Final answer:
To find the exact value of sin 2θ, cos 2θ, and tan 2θ, one must use the double-angle identities; however, without knowing the value of θ, it is impossible to give the exact values. The quadrant of 2θ depends on the value of θ.
Step-by-step explanation:
The student is asking for the exact value of sine, cosine, and tangent functions at double the angle θ (sin 2θ, cos 2θ, tan 2θ), as well as the quadrant in which the angle 2θ lies. If the angle θ is known, we can use the double-angle identities:
- sin 2θ = 2sin θ cos θ
- cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
- tan 2θ = (2 tan θ) / (1 - tan² θ)
However, without the exact value of θ, we cannot determine the exact values of these functions. As for the quadrant, angles are typically measured from the positive x-axis in the counter-clockwise direction. Any angle that is a multiple of 360° is effectively the same as an angle of 0°. Therefore, by determining θ, we could find 2θ and then use the ranges of 0°-90° (Quadrant I), 90°-180° (Quadrant II), 180°-270° (Quadrant III), or 270°-360° (Quadrant IV) to find the quadrant in which 2θ lies.