Find all the values of theta between 0 degree and 360 degree, and state the quadrants in which they exist. (a)cos theta=-0.71 (b)tan theta=1.732
(a) cos 44.76° = .71
cos < 0 in QII,QIII
similarly for (b)
To find all the values of theta between 0 degrees and 360 degrees for the given trigonometric equations, we can use the unit circle and the properties of trigonometric functions.
(a) For the equation cos(theta) = -0.71:
Step 1: Find the reference angle.
The reference angle is the acute angle between the terminal side of theta and the x-axis.
By taking the inverse cosine (cos^(-1)) of -0.71, we can find the reference angle.
cos^(-1)(-0.71) ≈ 136.87 degrees
Step 2: Determine the quadrants.
Based on the sign of cosine, cos(theta) = -0.71 is negative. In the unit circle, cosine is negative in the second and third quadrants. So the values of theta will exist in those quadrants.
Step 3: Calculate the values of theta.
Since the reference angle is 136.87 degrees, we need to find the corresponding angles in the second and third quadrants.
In the second quadrant, theta = 180 degrees - reference angle ≈ 180 degrees - 136.87 degrees ≈ 43.13 degrees.
In the third quadrant, theta = 180 degrees + reference angle ≈ 180 degrees + 136.87 degrees ≈ 316.87 degrees.
Therefore, the values of theta between 0 degrees and 360 degrees, where cos(theta) = -0.71, are approximately 43.13 degrees and 316.87 degrees, and they exist in the second and third quadrants.
(b) For the equation tan(theta) = 1.732:
Step 1: Find the reference angle.
The reference angle is the acute angle between the terminal side of theta and the x-axis.
By taking the inverse tangent (tan^(-1)) of 1.732, we can find the reference angle.
tan^(-1)(1.732) ≈ 60 degrees
Step 2: Determine the quadrants.
Based on the sign of tangent, tan(theta) = 1.732 is positive. In the unit circle, tangent is positive in the first and third quadrants. So the values of theta will exist in those quadrants.
Step 3: Calculate the values of theta.
Since the reference angle is 60 degrees, we need to find the corresponding angles in the first and third quadrants.
In the first quadrant, theta ≈ reference angle ≈ 60 degrees.
In the third quadrant, theta = 180 degrees + reference angle ≈ 180 degrees + 60 degrees ≈ 240 degrees.
Therefore, the values of theta between 0 degrees and 360 degrees, where tan(theta) = 1.732, are approximately 60 degrees and 240 degrees, and they exist in the first and third quadrants.