Final answer:
In quadrant IV, the cosine function is positive while the tangent function is negative. So, if we let x be the given angle θ, then cos 2θ = cos^2(x) - sin^2(x) and tan 2θ = 2tan(x) / (1 - tan^2(x)).
Step-by-step explanation:
In quadrant IV, the cosine function is positive while the tangent function is negative. So, if we let x be the given angle θ, then cos 2θ = cos2(x) - sin2(x) and tan 2θ = 2tan(x) / (1 - tan2(x)).
If
�
θ is an angle in quadrant IV, the cosine of
2
�
2θ is positive, and the tangent of
2
�
2θ is negative.
In quadrant IV, cosine (cos) is positive because the x-coordinate is positive. For
�
θ,
cos
�
cosθ is positive, and when
�
θ is doubled to
2
�
2θ,
cos
2
�
cos2θ remains positive.
The tangent (tan) of
2
�
2θ is negative because, in quadrant IV, the y-coordinate is negative. Since
tan
�
=
sin
�
cos
�
tanθ=
cosθ
sinθ
, and
sin
�
sinθ is also negative in quadrant IV, the overall result is negative. Therefore, in quadrant IV,
cos
2
�
cos2θ is positive, and
tan
2
�
tan2θ is negative. Understanding the trigonometric functions in different quadrants is crucial for solving problems involving angles in the coordinate plane.