Find all angles between 0° and 360° that satisfy the following trig ratios.
a) sin(theta)=-1 over square root of 2
b) ccs(theta)=2
c) cot(theta)=-square root of 3 over 1
a) To find the angles that satisfy sin(theta) = -1/√2, we need to find the reference angle first. Since sin(theta) is negative, the angle theta lies in the third or fourth quadrant. The reference angle is the positive acute angle between the terminal side of theta and the x-axis. The reference angle for sin(theta) = 1/√2 is 45°.
In the third quadrant, the sine of an angle is negative. Therefore, the angle theta can be found by adding the reference angle to 180°. So one angle that satisfies sin(theta) = -1/√2 is 180° + 45° = 225°.
In the fourth quadrant, the sine of an angle is negative, but the reference angle is already negative (since it is in the third quadrant). Therefore, we subtract the reference angle from 360° to find the angle theta. The angle that satisfies sin(theta) = -1/√2 in the fourth quadrant is 360° - 45° = 315°.
So the angles between 0° and 360° that satisfy sin(theta) = -1/√2 are 225° and 315°.
b) To find the angles that satisfy csc(theta) = 2, we need to use the reciprocal relationship between csc(theta) and sin(theta). Csc(theta) is equal to 1/sin(theta). So we can write the equation as 1/sin(theta) = 2.
To solve for sin(theta), we take the reciprocal of both sides of the equation, which gives us sin(theta) = 1/2. To find the angles that satisfy sin(theta) = 1/2, we need to find the reference angle first. The reference angle for sin(theta) = 1/2 is 30°.
In the first quadrant, the csc(theta) is positive. Therefore, the angle theta can be found by looking for the positive angle that has a reference angle of 30°. One angle that satisfies csc(theta) = 2 is 30°.
We can find additional angles that satisfy csc(theta) = 2 by using symmetry. Since the sine function is positive in the first and second quadrants, we can take the supplement of the angle 30° to find another angle. The supplement of 30° is 180° - 30° = 150°.
So the angles between 0° and 360° that satisfy csc(theta) = 2 are 30° and 150°.
c) To find the angles that satisfy cot(theta) = -√3/1, we need to use the reciprocal relationship between cot(theta) and tan(theta). Cot(theta) is equal to 1/tan(theta). So we can write the equation as 1/tan(theta) = -√3/1.
To solve for tan(theta), we take the reciprocal of both sides of the equation, which gives us tan(theta) = -1/√3. To find the angles that satisfy tan(theta) = -1/√3, we need to find the reference angle first. The reference angle for tan(theta) = -1/√3 is 30°.
In the second quadrant, the tan(theta) is negative. Therefore, the angle theta can be found by looking for the negative angle that has a reference angle of 30°. One angle that satisfies cot(theta) = -√3/1 is 180° + 30° = 210°.
We can find additional angles that satisfy cot(theta) = -√3/1 by using symmetry. Since the tangent function is negative in the second and fourth quadrants, we can take the supplement of the angle 210° to find another angle. The supplement of 210° is 360° - 210° = 150°.
So the angles between 0° and 360° that satisfy cot(theta) = -√3/1 are 210° and 150°.