Unit Circle: Circular Functions and Trigonometric Functions PDF
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Uploaded by FascinatingCarbon8449
Pavia National High School
Engr. John Christopher L. Padios, RCE
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Summary
This document presents a comprehensive study of the unit circle, including the concepts of circular functions and trigonometric functions. It covers topics like degrees and radians, conversion factors, and the relationships between different trigonometric identities and the unit circle. The document also provides examples and exercises to help readers better understand the material. The document is aimed at undergraduate students studying mathematics.
Full Transcript
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
