Write an expression that gives all solutions to the equation
sin x + cos x = - radical 2
Square both sides. The answer we get for sin x + cos x may have the wrong sign, so we will have to chaeck that later.
sin^2 x + cos^2x + 2 sinx cosx = 2
1 + sin(2x) = 2
sin (2x) = 1
2x = pi/2 + 2 n pi where n is any integer
x = pi/4 + n pi
= pi/4, 5 pi/4
= 45 , 225 degrees
The first answer does not work because the sign is wrong, so the only answer between 0 and 360 degrees is 225 degees
To find the solutions to the equation sin(x) + cos(x) = -√2, we can use the following steps:
Step 1: Combine the terms on the left-hand side of the equation.
sin(x) + cos(x) = -√2
Step 2: Simplify the equation by using a trigonometric identity.
We can use the identity sin(x) = cos(π/2 - x) to rewrite the equation as:
cos(π/2 - x) + cos(x) = -√2
Step 3: Combine the cosine terms using the cosine addition formula.
cos(π/2 - x) + cos(x) = -√2
Using the cosine addition formula, we have:
cos(π/2 - x)cos(x) - sin(π/2 - x)sin(x) = -√2
Step 4: Simplify the equation further.
Rearranging the terms, we get:
cos(π/2 - x)cos(x) + sin(π/2 - x)sin(x) = -√2
Using the identities cos(π/2 - x) = sin(x) and sin(π/2 - x) = cos(x), we have:
sin(x)cos(x) + cos(x)sin(x) = -√2
2sin(x)cos(x) = -√2
Step 5: Divide both sides of the equation by 2.
sin(x)cos(x) = -√2/2
Multiplying both sides by √2, we get:
√2sin(x)cos(x) = -1
Step 6: Use the double angle formula for sine.
The double angle formula for sine states that sin(2x) = 2sin(x)cos(x). Applying this formula to our equation, we have:
√2sin(2x) = -1
Step 7: Divide both sides of the equation by √2.
sin(2x) = -1/√2
This equation can now be solved by finding the angles whose sine value is -1/√2.
Step 8: Determine the reference angle.
The reference angle is the positive acute angle between the terminal side of the angle and the x-axis.
Since sin is negative, the reference angle is π/4.
Step 9: Find the quadrants.
We know that the sine function is negative in the third and fourth quadrants. Therefore, we need to find the angles in these quadrants that have a reference angle of π/4.
Step 10: Solve for the angles.
In the third quadrant, the angle is π + reference angle = π + π/4 = (4π + π)/4 = 5π/4.
In the fourth quadrant, the angle is 2π - reference angle = 2π - π/4 = (8π - π)/4 = 7π/4.
Therefore, the solutions to the equation sin(x) + cos(x) = -√2 are x = 5π/4 and x = 7π/4.