1. **Problem Statement:** Find the sine, cosine, and tangent of the given angles using the quadrant system. 2. **Formula and Rules:** - Sine, cosine, and tangent values depend on the quadrant of the angle. - Quadrants and signs: - Q1 (0° to 90°): sin > 0, cos > 0, tan > 0 - Q2 (90° to 180°): sin > 0, cos < 0, tan < 0 - Q3 (180° to 270°): sin < 0, cos < 0, tan > 0 - Q4 (270° to 360°): sin < 0, cos > 0, tan < 0 - Reference angle = angle's distance from the nearest x-axis (0°, 90°, 180°, 270°, or 360°) - Use reference angle to find sine, cosine, tangent values from known angles. 3. **Calculations:** **i. 260°** - Quadrant: Q3 (180° < 260° < 270°) - Reference angle: 260° - 180° = 80° - sin(260°) = -sin(80°) - cos(260°) = -cos(80°) - tan(260°) = tan(80°) **ii. 310°** - Quadrant: Q4 (270° < 310° < 360°) - Reference angle: 360° - 310° = 50° - sin(310°) = -sin(50°) - cos(310°) = cos(50°) - tan(310°) = -tan(50°) **iii. 98°** - Quadrant: Q2 (90° < 98° < 180°) - Reference angle: 180° - 98° = 82° - sin(98°) = sin(82°) - cos(98°) = -cos(82°) - tan(98°) = -tan(82°) **iv. 146°** - Quadrant: Q2 - Reference angle: 180° - 146° = 34° - sin(146°) = sin(34°) - cos(146°) = -cos(34°) - tan(146°) = -tan(34°) **v. 128°** - Quadrant: Q2 - Reference angle: 180° - 128° = 52° - sin(128°) = sin(52°) - cos(128°) = -cos(52°) - tan(128°) = -tan(52°) **vi. 75°** - Quadrant: Q1 - Reference angle: 75° (since in Q1) - sin(75°), cos(75°), tan(75°) are positive **vii. 290°** - Quadrant: Q4 - Reference angle: 360° - 290° = 70° - sin(290°) = -sin(70°) - cos(290°) = cos(70°) - tan(290°) = -tan(70°) **viii. 335°** - Quadrant: Q4 - Reference angle: 360° - 335° = 25° - sin(335°) = -sin(25°) - cos(335°) = cos(25°) - tan(335°) = -tan(25°) **ix. 203°** - Quadrant: Q3 - Reference angle: 203° - 180° = 23° - sin(203°) = -sin(23°) - cos(203°) = -cos(23°) - tan(203°) = tan(23°) **x. 40°** - Quadrant: Q1 - Reference angle: 40° - sin(40°), cos(40°), tan(40°) are positive 4. **Final values (approximate):** - sin(80°) ≈ 0.9848, cos(80°) ≈ 0.1736, tan(80°) ≈ 5.6713 - sin(50°) ≈ 0.7660, cos(50°) ≈ 0.6428, tan(50°) ≈ 1.1918 - sin(82°) ≈ 0.9903, cos(82°) ≈ 0.1392, tan(82°) ≈ 7.1154 - sin(34°) ≈ 0.5592, cos(34°) ≈ 0.8290, tan(34°) ≈ 0.6745 - sin(52°) ≈ 0.7880, cos(52°) ≈ 0.6157, tan(52°) ≈ 1.2809 - sin(75°) ≈ 0.9659, cos(75°) ≈ 0.2588, tan(75°) ≈ 3.7321 - sin(70°) ≈ 0.9397, cos(70°) ≈ 0.3420, tan(70°) ≈ 2.7475 - sin(25°) ≈ 0.4226, cos(25°) ≈ 0.9063, tan(25°) ≈ 0.4663 - sin(23°) ≈ 0.3907, cos(23°) ≈ 0.9205, tan(23°) ≈ 0.4245 - sin(40°) ≈ 0.6428, cos(40°) ≈ 0.7660, tan(40°) ≈ 0.8391 **Results:** | Angle | sin | cos | tan | |-------|-----|-----|-----| | 260° | -0.9848 | -0.1736 | 5.6713 | | 310° | -0.7660 | 0.6428 | -1.1918 | | 98° | 0.9903 | -0.1392 | -7.1154 | | 146° | 0.5592 | -0.8290 | -0.6745 | | 128° | 0.7880 | -0.6157 | -1.2809 | | 75° | 0.9659 | 0.2588 | 3.7321 | | 290° | -0.9397 | 0.3420 | -2.7475 | | 335° | -0.4226 | 0.9063 | -0.4663 | | 203° | -0.3907 | -0.9205 | 0.4245 | | 40° | 0.6428 | 0.7660 | 0.8391 |