Topicm476c868b38f0ffe2_1528449000663_0Topic

Calculation of numerical values of trigonometric expressions

Levelm476c868b38f0ffe2_1528449084556_0Level

Third

Core curriculumm476c868b38f0ffe2_1528449076687_0Core curriculum

VII. Trigonometry. The student:

4) uses formulas ${sin}^2\alpha +{cos}^2\alpha =1$, $tg\alpha =\frac{sin\alpha }{cos\alpha }$.

Timingm476c868b38f0ffe2_1528449068082_0Timing

45 minutes

General objectivem476c868b38f0ffe2_1528449523725_0General objective

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

Specific objectivesm476c868b38f0ffe2_1528449552113_0Specific objectives

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

2. Calculation of the values of other trigonometric functionstrigonometric functionstrigonometric functions when one trigonometric function is given.

3. Calculation of numerical values of trigonometric expressions.

Learning outcomesm476c868b38f0ffe2_1528450430307_0Learning outcomes

The student:

- calculates the values of other trigonometric functionstrigonometric functionstrigonometric functions when one trigonometric function is given,

- calculates numerical values of trigonometric expressions.

Methodsm476c868b38f0ffe2_1528449534267_0Methods

1. Incomplete sentences.

2. Case study.

Forms of workm476c868b38f0ffe2_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm476c868b38f0ffe2_1528450127855_0Introduction

Students, working in groups, using the method of incomplete sentences review their knowledge of trigonometric functionstrigonometric functionstrigonometric functions.

Task
Sentences that students must complete.

- The sine of the acute angleacute angleacute angle in a right‑angled triangleright‑angled triangleright‑angled triangle is called ...

- The cosine of the acute angle in a right‑angled triangle is called ...

- The tangent of the acute angleacute angleacute angle in a right‑angled triangleright‑angled triangleright‑angled triangle is called ...

- The value of the sine and the cosine of the acute angle belongs to the range of ...

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

Procedurem476c868b38f0ffe2_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to calculate the numerical values of trigonometric expressions.

Task
Students, working in pairs, analyze the material presented in the Interactive illustration. Can the values of other trigonometric functionstrigonometric functionstrigonometric functions be determined when one function is given? They hypothesize and formulate appropriate conclusions.

[Interactive illustration]

Conclusion:

- If we know the value of one trigonometric function, using the trigonometric properties we can determine the values of the other functions.m476c868b38f0ffe2_1527752263647_0- If we know the value of one trigonometric function, using the trigonometric properties we can determine the values of the other functions.

Using the information learned students solve the tasks on their own.

Task
Angle $\alpha$ is an acute angleacute angleacute angle, $\sin \alpha =0,4.$ Calculate the values of the other trigonometric functions of this angle.

Task
Angle $\alpha$ is an acute angle, $\cos \alpha =0,8.$ Calculate the values of the other trigonometric functionstrigonometric functionstrigonometric functions of this angle.

Task
Angle $\alpha$ is an acute angleacute angleacute angle, $tg\alpha =5.$ Calculate the values of the other trigonometric functions of this angle.

Discussion - Do we always have to use trigonometric function tables to calculate the value of a trigonometric expression? Students make hypotheses and formulate conclusions.

Conclusion:

- For any acute angle $\alpha$, the following equalities are true:m476c868b38f0ffe2_1527752256679_0- For any acute angle $\alpha$, the following equalities are true:

Using the information learned students solve the tasks on their own.

Task
Calculate the values of the expressions.

a) $(sin18°{)}^2+(sin72°{)}^2+3$

b) $sin{35}^{\circ }·cos{55}^{\circ }+cos{35}^{\circ }·sin{55}^{\circ }$

c) $\frac{{({(sin{60}^{\circ })}^2+{(cos{30}^{\circ })}^2)}^2-({(sin{20}^{\circ })}^2+{(sin{70}^{\circ })}^2)}{(sin{45}^{\circ }·cos{45}^{\circ })}$

Task
Check if there is an acute angleacute angleacute angle $\alpha$ that satisfies the given condition.

a) $\cos \alpha =4-\cos \alpha$

b) $6+3·sin\alpha =3$

c) $\sqrt{8}-3\cdot sin\alpha +5=8\cdot sin\alpha$

Task
Knowing, that $sin\alpha +cos\alpha =1,25$, calculate $sin\alpha ·cos\alpha .$

Students check the results of their work and the teacher explains any doubts.

An extra task:
Prove, that there is no acute $\alpha$ such that $\frac{\sin \alpha }{tg\alpha }+\cos \alpha =5.$

Lesson summarym476c868b38f0ffe2_1528450119332_0Lesson summary

Students do the revision exercisesand formulate conclusions to remember.

- If we know the value of one trigonometric function, using the trigonometric properties we can determine the values of the other functions.

- For any acute angleacute angleacute angle $\alpha$, the following equalities are true:

Selected words and expressions used in the lesson plan

acute angleacute angleacute angle

relations between trigonometric functions of the same anglerelations between trigonometric functions of the same anglerelations between trigonometric functions of the same angle

right‑angled triangleright‑angled triangleright‑angled triangle

trigonometric functionstrigonometric functionstrigonometric functions

trigonometric propertytrigonometric propertytrigonometric property