Explanation:
we know :
sin(2t) = 2×sin(t)×cos(t)
cos(2t) = cos²(t) – sin²(t) = 2×cos²(t) – 1 = 1 – 2×sin²(t)
tan(t) = sin(t)/cos(t)
tan(2t) = sin(2t)/cos(2t) = (2×tan(t))/(1 – tan²(t))
sin²(t) + cos²(t) = 1
as t is in quadrant I, 2t must be in either quadrant I or II.
sin(t) = 7/9
sin²(t) = (7/9)² = 49/81
cos²(t) = 1 - sin²(t) = 1 - 49/81 = 81/81 - 49/81 = 32/81
cos(t) = sqrt(32)/9
sin(2t) = 2×sin(t)×cos(t) = 2×7/9 × sqrt(32)/9 =
= 14×sqrt(32)/81 ≈ 0.98
cos(2t) = cos²(t) - sin²(t) = 32/81 - 49/81 = -17/81 ≈
≈ -0.21
tan(2t) = sin(2t)/cos(2t) =
= 14×sqrt(32)/81 / -17/81 =
= 14×sqrt(32)/81 × -81/17 =
= -14×sqrt(32)/17 ≈ -4.66