Topicmd102c2fbe2dd6d9a_1528449000663_0Topic

The use of trigonometric functionstrigonometric functionstrigonometric functions to calculate area of polygons

Levelmd102c2fbe2dd6d9a_1528449084556_0Level

Third

Core curriculummd102c2fbe2dd6d9a_1528449076687_0Core curriculum

VII. Trigonometry. The student:

5) applies the sine and cosine theorems and the formula for the area of a trianglearea of a trianglearea of a triangle: $P=\frac{1}{2}a\cdot b\cdot sin\gamma$.

Timingmd102c2fbe2dd6d9a_1528449068082_0Timing

45 minutes

General objectivemd102c2fbe2dd6d9a_1528449523725_0General objective

Interpreting and manipulating information presented both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesmd102c2fbe2dd6d9a_1528449552113_0Specific objectives

1. Communicating in English, developing mathematical, scientific, technical and IT competences, developing learning skills.

2. Learning formulas for area of polygons using trigonometric functionstrigonometric functionstrigonometric functions.

3. The use of a formula for the triangle area for calculating the side lengths and measures of angles in triangles.

Learning outcomesmd102c2fbe2dd6d9a_1528450430307_0Learning outcomes

The student:

- knows the formulas for area of polygons using trigonometric functions,

- uses a formula for the area of triangle to calculate the measures of segments and angles in a triangle.

Methodsmd102c2fbe2dd6d9a_1528449534267_0Methods

1. Priority pyramid.

2. Case study.

Forms of workmd102c2fbe2dd6d9a_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmd102c2fbe2dd6d9a_1528450127855_0Introduction

Students, working in small groups, create a pyramid of priorities containing the knowledge they have learned about trigonometric functionstrigonometric functionstrigonometric functions and how to calculate the area of a trianglearea of a trianglearea of a triangle. After finishing the task, they present their boards.

The teacher verifies the students' statements, explains the doubts.

Proceduremd102c2fbe2dd6d9a_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to get to know the formula for calculating the area of a triangle using trigonometric functionstrigonometric functionstrigonometric functions.

Students working in groups analyze the material presented in the Interactive illustration. They formulate their conclusions.

[Interactive illustration]

Conclusion:
- The area of the triangle is equal to the half of the product of the length of two sides of the triangle by the sine of the angle contained between these sides.md102c2fbe2dd6d9a_1527752256679_0Conclusion:
- The area of the triangle is equal to the half of the product of the length of two sides of the triangle by the sine of the angle contained between these sides.

Task
In the isosceles triangle, the base length is a = 8 cm and the angle measure at the base $\alpha =30$°. Calculate the area of this triangle.md102c2fbe2dd6d9a_1527712094602_0In the isosceles triangle, the base length is a = 8 cm and the angle measure at the base $\alpha =30$°. Calculate the area of this triangle.

Discussion - Can we use the formula for calculating the area of a trianglearea of a trianglearea of a triangle to calculate the area of other polygons? Students make hypotheses, check them and formulate a conclusion.

Conclusion:

- Using the formula for the area of the triangle, you can calculate the area of a parallelogram when we know the length of its sides and the measure of the angle contained between them.

The area of parallelogram equals the product of the length of the sides by the sine of the angle between them.

Using the information learnt students solve the tasks.

Task
In an isosceles triangle with a field equal to $16\sqrt{3}$ cmIndeks górny 2², the ratio of height lowered to the base, to the base length is equal to $\frac{\sqrt{3}}{6}$. Calculate the angle measures and the perimeter of this triangle.

Task
In an ABCD square with a side length of 10 cm, point E is the center of the side AD, and point F is the center of the side AB. Calculate the area and the perimeter of the EFC triangle.

Task
One of the sides of the parallelogram has a length of 18 cm and forms with the second side an angle $\alpha$ such that $\sin \alpha =0,8$. Calculate the perimeter of this parallelogram if its field is 72 cmIndeks górny 2².

Task
Calculate the area and perimeter of a rectangular trapezoidrectangular trapezoidrectangular trapezoid in which the shorter base is 6 cm long, 8 cm high, and the longer diagonal forms a 30° angle with the longer trapezium base.

After solving all tasks, the students evaluate their work. The teacher verifies the answers and explains all doubts.

Task for volunteers:
Prove that, if the diagonals of the parallelogram have lengths c, d and they intersect at an angle $\alpha$, then the area of a parallelogram is given the following formula: $P=\frac{1}{2}cdsin\alpha$.

Lesson summarymd102c2fbe2dd6d9a_1528450119332_0Lesson summary

Students do the revision exercises.

Together, they formulate conclusions to remember.

- The area of the triangle is equal to the half of the product of the length of two sides of the triangle by the sine of the angle contained between these sides.

- The area of a parallelogram equals the product of the length of the sides by the sine of the angle between them.

Selected words and expressions used in the lesson plan

any triangleany triangleany triangle

area of a trianglearea of a trianglearea of a triangle

area of a quadrilateralarea of a quadrilateralarea of a quadrilateral

diagonals of a parallelogramdiagonals of a parallelogramdiagonals of a parallelogram

rectangular trapezoidrectangular trapezoidrectangular trapezoid

right‑angled triangleright‑angled triangleright‑angled triangle

the angle between the diagonals of the parallelogramthe angle between the diagonals of the parallelogramthe angle between the diagonals of the parallelogram

the angle between the sides of the trianglethe angle between the sides of the trianglethe angle between the sides of the triangle

trigonometric functionstrigonometric functionstrigonometric functions