exact values of all the solutions in the interval 0<or equal to x<2pi for sin(x+pi/4)+sin(x-pi/4)= -1?

Question

Please please help

Answer 1

Use the sum/product identity. sin(a) + sin(b) = 2 sin((a+b)/2)cos((a-b)/2).
So sin(x+pi/4)+sin(x-pi/4) = 2sin(x)cos(pi/4) = -1. So 2sin(x)*sqrt(2)/2 = -1, so sin(x) = -1/sqrt(2).
That gives x = 5pi/4 or 7pi/4.

Answer 2

answer: submit to in thoughts the values of sin x, while x lies between [0, 2pi]: Sin 0 = 0, Sin pi/6 = a million/2, Sin pi/4 = a million/sqrt(2) Sin pi/3 = [sqrt(3)]/2 Sin pi/2 = a million you may instruct those values by utilising utilising a calculator or by utilising an equilateral triangle or an isosceles triangle. Sin x = 0 The sine curve cuts the x-axis at pi in between 0 and 2 pi. Sin x = a million/2 all of us be attentive to Sin pi/6 = a million/2 lower back, sin(pi - pi/6) = sin (pi/6) = a million/2 sin(5pi/6) = a million/2

Answer 3

sin(x + (pi/4)) + sin(x - (pi/4)) = -1
(sin(x)cos(pi/4) + sin(pi/4)cos(x)) + (sin(x)cos(pi/4) - sin(pi/4)cos(x)) = -1

sin(x)cos(pi/4) + sin(pi/4)cos(x) + sin(x)cos(pi/4) - sin(pi/4)cos(x) = -1

sin(x)cos(pi/4) + sin(x)cos(pi/4) = -1
2sin(x)cos(pi/4) = -1
sin(x)cos(pi/4) = (-1/2)
sin(x) * (sqrt(2)/2) = (-1/2)
sin(x) = (-1/2)/(sqrt(2)/2)
sin(x) = (-1/2)*(2/sqrt(2))
sin(x) = -2/(2sqrt(2))
sin(x) = (-1/sqrt(2))
sin(x) = (-sqrt(2)/2)

x = (-pi/4) = (7pi/4) or (-3pi/4) = (5pi/4)

It depends on if you want this to be negative or positive.

By the way, both (-pi/4) or 315° and (-3pi/4) or 225° will both give you a (-sqrt(2)/2).

If you go to www.quickmath.com, click on Solve under Equations, and type in sin(x + (pi/4)) + sin(x - (pi/4)) = -1, it will give you (-pi/4) as the answer.