Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.)
sin θ = root 2/2 and List six specific solutions.
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sam i mean
you know that sin pi/4 = 1/√2
place in proper quadrants, and add multiples of 2pi
Well said Jman!
To solve the equation sin θ = √2/2, we need to find the values of θ that satisfy this equation.
To find the solutions, we need to determine the angles whose sine is equal to √2/2.
The sine of an angle is equal to the opposite side divided by the hypotenuse in a right triangle. So, in this case, √2/2 would be the ratio between the length of the opposite side and the hypotenuse.
We know that √2 = 1.414, so we can simplify the equation to sin θ = 1.414/2.
Now, we need to look at the unit circle to find the angles whose sine is 1.414/2. The unit circle is a circle with a radius of 1 unit, centered at the origin (0, 0) of a coordinate plane.
The unit circle has key angles with known values for sine, cosine, and tangent. These key angles are 0°, 30°, 45°, 60°, and 90°.
Looking at the unit circle, we can see that the angle θ that satisfies sin θ = 1.414/2 is 45°.
Since sine is positive in the first and second quadrants, we can add or subtract multiples of 360° to get additional solutions in those quadrants.
So, the first solution is θ = 45°.
To find additional solutions, we can add or subtract multiples of 360° or 180°.
Adding 360° to 45° gives us 405°, subtracting 360° gives us -315°.
Adding 180° to 45° gives us 225°, subtracting 180° gives us -135°.
Adding 360° again gives us 765°, subtracting 360° gives us -675°.
So, the six specific solutions are θ = 45°, 225°, 405°, -135°, -315°, -675°.