Find two values for theta such that theta is greater than or equal to 0 but less than or equal to 2pi.
1. Tan theta=1.1071
2. Sin theta = -0.8818
To find the values of theta within the specified range, we'll use the inverse trigonometric functions.
1. For the equation tan(theta) = 1.1071, we need to find the inverse tangent of 1.1071. In most calculators, this is represented as atan(1.1071). Evaluating this, we get approximately 0.8532 radians or 48.82 degrees. However, since we are interested in values of theta between 0 and 2π, we can add 2π to the angle until we obtain a value within the desired range.
By adding 2π to the initial angle (0.8532 radians), we get:
0.8532 + 2π = 6.99615 radians or approximately 400.88 degrees.
Thus, one value of theta is approximately 0.8532 radians (48.82 degrees) and the other value is approximately 6.99615 radians (400.88 degrees), within the range specified (0 ≤ θ ≤ 2π).
2. For the equation sin(theta) = -0.8818, we need to find the inverse sine of -0.8818. In most calculators, this is represented as asin(-0.8818). Evaluating this, we get approximately -1.0868 radians or -62.36 degrees.
Similarly, since we want theta in the range 0 to 2π, we can add 2π until we obtain a value within the desired range.
By adding 2π to the initial angle (-1.0868 radians), we get:
-1.0868 + 2π = 5.19615 radians or approximately 297.75 degrees.
Thus, one value of theta is approximately -1.0868 radians (-62.36 degrees) and the other value is approximately 5.19615 radians (297.75 degrees), within the range specified (0 ≤ θ ≤ 2π).
To find the two values for theta, we can use the inverse trigonometric functions.
1. Tan theta = 1.1071:
Using the inverse tangent function, we can find the value of theta.
theta = arctan(1.1071)
theta ≈ 0.832 radians (approximately)
To find the second value, we need to add pi (180 degrees) to the first value since tan has a period of pi.
Second value: theta + pi ≈ 0.832 + pi ≈ 3.974 radians (approximately)
2. Sin theta = -0.8818:
Using the inverse sine function, we can find the value of theta.
theta = arcsin(-0.8818)
theta ≈ -1.062 radians (approximately)
The second value will be found by taking pi minus the first value since sin has a period of 2pi.
Second value: pi - (-1.062) ≈ 4.203 radians (approximately)
Therefore, the two values for theta are approximately 0.832 and 4.203 radians.