Problems with trigonometry and the area of a triangle
Lessons 26–27 · 2 lessons · B. Xaydarov, E. Sariqov, A. Qo‘chqorov. Geometry, Grade 9 (textbook for general secondary schools), revised 4th edition. Huquq va Jamiyat Publishing, Tashkent, 2019
27
Area of a triangle using the sine of an angle
Textbook: pp. 82–83
GoalCalculate the area of a triangle, parallelogram and quadrilateral using the sine of an angle.
New words
two sides and the included angle · ikki tomon va orasidagi burchakangle between the diagonals · diagonallar orasidagi burchakarea via sine · sinus bilan yuzheight · balandlik
Explanation
Theorem 1: the area of a triangle equals half the product of two sides and the sine of the angle between them: S = ½·ab·sinC. Proof: the height is h = a·sinC, and S = ½·b·h. Similarly S = ½·bc·sinA = ½·ac·sinB. The area of a parallelogram is the product of adjacent sides and the sine of the angle between them: S = ab·sinα. Theorem 2: the area of a convex quadrilateral equals half the product of its diagonals and the sine of the angle between them: S = ½·d₁·d₂·sinφ. The formula also works for obtuse angles since sin(180° − α) = sinα. These formulas are handy when the height is unknown.
Worked examples
△ABC: AB = 10, AC = 6, ∠A = 30°. S = ½·10·6·sin30° = ½·10·6·½ = 15.
A parallelogram has sides 8 and 5 and an angle of 150°: S = 8·5·sin150° = 40·½ = 20. A quadrilateral with diagonals 10 and 12 and a 30° angle between them: S = ½·10·12·½ = 30.
Class activity
“Area two ways”: for a triangle with sides 6 and 8 and a 90° angle between them, compute the area with the simple formula and with the sine formula and check that they agree.
Practice
1
Two sides of a triangle are 10 and 14 with a 30° angle between them. What is the area?
35
2
A triangle has area 18, AC = 9 and ∠A = 150°. How long is AB?
8
3
A parallelogram has sides 4 and 6 and an angle of 120°. What is its area?
12√3
4
Why does the sine formula reduce to S = ½ab when the angle is 90°?