1. A Trig Formula for the Area of a Triangle

2. Trigonometry In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: 3 Trig Ratios: A reminder opposite hypotenuse x

3. Trigonometry In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: hypotenuse x adjacent 3 Trig Ratios: A reminder

4. Trigonometry In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: opposite x adjacent 3 Trig Ratios: A reminder

5. Trigonometry Using the trig ratios we can find unknown angles and sides of a right angled triangle, provided that, as well as the right angle, we know the following: either 1 side and 1 angle or 2 sides 3 Trig Ratios: A reminder

6. Trigonometry 7 y e.g. 1 e.g. 2 10 8 (3 s.f.) Tip: Always start with the trig ratio, whether or not you know the angle. 3 Trig Ratios: A reminder

7. Trigonometry Scalene Triangles We will now find a formula for the area of a triangle that is not right angled, using 2 sides and 1 angle.

8. Trigonometry a, b and c are the sides opposite angles A, B and C respectively. ( This is a conventional way of labelling a triangle ). ABC is a non-right angled triangle. A B C b a c Area of a Triangle

9. Trigonometry Draw the perpendicular, h, from C to BA. N h C b a c A B Area of a Triangle ABC is a non-right angled triangle.

10. Trigonometry Draw the perpendicular, h, from C to BA. N h - - - - - (1) In C b a c A B Area of a Triangle ABC is a non-right angled triangle.

11. Trigonometry Draw the perpendicular, h, from C to BA. N h - - - - - (1) In C b a c A B Area of a Triangle ABC is a non-right angled triangle.

12. Trigonometry h b a c c C N A B - - - - - (1) In Draw the perpendicular, h, from C to BA. Area of a Triangle ABC is a non-right angled triangle.

13. Trigonometry h b a c a c C N B Substituting for h in (1) A - - - - - (1) In Draw the perpendicular, h, from C to BA. Area of a Triangle ABC is a non-right angled triangle.

14. Trigonometry c b a a C B A Substituting for h in (1) - - - - - (1) In Draw the perpendicular, h, from C to BA. Area of a Triangle ABC is a non-right angled triangle.

15. Trigonometry Any side can be used as the base, so Area of a Triangle The formula always uses 2 sides and the angle formed by those sides Area = = =

16. Trigonometry Any side can be used as the base, so Area of a Triangle The formula always uses 2 sides and the angle formed by those sides c b a a C B A Area = = =

17. Trigonometry Any side can be used as the base, so Area of a Triangle The formula always uses 2 sides and the angle formed by those sides c b a a C B A Area = = =

18. Trigonometry Any side can be used as the base, so Area of a Triangle The formula always uses 2 sides and the angle formed by those sides c b a a C B A Area = = =

19. Trigonometry 1.Find the area of the triangle PQR. Example 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths.

20. Trigonometry 1.Find the area of the triangle PQR. Example 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q

21. Trigonometry 1.Find the area of the triangle PQR. Example 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q cm 2 (3 s.f.)

22. Trigonometry A useful application of this formula occurs when we have a triangle formed by 2 radii and a chord of a circle. Area of a Triangle r B A C r

23. Trigonometry  The area of triangle ABC is given by SUMMARY  The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by where is the angle between the radii. or

24. Trigonometry 1.Find the areas of the triangles shown in the diagrams. Exercises radius = 4 cm., (a) (b) X 12 cm 9 cm B A C Y O (a) cm 2 (3 s.f.) (b) cm 2 (3 s.f.) Ans:

25. Trigonometry Any side can be used as the base, so Area of a Triangle The formula always uses 2 sides and the angle formed by those sides c b a a C B A Area = = =

26. Trigonometry e.g. Find the area of the triangle PQR. 7 cm 8 cm R Q P Solution: We must use the angle formed by the 2 sides with the given lengths. We know PQ and RQ so use angle Q cm 2 (3 s.f.)

27. Trigonometry  The area of triangle ABC is given by SUMMARY  The area of a triangle formed by 2 radii of length r of a circle and the chord joining them is given by where is the angle between the radii. or