Topicm998d5edd3cbba42a_1528449000663_0Topic

The values of trigonometric functions for selected acute angles.

Levelm998d5edd3cbba42a_1528449084556_0Level

Third

Core curriculumm998d5edd3cbba42a_1528449076687_0Core curriculum

VII. Trigonometry. The student:

1) applies the definitions of the sine, cosine and tangent function of angles between 0° and 180°, in particular finds the value of trigonometric functions for angles 30°, 45°, 60°.

Timingm998d5edd3cbba42a_1528449068082_0Timing

45 minutes

General objectivem998d5edd3cbba42a_1528449523725_0General objective

Interpretation and the use of information presented both in a mathematical and popular science texts also using graphs, diagrams and tables.

Specific objectivesm998d5edd3cbba42a_1528449552113_0Specific objectives

1. Communication in English, developing mathematical, IT and basic scientific and technical competence, developing learning skills.

2. Calculating the value of trigonometric functions for selected acute angles.

3. Calculating the lengths of sides and measures of angles in a right triangle.

Learning outcomesm998d5edd3cbba42a_1528450430307_0Learning outcomes

The student:

- calculates the value of trigonometric functions for selected acute angles,

- calculates the lengths of sides and measures of angles in a right triangle.

Methodsm998d5edd3cbba42a_1528449534267_0Methods

1. Task competition.

2. Situational analysis.

Forms of workm998d5edd3cbba42a_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm998d5edd3cbba42a_1528450127855_0Introduction

The students take part in a short task competition. Thea im of the competition is recollecting the definition of trigonometric functionstrigonometric functionstrigonometric functions of acute angles in a right triangle.

Competition task
A right triangle with legs 3 cm and 8 cm long is given. Calculate.

a) The value of the sinesinesine function for both acute angles of this triangle.

b) The value of the cosinecosinecosine function for both acute angles of this triangle.

c) The value of the tangenttangenttangent function for both acute angles of this triangle.

Having solved the tasks, the teacher assesses the students’ work and explains the doubts. The fastest students are awarded with “pluses”.

Procedurem998d5edd3cbba42a_1528446435040_0Procedure

The teacher informs the students that the aim of this class is getting to know the value of trigonometric functionstrigonometric functionstrigonometric functions of 30°, 45° and 60° angles.

Discussion – does a right triangle with one angle measuring 30° or 60° exist? If so, what is the relationship between the lengths of the sides in this triangle? The students formulate hypotheses and conclusions.

Conclusions that should be formed by the students:

- The altitude divides an equilateral triangleequilateral triangleequilateral triangle into two right triangles. The measures of acute angles in each of the triangles equals 30° and 60°.

- If the length of a side in an equilateral triangleequilateral triangleequilateral triangle is 1, in the formed right triangle the lengths of the legs are $\frac{1}{2}$ and $\frac{\sqrt{3}}{2}$, the length of the hypotenuse is 1.

Using the conclusion, the students calculate the value of trigonometric functionstrigonometric functionstrigonometric functions of 30° and 60° angles.

Task
They make a table containing the results.

[Table 1]

Task
The students check the correctness of the table analyzing the material presented in the Interactive illustration.

[Interactive illustration 1]

Discussion – does a right triangle with one angle measuring 45° exist? If so, what is the relationship between the lengths of the sides in this triangle? The students formulate hypotheses and conclusions.

Conclusions that should be formed by the students:

- The diagonal divides a squaresquaresquare into two equilateral triangles. The measures of acute angles in each of the triangles equal 45°.

- If the length of the side of the squaresquaresquare equals 1, the legs of the triangles also equal 1 and the hypotenuse $\sqrt{2}$.

Using the conclusion, the students calculate the value of trigonometric functionstrigonometric functionstrigonometric functions of 45° angle.

Task
They make a table containing the results.

[Table 2]

Task
The students check the correctness of the table analyzing the material presented in the Interactive illustration.

[Interactive illustration 2]

Using the information the students solve the tasks individually .

Task
Calculate.

a) 5 ∙ cos60° ∙ sin30° - cos30° ∙ sin60°

b) (sin45° + tg45°) ∙ (3∙sin60° – tg60°)

c) (cos45° – cos30°) ∙ (cos45° + cos30°)

d) (sin60° + cos30°)Indeks górny 2² – (sin30° + cos60°)Indeks górny 2²

Task
In an equilateral triangleequilateral triangleequilateral triangle a leg has the length of 30 cm, and the base angle measures 30°. Calculate the lengths of all the altitudes of this triangle.

Task
The diagonal of a rectangle measures 10 cm and one of the angles between the diagonals measures 60°. Calculate the lengths of the sides of the rectangle.

Task
A ladder whose length is 8 m leans against the wall of a house at the angle measuring 30°. Does the ladder reach a window situated 6 m above the ground?

Having solved all the tasks the students present their results. The teacher assesses their work and explains the doubts.

An extra task:
The altitude of a quadrangular right regular prism equals $6\sqrt{2}$, and the area of its base is 36. Calculate the measure of the angle between the diagonal of the prism and its base.m998d5edd3cbba42a_1527752263647_0An extra task:
The altitude of a quadrangular right regular prism equals $6\sqrt{2}$, and the area of its base is 36. Calculate the measure of the angle between the diagonal of the prism and its base.

Lesson summarym998d5edd3cbba42a_1528450119332_0Lesson summary

The students do the consolidation tasks.

They formulate the conclusions to memorize.

The table of the values of trigonometric functionstrigonometric functionstrigonometric functions of 30°, 45°, 60° angles.

[Table 3]

Selected words and expressions used in the lesson plan

cosinecosinecosine

equilateral triangleequilateral triangleequilateral triangle

sinesinesine

squaresquaresquare

tangenttangenttangent

trigonometric functionstrigonometric functionstrigonometric functions