Topicmd74a1cc86b510e61_1528449000663_0Topic

Trigonometric functions in the right‑angled triangle – doing exercises

Levelmd74a1cc86b510e61_1528449084556_0Level

Third

Core curriculummd74a1cc86b510e61_1528449076687_0Core curriculum

VII. Trygonometry.

The basic level. The student:

1. Applies definitions of functions: sine, cosine, tangent and cotangent for angles from 0° to 180°, especially identifies values of trigonometric functions for angles 30°, 45°, 60°;

2. Finds approximate values of trigonometric functions using tables or the calculator;

3. Finds the approximate value of an angle if the value of the trigonometric function is given;

6. Calculates angles of triangles and lengths of its sides while having appropriate data given (solves triangles).

Timingmd74a1cc86b510e61_1528449068082_0Timing

45 minutes

General objectivemd74a1cc86b510e61_1528449523725_0General objective

Choosing and creating mathematical models to solve practical and theoretical problems.

Specific objectivesmd74a1cc86b510e61_1528449552113_0Specific objectives

1. Reading approximate values of trigonometric functions (using tables of the calculator) and approximate values of angles (based on given values of trigonometric functions).

2. Calculating angles and sides of triangles while having appropriate data given (“solving triangles”).

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesmd74a1cc86b510e61_1528450430307_0Learning outcomes

The Students:

- reads approximate values of trigonometric functions (using tables of the calculator) and approximate values of angles (based on given values of trigonometric functions),

- calculates angles and sides of triangles while having appropriate data given (“solving triangles”).

Methodsmd74a1cc86b510e61_1528449534267_0Methods

1. Situational analysis.

2. JIGSAW.

Forms of workmd74a1cc86b510e61_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionmd74a1cc86b510e61_1528450127855_0Introduction

Students revise definitions of function: sine, cosine, tangent and cotangent for a right‑angled triangle
and rules of reading values of trigonometric functions from mathematical tables.

Proceduremd74a1cc86b510e61_1528446435040_0Procedure

Students work individually, using computers. They open the slideshow and observe how we can calculate values of trigonometric functions of acute angles in right‑angled triangles 90°, 45°, 45° and 90°,60°, 30°.

[Slideshow]

After having completed the exercise, students fill in the table:

[Table 1]

Students use obtained information in exercises, using the JIGSAW method.

The teacher divides students into 3 persons groups. Each member of the group gets different task from
the tasks below. After solving the tasks, students gather in groups that were doing the same task. They discuss the solutions and clarify any doubts. Then, they return to the initial groups and present
the solutions to other members.

Task 1

Calculate:
a) $(\cos 60°+\sin 45°)(\cos 60°-\sin 45°)$
b) $(\mathrm{ctg} 60°·\cos 45°·\sin 30°):(\mathrm{ctg} 30°·\sin 45°)$md74a1cc86b510e61_1527752263647_0Calculate:
a) $(\cos 60°+\sin 45°)(\cos 60°-\sin 45°)$
b) $(\mathrm{ctg} 60°·\cos 45°·\sin 30°):(\mathrm{ctg} 30°·\sin 45°)$

Task 2

Identify acute angles α and β in right‑angled triangles ABC knowing that:

a) the length of the hypotenusehypotenusehypotenuse is 2 and the cathetuscathetuscathetus opposite to the angle β is equal to $\sqrt{2}$;

b) the cathetus opposite to the angle α is equal to $\sqrt{6}$, the cathetus opposite to the angle β is equal to $\sqrt{2}$.

Task 3

In the isosceles triangle KLM, $\left|MK\right|=\left|LM\right|=15$ cm, and $|\sphericalangle MKL|=30°.$Calculate lengths of all altitudes of this triangle.md74a1cc86b510e61_1527752256679_0In the isosceles triangle KLM, $\left|MK\right|=\left|LM\right|=15$ cm, and $|\sphericalangle MKL|=30°.$Calculate lengths of all altitudes of this triangle.

An extra task:

Two longest sides of the right‑angled triangles are equal to $\sqrt{30}$ and $\sqrt{45}$. Identify measures of acute angles of this triangle.

Lesson summarymd74a1cc86b510e61_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

[Table 2]

Selected words and expressions used in the lesson plan

cathetuscathetuscathetus

cosine of the anglecosine of the anglecosine of the angle

cotangent of the anglecotangent of the anglecotangent of the angle

hypotenusehypotenusehypotenuse

sine of the anglesine of the anglesine of the angle

tangent of the angletangent of the angletangent of the angle