Trigonometric identities
•Descargar como PPTX, PDF•
0 recomendaciones•94 vistas
Derivations of trigonometric Identities
Recomendados
Más contenido relacionado
Destacado
Destacado (20)
Trigonometric identities
- 2. www.harekrishnahub.com We have the following trigonometric identities: 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐2 𝜃 1 + 𝑐𝑜𝑡2 𝜃 = 𝑐𝑜𝑠𝑒𝑐2 𝜃 Let us prove one by one. Note: 𝑠𝑖𝑛 𝜃 2 , 𝑐𝑜𝑠 𝜃 2 , etc., are written as 𝑠𝑖𝑛2 𝜃and 𝑐𝑜𝑠2 𝜃 etc. and read as sine squared θ
- 3. www.harekrishnahub.com Consider a right angle triangle ABC as shown below. Let ∠𝐴𝐵𝐶 = 90° and ∠𝐶𝐴𝐵 = 𝜃 𝐴𝐵 = 𝑥 , 𝐵𝐶 = 𝑦 , 𝐴𝐶 = 𝑟
- 4. www.harekrishnahub.com we have, 𝒙 𝟐 + 𝒚 𝟐 = 𝒓 𝟐 Dividing the above equation throughout by 𝑟2 , we get 𝒙 𝟐 𝒓 𝟐 + 𝒚 𝟐 𝒓 𝟐 = 𝒓 𝟐 𝒓 𝟐 𝒙 𝒓 𝟐 + 𝒚 𝒓 𝟐 = 𝟏 But 𝑥 𝑟 = 𝑐𝑜𝑠 𝜃 and 𝑦 𝑟 = 𝑠𝑖𝑛 𝜃 ∴ 𝑐𝑜𝑠 𝜃 2 + 𝑠𝑖𝑛 𝜃 2 = 1 ∴ we have the identity 𝒔𝒊𝒏 𝟐 𝜽 + 𝒄𝒐𝒔 𝟐 𝜽 = 𝟏
- 5. www.harekrishnahub.com we have, 𝒙 𝟐 + 𝒚 𝟐 = 𝒓 𝟐 Dividing the above equation throughout by 𝑥2 (for 𝒙 ≠ 𝟎) , we get 𝒙 𝟐 𝒙 𝟐 + 𝒚 𝟐 𝒙 𝟐 = 𝒓 𝟐 𝒙 𝟐 𝟏 + 𝒚 𝒙 𝟐 = 𝒓 𝒙 𝟐 But 𝑟 𝑥 = 𝑠𝑒𝑐 𝜃 and 𝑦 𝑥 = 𝑡𝑎𝑛 𝜃 ∴ we have 𝟏 + 𝒕𝒂𝒏 𝜽 𝟐 = 𝒔𝒆𝒄 𝜽 𝟐 ∴ we have the identity 1 + 𝒕𝒂𝒏2 𝜽 = 𝒔𝒆𝒄2 𝜽
- 6. www.harekrishnahub.com we have, 𝒙 𝟐 + 𝒚 𝟐 = 𝒓 𝟐 Dividing the above equation throughout by 𝑦2(for 𝒙 𝟐 𝒚 𝟐 + 𝒚 𝟐 𝒚 𝟐 = 𝒓 𝟐 𝒚 𝟐 𝒙 𝒚 𝟐 + 𝟏 = 𝒓 𝒚 𝟐 𝒓 𝒚 = 𝒄𝒐𝒔𝒆𝒄 𝜽 𝒙 𝒚 = 𝒄𝒐𝒕 𝜽 ∴ we have 𝒄𝒐𝒕 𝜽 𝟐 + 𝟏 = 𝒄𝒐𝒔𝒆𝒄 𝜽 𝟐 ∴ we have the identity 1 + 𝒄𝒐𝒕2 𝜽 = 𝒄𝒐𝒔𝒆𝒄2 𝜽