A Trig Formula for the Area of a Triangle. Trigonometry In a right angled triangle, the 3 trig ratios for an angle x are defined as follows: 3 Trig Ratios:

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34: A Trig Formula for the Area of a Triangle

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Presentation transcript:

A Trig Formula for the Area of a Triangle

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

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

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

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

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

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.

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

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.

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.

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.

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.

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.

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.

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 = = =

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 = = =

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 = = =

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 = = =

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.

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

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.)

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

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

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:

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 = = =

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.)

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