
Thus, cos(x) = 12/13.
Step 5: Substitute the value of cos(x) back into the expression:
sin(x - π/3) = (-5/13)(1/2) - (12/13)(√3/2)
= -5/26 - 12√3/26
= (-5 - 12√3)/26
Therefore, the exact value of sin(x - π/3) is (-5 - 12√3)/26 or ((12√3) - 5)/26 (same value, just rearranged).