Topicma98d77a6808ea6a9_1528449000663_0Topic

Basic trigonometric identitiestrigonometric identitiestrigonometric identities

Levelma98d77a6808ea6a9_1528449084556_0Level

Third

Core curriculumma98d77a6808ea6a9_1528449076687_0Core curriculum

VII. Trigonometry. The student:

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

Timingma98d77a6808ea6a9_1528449068082_0Timing

45 minutes

General objectivema98d77a6808ea6a9_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 objectivesma98d77a6808ea6a9_1528449552113_0Specific objectives

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

2. Getting to know basic relationships between the sine, cosinecosinecosine and tangenttangenttangent.

3. Applying basic trigonometric identities in tasks.

Learning outcomesma98d77a6808ea6a9_1528450430307_0Learning outcomes

The student:

- knows basic relationships between the sinesinesine, cosine and tangenttangenttangent,

- applies basic trigonometric identitiestrigonometric identitiestrigonometric identities in tasks.

Methodsma98d77a6808ea6a9_1528449534267_0Methods

1. Educational game.

2. Situational analysis.

Forms of workma98d77a6808ea6a9_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionma98d77a6808ea6a9_1528450127855_0Introduction

A short test – recollecting the information about trigonometric functionstrigonometric functionstrigonometric functions. The students work individually. One by one, the teacher show flashcards with the following inscriptions: sinesinesine, cosinecosinecosine, tangenttangenttangent and later the values of functions for 30°, 45°, 60° angles. The students write down definitions of the functions and the measure of the angle for which the value of the function is given (e.g. sin $\alpha$ = 0,5, this $\alpha$ = ?). Having finished the activity, the teacher gives the correct answers and explains the doubts. The students self‑assess their results.

Procedurema98d77a6808ea6a9_1528446435040_0Procedure

The teacher informs the students that the aim of the class is getting to know the relationships between trigonometric functionstrigonometric functionstrigonometric functions of the same angle.

Task

Analyze carefully the material presented in the slideshow in which relations between trigonometric functions of the same angle were derived.

[Slideshow]

The students should notice that:

- For any acute angle $\alpha$ the relationships are true:

${\left(\sin \alpha \right)}^2+{\left(\cos \alpha \right)}^2=1$

and

$\mathrm{tg}\mathrm{}\alpha =\frac{\sin \alpha }{\cos \alpha }$

The teacher informs the students that the relationships above are called trigonometric identitiestrigonometric identitiestrigonometric identities. The first of them is often called Pythagorean trigonometric identitypythagorean trigonometric identityPythagorean trigonometric identity.

Discussion – is it possible to find all trigonometric functions of the given angle when only one of them is known? The students formulate hypotheses, check them and formulate an appropriate conclusion.

The conclusion:

- If the value of one trigonometric function is given, it is possible to find the values of the other trigonometric functionstrigonometric functionstrigonometric functions by applying trigonometric identities.

Working in groups, the students use the information to solve the tasks.

Task for group 1
Calculate the values of the other trigonometric functions of an acute angle if $\sin \alpha =\frac{1}{5}$.

Task for group 2
Calculate the values of the other trigonometric functions of an acute angle if $\cos \alpha =\frac{2}{3}$.

Task for group 3
Calculate the values of the other trigonometric functions of an acute angle if $\mathrm{tg}\mathrm{}\alpha =3$.

Having solved the tasks, the representatives of groups present their results and the method of obtaining them.

The teacher explains the doubts and assesses the students’ work.

Using the information the students solve the tasks individually.

Task
Write down in a simpler form.

a) $cos\alpha ·tg\alpha$

b) $sin\alpha ·{(cos\alpha )}^2+{(sin\alpha )}^3$

c) $\frac{1}{{(sin\alpha )}^2}·[1-{[cos\alpha ]}^2]$

Task
Calculate the value of $\frac{5\sin \alpha -4\cos \alpha }{\cos \alpha +5\sin \alpha }$ if you know that $tg\alpha =5$.ma98d77a6808ea6a9_1527752256679_0Calculate the value of $\frac{5\sin \alpha -4\cos \alpha }{\cos \alpha +5\sin \alpha }$ if you know that $tg\alpha =5$.

Task
Calculate $\mathrm{tg}\mathrm{}\alpha$, if $\alpha \in (0^{\circ },{90}^{\circ })$ and $4{(sin\alpha )}^2=3{(cos\alpha )}^2$.ma98d77a6808ea6a9_1527712094602_0Calculate $\mathrm{tg}\mathrm{}\alpha$, if $\alpha \in (0^{\circ },{90}^{\circ })$ and $4{(sin\alpha )}^2=3{(cos\alpha )}^2$.

Task
Calculate the value of${\left(\sin \alpha \right)}^3+{\left(\cos \alpha \right)}^3$, if you know that $sin\alpha +cos\alpha =0,8$.

Task
Check if there is angle $\alpha \in (0°,90°)$, which fulfills the following conditions.

a) $\sin \alpha =\frac{\sqrt{3}}{5}$ and $\cos \alpha =\frac{4}{5}$

b) $\cos \alpha =\frac{1}{3}$ and $\mathrm{tg}\mathrm{}\alpha =2\sqrt{2}$

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

Extra task:
Angle $\alpha$ is an acute angle and $\sin \alpha +\cos \alpha =\frac{7}{5}$. Calculate ${\left(\sin \alpha \right)}^4+{\left(\cos \alpha \right)}^4$.

Lesson summaryma98d77a6808ea6a9_1528450119332_0Lesson summary

The students do the consolidation tasks.

They formulate the conclusions to memorize.

- If the value of one trigonometric function is given, it is possible to find the values of the other trigonometric functionstrigonometric functionstrigonometric functions by applying trigonometric identitiestrigonometric identitiestrigonometric identities.

Selected words and expressions used in the lesson plan

cosinecosinecosine

pythagorean trigonometric identitypythagorean trigonometric identitypythagorean trigonometric identity

sinesinesine

tangenttangenttangent

trigonometric functionstrigonometric functionstrigonometric functions

trigonometric identitiestrigonometric identitiestrigonometric identities