Basic trigonometric identities

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Podstawowe tożsamości trygonometryczne

Learning objectives

You will develop the following competences: communicating in English, mathematical, IT and basic scientific and technical competence, your learning skills.

You will get to know basic relationships between the sine, cosine and tangent.

You will get to know the method of applying basic trigonometric identities in tasks.

Learning effect

You will learn the basic relationships between the sine, cosine and tangent.
You will learn to apply basic trigonometric identities in tasks.

Recollect the information about trigonometric functionstrigonometric functionstrigonometric functions and the table with the values of functions for 30°, 45°, 60° angles.

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

Task 1

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

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nagranie abstraktu

You should notice that:

for any acute angle $α$ the relationships are true:
${(sin α)}^{2} + {(cos α)}^{2} = 1$
and
$t g α = \frac{sin α}{cos α}$

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nagranie abstraktu

Find the names of the relationships above in any available sources.

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

Think if it is possible to find all trigonometric functionstrigonometric functionstrigonometric functions of the given angle when only one of them is known. Check your hypotheses and formulate an appropriate conclusion.

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nagranie abstraktu

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 identitiestrigonometric identitiestrigonometric identities.

Use the information to solve the tasks.

Task 2

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nagranie abstraktu

Calculate the values of the other trigonometric functionstrigonometric functionstrigonometric functions of an acute angle if

$sin α = \frac{1}{5} .$

Task 3

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nagranie abstraktu

Calculate the values of the other trigonometric functionstrigonometric functionstrigonometric functions of an acute angle if

$t g α = 3 .$

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nagranie abstraktu

Using the information, solve the tasks.

Task 4

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nagranie abstraktu

Write down in a simpler form:

a) $cos α \cdot t g α$

b) $sin α \cdot {(cos α)}^{2} + {(sin α)}^{3}$

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

Task 5

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nagranie abstraktu

Calculate the value of $\frac{5 sin α - 4 cos α}{cos α + 5 sin α}$ if you know that $t g α = 5$ .

Task 6

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nagranie abstraktu

Calculate $t g α$ if $α \in (0^{\circ}, 90^{\circ})$ and $4 {(\sin α)}^{2} = 3 {(\cos α)}^{2}$ .

Task 7

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nagranie abstraktu

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

Task 8

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nagranie abstraktu

Check if there is angle $α \in (0^{\circ}, 90^{\circ})$ which fulfills the following conditions:

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

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

Task 9

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nagranie abstraktu

Extra task:

Angle $α$ is an acute angle and $\sin α + \cos α = \frac{7}{5}$ . Calculate ${(\sin α)}^{4} + {(\cos α)}^{4}$ .

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nagranie abstraktu

Having finished all the tasks, do the consolidation tasks. 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.

Exercises

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Exercise 1

If for an angle $β$ there is equality $(s i n β + c o s β)^{2} = 1, 4$ , product $\sin β \cdot \cos β$ equals.

Exercise 2

For which acute angle $α$ numbers $cos α, \frac{1}{t g α}, \frac{1}{sin α}$ are three consecutive terms of a geometric sequence?

Exercise 3

Describe in English the relationships between trigonometric functions of the same acute angle.

Exercise 4

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Indicate which pairs of expressions or words are translated correctly.

funkcje trygonometryczne - trigonometric functions
tożsamości trygonometryczne - trigonometric identities
sinus - sine
cosinus - cosine
tożsamości trygonometryczne - trigonometric functions
funkcje trygonometryczne - pythagorean trigonometric identity

zadanie

Source: GroMar, licencja: CC BY 3.0.

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Match Polish terms with their English equivalents.

cosinus
trigonometric identities
jedynka trygonometryczna
trigonometric functions
pythagorean trigonometric identity
tożsamości trygonometryczne
cosine
sine
sinus
funkcje trygonometryczne

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

cosine

cosinus

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wymowa w języku angielskim: cosine

pythagorean trigonometric identity

jedynka trygonometryczna

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wymowa w języku angielskim: Pythagorean trigonometric identity

sine

sinus

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wymowa w języku angielskim: sine

tangent

tangens

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wymowa w języku angielskim: tangent

trigonometric functions

funkcje trygonometryczne

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wymowa w języku angielskim: trigonometric functions

trigonometric identities

tożsamości trygonometryczne

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wymowa w języku angielskim: trigonometric identities

Keywords

cosinecosinecosine

pythagorean trigonometric identitypythagorean trigonometric identitypythagorean trigonometric identity

sinesinesine

trigonometric functionstrigonometric functionstrigonometric functions

trigonometric identitiestrigonometric identitiestrigonometric identities

Scenariusz

Exercises

Glossary

Keywords