# Hello,  I do not know how to solve for the smallest positive solutions. Can anyone please help? Relevance

sin(3x)cos(8x) - cos(3x)sin(8x) = -0.85

remember

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

so the 15 equation is equivalent to

sin(3x-8x) = sin(-5x) = -0.85

sin(-5x) =-sin(5x) = -0.85

sin(5x) = 0.85

asin(0.85)  =  1.015985294

plus since

sin(x) = sin(pi-x)

asin(0.85) gvies   1.015985294    or

2.12560736

Remember sine and cosine

have a period of 2pi

5x = 1.015985294 + 2pik   where k is an integer

x = (1.015985294/5) + (2/5)pik

x=     0.203197059    + (2/5)*pik

since (2/5)pi is greater than 0.2032

k = 0 is the smallest

x =  0.203197059

or

5x =  2.12560736   + 2pik

x = (2.12560736/5) + (2/5)pik

x=  0.425121472 + (2/5)*pik

since (2/5)pi is greater than 0.42512

x= 0.425121472

so

x = 0.203197059

is the smallest solution

• Thank you sooooo much Alvin!

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• By Trig. Identity

Sin( 3 x - 8x) =>

Sin (-5x) = -0.85

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