Find all values of theta where 0degrees<theta<360 degrees when csc theta = square root 2
well, where is sin x = 1/√2 ?
which quadrants is sin > 0?
To find all values of theta satisfying the equation csc(theta) = √2, we first note that csc(theta) is the reciprocal of the sine function:
csc(theta) = 1 / sin(theta)
So, we can rewrite the equation as:
1 / sin(theta) = √2
To eliminate the fraction, we can multiply both sides of the equation by sin(theta):
sin(theta) * (1 / sin(theta)) = √2 * sin(theta)
This simplifies to:
1 = √2 * sin(theta)
Now, isolate sin(theta) by dividing both sides of the equation by √2:
1 / √2 = sin(theta)
Next, simplify the left-hand side:
1 / √2 = (√2 / √2) * (1 / √2) = √2 / 2
So, we have:
√2 / 2 = sin(theta)
Now, find the reference angle for sin(theta) = √2 / 2. The reference angle is the angle formed between the terminal side of theta and the x-axis. For sin(theta) = √2 / 2, the reference angle is π/4 (45 degrees).
Since sin(theta) is positive in the first and second quadrants, we have two sets of solutions:
First quadrant: theta = reference angle = π/4 (45 degrees)
Second quadrant: theta = π - reference angle = π - π/4 = 3π/4 (135 degrees)
To find the angles in the third and fourth quadrants, we need to subtract the reference angle from a full rotation of 2π (360 degrees):
Third quadrant: theta = π + reference angle = π + π/4 = 5π/4 (225 degrees)
Fourth quadrant: theta = 2π - reference angle = 2π - π/4 = 7π/4 (315 degrees)
Therefore, the values of theta that satisfy csc(theta) = √2 for 0 degrees < theta < 360 degrees are:
theta = 45 degrees, 135 degrees, 225 degrees, and 315 degrees.