Question

The measure of angle \theta is (7\pi )/(4). The measure of its reference angle is \deg , and tan \theta is .

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the reference angle, we need to find the acute angle that has the same trigonometric values as the given angle.

Since the measure of angle θ is (7π)/(4), and a full circle is 2π radians, we can subtract 2π from θ to obtain an angle in the same quadrant with a measure less than τ/2 radians.

(7π)/(4) - 2π = (7π)/(4) - (8π)/(4) = -π/(4)

Since -π/4 is in the fourth quadrant, we add π to obtain the reference angle in the first quadrant.

Reference angle = -π/(4) + π = (3π)/(4) = 135 degrees

To find the tangent of θ, we use the fact that the tangent function equals the ratio of the opposite side to the adjacent side in a right triangle. Let's assume the triangle is a 45-45-90 triangle.

We can draw a right triangle in the first quadrant with angle θ, and let's assume that the lengths of the legs are equal to 1.

In this triangle, the opposite side is equal to 1 and the adjacent side is also equal to 1.

Tangent of θ = opposite/adjacent = 1/1 = 1.

Therefore, the measure of the reference angle is 135 degrees, and the tan θ is 1.