Ratios of Complementary Angles and Trigonometric Co-functions

1. Lesson 2 (Seah Shao Xuan, Tan Yong Yi, Vincent Chung, Quek Yong Jie) 3rd May 2011 Ratios of Complementary Angles and Trigonometric Co-functions

2. Part 1 Recap

3. The adjacent, opposite and hypotenuse are the 3 different sides we can find on a right-angled triangle. The hypotenuse is the longest side of the triangle. The angle measured determines the position of the adjacent and opposite sides. The adjacent side is the line touching the angleother than the hypotenuse. The opposite side is the line which the angle does not use. Hypotenuse, Adjacent and Opposite hypotenuse opposite adjacent

4. Sine, cosine and tangent are the 3 basic functions of calculating the ratio of two different sides of a right-angled triangle by inputting its angle. Sine is defined by the opposite over the hypotenuse, cosine is defined by the adjacent over the hypotenuse and the tangent is defined by the opposite over the adjacent. Sine, cosine and tangent is usually spelt as sin, cosand tan respectively when in math sums. Sine, Cosine and Tangent

5. Part 2 Trigonometric Co-functions

6. The cosecant (or csc) is the reciprocal of the sine function. = Thus, as the sine is , the cosecant is . When we use cosecant for the angle of a right-angled triangle, the ratio we get is the hypotenuse over the opposite. That means that the value we get is the unit length of the hypotenuse for 1 unit length on the opposite. Cosecant

7. The secant (or sec) is the reciprocal of the cosine function. = Thus, as the sine is , the cosecant is . When we use secant for the angle of a right-angled triangle, the ratio we get is the hypotenuse over the adjacent. That means that the value we get is the unit length of the hypotenuse for 1 unit length on the adjacent. secant

8. The cotangent (or cot) is the reciprocal of the tangent function. = Thus, as the tangent is , the cosecant is . When we use cotangent for the angle of a right-angled triangle, the ratio we get is the adjacent over the opposite. That means that the value we get is the unit length of the adjacent for 1 unit length on the opposite. This is also known as run over rise. Cotangent

9. Secant, Cosecantand Cotangent sec = csc = cot =

10. Part 2 Complementary Angles

11. Complementary angles are two angles that make up a total of 90°. The complementary angle of a right-angled triangle is the last angle, other than the right-angle and the designated angle. Complementary Angles opposite hypotenuse adjacent

12. Part 3a Sine and Cosine: Relationships

13. Functions of Complementary Angles (Sine and Cosine) 1 opposite (y) hypotenuse (x) a adjacent (z) sin(a) = =

14. Functions of Complementary Angles (Sine and Cosine) 2 adjacent (y) hypotenuse (x) b opposite (z) cos(b) = =

15. Functions of Complementary Angles (Sine and Cosine) 3 x b y a z cos(b) = = sin(a) = = sin(a) = cos(b)

16. Part 3b Secant and Cosecant: Relationships

17. Functions of Complementary Angles (Secant and Cosecant) 1 opposite (y) hypotenuse (x) a adjacent (z) sec(a) = =

18. Functions of Complementary Angles (Secant and Cosecant) 2 adjacent (y) hypotenuse (x) b opposite (z) csc(b) = =

19. Functions of Complementary Angles (Secant and Cosecant) 3 x b y a z sec(a) = = csc(b) = = sec(a) = csc(b)

20. Part 3c Tangent and Cotangent: Relationships

21. Functions of Complementary Angles (Tangent and Cotangent) 1 opposite (y) hypotenuse (x) a adjacent (z) tan(a) = =

22. Functions of Complementary Angles (Tangent and Cotangent) 2 adjacent (y) hypotenuse (x) b opposite (z) cot(b) = =

23. Functions of Complementary Angles (Tangent and Cotangent) 3 x b y a z tan(a) = = csc(b) = = tan(a) = cot(b)

24. Relationships (All Summed Up in a Table)

25. Part 4 Take Note

26. When we measure the angle, if it is measured anticlockwise from the right, it is positive. If the angle is measured clockwise from the right, it is negative. This angle can affect your results! Things to take note

27. The End