Find two values of 0, 0 ≤ 0 < 2π, that satisfy the following equation. Show your work.
tan 0 = 1
To find two values of θ that satisfy the equation tan θ = 1, we need to find the values of θ where the tangent function equals 1.
First, we need to recall the unit circle and the important points on it. The unit circle is a circle with a radius of 1, centered at the origin (0,0) in the coordinate plane.
We know that the tangent of an angle is equal to the y-coordinate divided by the x-coordinate of a point on the unit circle where the terminal side of the angle intersects the circle. For tan θ = 1, the y-coordinate must be 1 and the x-coordinate must be 1.
One point on the unit circle where the y-coordinate is 1 and the x-coordinate is 1 is (1, 1). Another point with the same coordinates lies in the fourth quadrant, which can be found by subtracting 2π from the angle of the first point.
Therefore, we have two values for θ that satisfy the equation tan θ = 1:
θ₁ = 45° or π/4 (in radians)
θ₂ = 45° + 180° or π/4 + π = 225° or 5π/4 (in radians)
Note: 45° is in the first quadrant and 225° is in the fourth quadrant. Both angles have tangent values of 1.