Unit Circle: Circular Functions and Trigonometric Functions PDF

Unit Circle: CIRCULAR FUNCTIONS AND TRIGONOMETRIC FUNCTIONS UNIT CIRCLE: LEARNING OBJECTIVES UNIT CIRCLE: DEGREES AND RADIANS 𝐶 𝑦 𝜃 90°...

Unit Circle: CIRCULAR FUNCTIONS AND TRIGONOMETRIC FUNCTIONS UNIT CIRCLE: LEARNING OBJECTIVES UNIT CIRCLE: DEGREES AND RADIANS 𝐶 𝑦 𝜃 90° (0, 1) 1 𝑟𝑎𝑑 𝜃𝑟𝑎𝑑 𝑠 = 1 𝑟𝑎𝑑 𝑠 𝑟 180° 𝐶(0,0) 𝜃 = 57.3° (1, 0) 𝑥 𝑠 (−1, 0) 𝑟 = 1 𝑢𝑛𝑖𝑡 360° 𝜃𝑟𝑎𝑑 = 𝑟 360° = 2𝜋 𝑟𝑎𝑑 (0, −1) 270° 180° = 𝜋 𝑟𝑎𝑑 180° 360° = 2𝜋 𝑟𝑎𝑑 1 𝑟𝑎𝑑 = 𝜋 1 𝑟𝑎𝑑 = 57.30°  UNIT CIRCLE: DEGREES AND RADIANS 360° = 2𝜋 𝑟𝑎𝑑 ≈ 180° = 𝜋 𝑟𝑎𝑑 𝜋 𝑟𝑎𝑑 180° ° ° 𝜋 𝑟𝑎𝑑 75° × 𝜋 𝑟𝑎𝑑 3 ∙ 5 ∙ 5 ∙ 𝜋 𝑟𝑎𝑑 5 ∙ 𝜋 𝑟𝑎𝑑 𝟓𝝅 𝒓𝒂𝒅 𝜋 𝑟𝑎𝑑 240° × 𝜋 𝑟𝑎𝑑 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ∙ 5 ∙ 𝜋 𝑟𝑎𝑑 2 ∙ 2 ∙ 𝜋 𝑟𝑎𝑑 𝟒𝝅 𝒓𝒂𝒅 75° = = = = 240° = = = = 180° 180° 2∙2∙3∙3∙5 2∙2∙3 𝟏𝟐 180° 180° 2∙2∙3∙3∙5 3 𝟑 180° 𝜋 𝑟𝑎𝑑 𝜋 5𝜋 𝑟𝑎𝑑 𝑟𝑎𝑑 18 3 𝜋 180° 180° 5𝜋 180° 900° 𝑟𝑎𝑑 = = 𝟏𝟎° 𝑟𝑎𝑑 = = 𝟑𝟎𝟎° 18 𝜋 𝑟𝑎𝑑 18 3 𝜋 𝑟𝑎𝑑 3°  UNIT CIRCLE: CIRCULAR FUNCTIONS ° ° ° ° 𝜃 𝑦 𝜃 90° (0, 1) 𝑃(𝑥, 𝑦) 𝑠𝑖𝑛(𝜃) 𝑦 𝒔𝒊𝒏 𝜽 = 𝒚 𝟏 180° 𝜃 (1, 0) 𝒄𝒔𝒄 𝜽 = ;𝒚 ≠ 𝟎 𝑥 𝒚 (−1, 0) 𝐶(0,0) 𝑥 360° 𝑐𝑜𝑠(𝜃) 𝒄𝒐𝒔 𝜽 = 𝒙 𝟏 𝒔𝒆𝒄 𝜽 = ; 𝒙 ≠ 𝟎 𝒙 𝒚 𝒙 𝒕𝒂𝒏 𝜽 = ; 𝒙 ≠ 𝟎 𝒄𝒐𝒕 𝜽 = ; 𝒚 ≠ 𝟎 𝒙 𝒚 (0, −1) 270°  UNIT CIRCLE: CIRCULAR FUNCTIONS 𝑜𝑝𝑝 𝑎𝑑𝑗 𝑜𝑝𝑝 1 1 1 𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 𝜃 = 𝑡𝑎𝑛 𝜃 = 𝑐𝑠𝑐 𝜃 = 𝑠𝑒𝑐 𝜃 = 𝑐𝑜𝑡 𝜃 = ℎ𝑦𝑝 ℎ𝑦𝑝 𝑎𝑑𝑗 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 𝜃 𝑜𝑝𝑝 = 𝑦 𝑎𝑑𝑗 = 𝑥 ℎ𝑦𝑝 = 1 𝑦 𝑜𝑝𝑝 𝑦 𝑎𝑑𝑗 𝑥 𝑜𝑝𝑝 𝑦 𝑠𝑖𝑛 𝜃 = = 𝑐𝑜𝑠 𝜃 = = 𝑡𝑎𝑛 𝜃 = = ℎ𝑦𝑝 1 ℎ𝑦𝑝 1 𝑎𝑑𝑗 𝑥 𝑃(𝑥, 𝑦) 𝑠𝑖𝑛(𝜃) 𝒚 𝒔𝒊𝒏 𝜽 = 𝒚 𝒄𝒐𝒔 𝜽 = 𝒙 𝒕𝒂𝒏 𝜽 = 𝑦 𝒙 𝜃 𝟏 𝟏 𝒙 𝑥 𝒄𝒔𝒄 𝜽 = 𝒔𝒆𝒄 𝜽 = 𝒄𝒐𝒕 𝜽 = 𝐶(0,0) 𝑥 𝒚 𝒙 𝒚 𝑐𝑜𝑠(𝜃) UNIT CIRCLE: CIRCULAR FUNCTIONS (EXAMPLES) 𝑦 𝑠𝑖𝑛 90° 𝑐𝑜𝑠 270° 𝑡𝑎𝑛 180° 𝑠𝑖𝑛 𝜃 = 𝑦 𝑐𝑜𝑠 𝜃 = 𝑥 𝑦 𝑡𝑎𝑛 𝜃 = 90° (0, 1) 𝒔𝒊𝒏 𝟗𝟎° = 𝟏 𝒄𝒐𝒔 𝟐𝟕𝟎° = 𝟎 𝑥 𝑃(𝑥, 𝑦) 0 𝑠𝑖𝑛(𝜃) 𝑡𝑎𝑛 180° = −1 𝑦 𝒕𝒂𝒏 𝟏𝟖𝟎° = 𝟎 180° 𝜃 (1, 0) 𝑥 (−1, 0) 𝐶(0,0) 𝑥 360° 𝑠𝑖𝑛 360° 𝑐𝑜𝑠 180° 𝑡𝑎𝑛 90° 𝑐𝑜𝑠(𝜃) 𝑐𝑜𝑠 𝜃 = 𝑥 𝑦 𝑠𝑖𝑛 𝜃 = 𝑦 𝑡𝑎𝑛 𝜃 = 𝒔𝒊𝒏 𝟑𝟔𝟎° = 𝟎 𝒄𝒐𝒔 𝟑𝟔𝟎° = −𝟏 𝑥 1 𝑡𝑎𝑛 90° = (0, −1) 270° 0 𝒕𝒂𝒏 𝟗𝟎° = 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅 UNIT CIRCLE: CIRCULAR FUNCTIONS (EXERCISES) 𝑦 𝑐𝑠𝑐 270° 𝑠𝑒𝑐 180° 𝑐𝑜𝑡 90° 90° (0, 1) 𝑃(𝑥, 𝑦) 𝑠𝑖𝑛(𝜃) 𝑦 𝜃 180° (1, 0) 𝑥 (−1, 0) 𝐶(0,0) 𝑥 360° 𝑐𝑠𝑐 270° 𝑠𝑒𝑐 360° 𝑐𝑜𝑡 180° 𝑐𝑜𝑠(𝜃) (0, −1) 270° UNIT CIRCLE: CIRCULAR FUNCTIONS (EXERCISES) 𝜋 1 3 𝑃 , 3 2 2 𝜋 𝜋 𝜋 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑡𝑎𝑛 3 3 3 𝜋 𝜋 𝜋 𝑐𝑠𝑐 3 𝑠𝑒𝑐 3 𝑐𝑜𝑡 3