1. Lesson 2: Trigonometric Ratios of Complementary Angles Trigonometric Co-functions Prepared By: Nathan Ang (19) Ivan Yeo (13) Nicholas Tey (22) Yeo Kee Xuan (25)

2. In lesson 1, you learned that : Sine A = Y/Z Cosine A = X/Z Tangent A = Y/X X Y Z Angle A

3. Cosecant A = 1 / Sin A Secant A = 1 / Cos A Cotangent A = 1 / Tan A Introduce Trigonometric Co-functions Short form : Cosec, Sec and Cot respectively

4. In lesson 2, you will learn that : Sin A = Cos (90 – A) Cos A = Sin (90 – A) Tan A = Cot (90 – A) Tan A = Sin A / Cos A Cot A = Cos A / Sin A known as quotient identities

5. In lesson 2, you will learn that : Sin 2 A + Cos 2 A = 1 Cosec 2 A = 1 + Cot 2 A Tan 2 A + 1 = Sec 2 A Known as the Pythagorean Identities

6. Take note the lesson is based on a right angled triangle as shown below : X Y Z Angle A Angle 90-A

7. Since Sin A = Y/Z Cos A = X/Z Tan A = Y / X  Tan A = Sin A / Cos A Proof : Tan A = Sin A / Cos A

8. Since Sin A = Y/Z Cos (90 – A) = Y/Z  Sin A = Cos (90 – A) Proof : Sin A = Cos (90 – A)

9. Since Cos A = X/Z Sin (90 – A) = X/Z  Cos A = Sin (90 – A) Proof : Cos A = Sin (90 – A)

10. From Pythagoras Theorem X 2 + Y 2 = Z 2 ------ (1) (1) / Z 2, (X 2 /Z 2 ) + (Y 2 /Z 2 ) = 1  cos 2 A + sin 2 A = 1 Proof : Cos 2 A + Sin 2 A = 1

11. From Pythagoras Theorem X 2 + Y 2 = Z 2 ---- (1) (1) / Y 2, (X 2 /Y 2 ) + 1 = Z 2 /Y 2  cot 2 A + 1= cosec 2 A Proof : 1 + Cot 2 A = Cosec 2 A

12. From Pythagoras Theorem X 2 + Y 2 = Z 2 ------ (1) (1)/ X 2, 1 + (Y 2 /X 2 ) =( Z 2 /X 2 )  1 + Tan 2 A = Sec 2 A Proof : 1 + Tan 2 A = Sec 2 A

13. E.g. cosec A = cot A / cos A cot A = cos A / sin A cosec A = sec A / tan A and many more... Many other identities can be derived :

14. How to remember …….. Tangent Sine Cosine Cotangent Cosecant Secant

15. 1.The trigonometric ratios at opposite ends of the same diagonal are reciprocals of one another 2. Any trigonometric ratio is equal to the product of its two immediate neighbours. E.g. sin A = cos A * tan A cosec A = sec A * cot A

16. Thank you for your kind attention !