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
26
Problem solving
Textbook: pp. 78–81
GoalSolve problems on right triangles, rhombuses, trapezoids and circle chords with trigonometric relations.
New words
opposite leg · qarama-qarshi katetadjacent leg · yopishgan katetangle of elevation · ko‘tarilish burchagichord · vatar
Explanation
In a right triangle with hypotenuse c and acute angle α, the leg opposite α is a = c·sinα, the adjacent leg is b = c·cosα, and a = b·tgα. These relations solve problems with rhombuses, trapezoids or roads with a given angle of elevation: for example the height of a rhombus is h = a·sinα (a is the side, α an angle). In a circle of radius R, a chord subtending an arc of θ measures 2R·sin(θ/2), because the perpendicular from the centre bisects the chord and the central angle. When solving, draw a figure, isolate a right triangle and choose which ratio (sin, cos, tan) connects the given and the unknown segments.
Worked examples
A rhombus has side 8 cm and acute angle 30°. Its height is h = 8·sin30° = 4 cm and its area is 8·4 = 32 cm².
An isosceles right triangle has hypotenuse 10 cm: each leg is 10·sin45° = 5√2 cm and the area is ½·(5√2)² = 25 cm².
Class activity
“Road slope”: if an imaginary road rises at 30° to the horizontal, calculate how many metres it climbs per 100 m of road and show it in a drawing.
Practice
1
A road rises at 30° to the horizontal. After 200 m along the road, how many metres has it climbed?
100
2
The height of an equilateral triangle is 5√3 cm. What is its perimeter (cm)?
30
3
In a circle of radius 8 cm, how long is the chord subtending a 120° arc?
8√3 cm
4
Why is the height of a rhombus h = a·sinα?
The height and the side form a right triangle in which h is the leg opposite angle α, so h = a·sinα.