Use the given conditions to find the exact value of the expression.

Question

sin α = − 5/13,   tan α > 0,  sin(α − π/3)

GIVE EXACT VALUE NO DECIMAL

Answer 1

sine is negative and tangent is positive in quadrant 3

With respect to angle A, opposite is -5, hypotenuse is 13.

Pythagorean says adjacent is -12.

Tangent is 5/12

sin ( A - pi/3) = siin (A + -pi/3) = sinA *cos (-pi/3) + cos(A) *sin (-pi/3)

= sinA * cos (300) + cos(A) * sin(300)

= sinA * (1/2) + cos(A) * -sqrt(3)/2

= (-5/13)(1/2) + (-12/13)(-sqrt(3))/2

= -5/26 + 12*Sqrt(3)/26

= [ 12*Sqrt(3) - 5]/26

Answer 2

Remember sin(a-b)=sinacosb-cosasinb and sin(π/3)=√(3)/2 and cos(π/3) = 1/2

Since sina = -5/13 and we are in quadrant III, y = -5, x = -12, and y = 13 So cosa = -12/13

Sin(a-π/3) = sina(cosπ/3) - cosa(sinπ/3)

Just plug in the numbers, and that's it!