1) To find the measures of the angles in each quadrant, we can use the reference angle of 82˚ and apply the properties of angles in each quadrant:
(a) The angle in Quadrant I:
The angle in Quadrant I has the same measure as the reference angle because both angles are positive and less than 90˚.
Therefore, the angle in Quadrant I is 82˚.
(b) The angle in Quadrant II:
The angle in Quadrant II is equal to 180˚ minus the reference angle, as angles in this quadrant are greater than 90˚ and less than 180˚.
Therefore, the angle in Quadrant II is 180˚ - 82˚ = 98˚.
(c) The angle in Quadrant III:
The angle in Quadrant III is equal to 180˚ plus the reference angle, as angles in this quadrant are greater than 180˚ and less than 270˚.
Therefore, the angle in Quadrant III is 180˚ + 82˚ = 262˚.
(d) The angle in Quadrant IV:
The angle in Quadrant IV is equal to 360˚ minus the reference angle, as angles in this quadrant are greater than 270˚ and less than 360˚.
Therefore, the angle in Quadrant IV is 360˚ - 82˚ = 278˚.
The measures of the angles in each quadrant are:
(a) Quadrant I: 82˚
(b) Quadrant II: 98˚
(c) Quadrant III: 262˚
(d) Quadrant IV: 278˚
(2) Given sin(θ) = 2/9, we can use the Pythagorean identity to find cos(θ) and then calculate tan(θ):
sin²(θ) + cos²(θ) = 1
(2/9)² + cos²(θ) = 1
4/81 + cos²(θ) = 1
cos²(θ) = 1 - 4/81
cos²(θ) = 77/81
cos(θ) = √(77/81)
Since sin(θ) and cos(θ) are both positive, we can determine that θ lies in the first quadrant.
Now, to find tan(θ):
tan(θ) = sin(θ) / cos(θ)
tan(θ) = (2/9) / √(77/81)
To simplify this expression, we can rationalize the denominator by multiplying both the numerator and denominator by √81:
tan(θ) = (2/9) / (√(77/81) * √(81/81))
tan(θ) = (2/9) / (√(77*81) / 81)
tan(θ) = (2/9) / (√6237 / 81)
tan(θ) = (2/9) * (81 / √6237)
tan(θ) = 162 / (9√6237)
tan(θ) ≈ 1.0809 (rounded to four decimal places)
Therefore, tan(θ) ≈ 1.0809.