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Vectors. Trigonometric functions

Trigonometric functions for acute angles

Two right angled triangles are similar if a pair of their acute angles are equal, therefore the ratio of the sides of any right angled triangle is uniquely determined by any of its acute angles. The trigonometric functions for acute angles are such ratios depending solely on the angle.

In the definition we make use of the notation of Figure 49.

DEFINITION: \latex{\sin \alpha =\frac{a}{c};\cos \alpha =\frac{b}{c};\tan \alpha =\frac{a}{b};\cot \alpha =\frac{b}{a}.}

The following equivalences are easy consequences of the definitions.

\latex{\tan \alpha =\frac{\sin \alpha }{\cos \alpha };\cot \alpha =\frac{\cos \alpha }{\sin \alpha };\tan \alpha =\frac{1}{\cot \alpha };}

\latex{\sin \alpha =\cos (90°-\alpha );\cos \alpha =\sin (90°-\alpha );}

\latex{\tan \alpha =\cot (90°-\alpha );\cot \alpha =\tan (90°-\alpha ).}

By using the Pythagorean theorem the so-called Pythagorean identity can be obtained:

\latex{\sin ^{2}\alpha +\cos ^{2}\alpha =1.}

The table below shows the values of the trigonometric functions for some notable acute angles.

\latex{ 30 }°

\latex{ 45 }°

\latex{ 60 }°

sin

cos

tan

cot

\latex{\frac{1}{2}}

\latex{\frac{\sqrt{2} }{2}}

\latex{\frac{\sqrt{3} }{2}}

\latex{\frac{\sqrt{2} }{2}}

\latex{\frac{1}{2}}

\latex{\frac{\sqrt{3} }{3}}

\latex{1}

\latex{\sqrt{3}}

\latex{1}

\latex{\frac{\sqrt{3} }{3}}

Vectors

I. The concept of vectors

DEFINITION: By distinguishing the two endpoints of a segment, calling one its initial-, the other one its terminal point, a directed segment is obtained. (Figure 51)

DEFINITION: Any directed segment uniquely determines a vector.

(Notation:\latex{\overrightarrow{AB},\overrightarrow{v},v.} )

The directed segments pointing from \latex{ A } to \latex{ B } and from \latex{ A’ } to \latex{ B’ } determines the same vector if there exists a translation which maps \latex{ A } to \latex{ A’ } and \latex{ B } to \latex{ B’ }. (Figure 52)

DEFINITION: The length of the directed segment determining a vector is called the absolute value of the vector.

(Notation:\latex{|\overrightarrow{AB}|,|\overrightarrow{a}|,|a|.} )

DEFINITION: The vector with an absolute value of \latex{ 0 } is called the zero vector.

(Notation:\latex{\overrightarrow{0}} or \latex{0}.)

DEFINITION: Two vectors are parallel, if the lines their directed segments incident to are parallel.

DEFINITION: The vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} are said to be unidirectional, if they are parallel and they point in the same direction, that is, by taking the directed segments representing \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} with a common initial point they are on the same line and their endpoints lie on the same ray determined by the initial point. (Figure 53)

The direction of the \latex{\overrightarrow{0}} is arbitrary, it is viewed as unidirectional to any other vector.

DEFINITION: Two vectors are in the opposite directions if they are parallel but not unidirectional. (Figure 54)

(The opposite of \latex{\overrightarrow{a}} is denoted by \latex{-\overrightarrow{a}}.) (Figure 55)

DEFINITION: If two vectors are in the opposite directions and have the same absolute value then they are called opposite vectors.

DEFINITION: Two vectors are said to be equal if they are unidirectional and they have the same absolute value.

II. Operations with vectors

SUM OF VECTORS

DEFINITION: The sum of vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} is the vector of the translation which can be replaced by consecutive translations determined by \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}}. (Notation:\latex{\overrightarrow{a}+\overrightarrow{b}.})

Geometrically two vectors can be added using the triangle law or parallelogram law.
(Figure 56)

\latex{\overrightarrow{a}+\overrightarrow{b}}

\latex{\overrightarrow{a}}

\latex{\overrightarrow{b}}

Figure 56

Addition of vectors is

(1) a commutative operation, that is,

\latex{\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}}

(2) an associative operation (Figure 57), that is

\latex{(\overrightarrow{a}+\overrightarrow{b} )+\overrightarrow{c}=\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c} )=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}.}

MULTIPLICATION OF VECTOR BY NUMBER

DEFINITION:

If \latex{\overrightarrow{a}\neq \overrightarrow{0}} and \latex{\alpha} is an arbitrary real number, then \latex{\alpha \times \overrightarrow{a}} is a vector whose absolute value is \latex{|\alpha |\times |\overrightarrow{a}|} and for \latex{\alpha \gt 0} it is unidirectional with \latex{\overrightarrow{a}} , for \latex{\alpha \lt 0} it is in the opposite direction to \latex{\overrightarrow{a}.}
If \latex{\overrightarrow{a}= \overrightarrow{0}} then \latex{\alpha \times \overrightarrow{a}= \overrightarrow{0}} for any real number \latex{\alpha}.

Multiplication with a real number (scalar) satisfies the following identities:

(1) \latex{\alpha\times \overrightarrow{a}+\beta \times \overrightarrow{a}=(\alpha +\beta )\times \overrightarrow{a};}

(2) \latex{\alpha \times (\beta \times \overrightarrow{a} )=(\alpha \times \beta )\times \overrightarrow{a};}

(3) \latex{\alpha \times (\overrightarrow{a}+\overrightarrow{b} )=\alpha \times \overrightarrow{a}+\alpha \times \overrightarrow{b}.}

DIFFERENCE OF VECTORS

DEFINITION: The difference of the vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} is the vector \latex{\overrightarrow{a}+(-\overrightarrow{b} ).} (Notation: \latex{\overrightarrow{a}-\overrightarrow{b}} ) (Figure 58)

Geometrically the difference of two vectors can be constructed as seen in Figure 59.

THE SCALAR PRODUCT OF TWO VECTORS

DEFINITION: The scalar product of the vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} is \latex{\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}|\times |\overrightarrow{b}|\times \cos \alpha ,}

where \latex{\alpha} is the angle between representations of the two vectors with a common initial point \latex{(0°\leq \alpha \leq 180°)} .

Properties of the scalar product:

it is commutative, that is, \latex{\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{a};}
for an arbitrary real number \latex{\lambda,(\lambda\times \overrightarrow{a} )\times \overrightarrow{b}=\lambda\times (\overrightarrow{a}\times \overrightarrow{b} )=\overrightarrow{a}\times (\lambda\times \overrightarrow{b} );}
\latex{(\overrightarrow{a}+\overrightarrow{b} )\times \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{c}} (distributivity);
\latex{\overrightarrow{a}\times \overrightarrow{a}=\overrightarrow{a^{2} }=|\overrightarrow{a}|^{2},} thus \latex{|\overrightarrow{a}|=\sqrt{\overrightarrow{a^{2} } }.}

THEOREM: The scalar product of two vectors is \latex{ 0 } if and only if the two vectors are perpendicular.

III. Division of vectors into components

THEOREM: If \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} are parallel vectors not equal to the zero vector, then there exists exactly one real number \latex{\alpha} not equal to \latex{ 0 }, for which \latex{\overrightarrow{b}=\alpha \times \overrightarrow{a}.}

If \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} are point in the same direction:
\latex{\alpha =\frac{|\overrightarrow{b}| }{|\overrightarrow{a}| }\gt 0,}

if they are point in opposite directions:
\latex{\alpha =\frac{|\overrightarrow{b}| }{|\overrightarrow{a}| }\lt 0.}

DEFINITION: If \latex{\overrightarrow{v}=\alpha \times \overrightarrow{a}+\beta \times \overrightarrow{b}} holds for planar vectors \latex{\overrightarrow{a},\overrightarrow{b}} and \latex{\overrightarrow{v}} with some real numbers \latex{\alpha} and \latex{\beta}, then \latex{\overrightarrow{v}} is said to be a linear combination of \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}.}

THEOREM: If the non-parallel, non-zero vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} are given, then any vector \latex{\overrightarrow{v}} of the plane determined by the two vectors can be expressed exactly one way as a linear combination of \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}}, that is there exist uniquely determined real number \latex{\alpha} and \latex{\beta} depending only on \latex{\overrightarrow{v}} for which

\latex{\overrightarrow{v}=\alpha \times \overrightarrow{a}+\beta \times \overrightarrow{b}.} (Figure 60)

The vectors \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}} from the theorem are usually called basis vectors.

*The \latex{ 3 }-dimensional equivalent of the above theorem is also true.

*THEOREM: If the pairwise non-parallel, non-zero vectors \latex{\overrightarrow{a},\overrightarrow{b}} and \latex{\overrightarrow{c}} are given which satisfy that none of them is incident to the plane defined the other two, then any vector \latex{\overrightarrow{v}} can be expressed exactly one way as a linear combination of \latex{\overrightarrow{a},\overrightarrow{b}} and \latex{\overrightarrow{c}} hat is, there exists uniquely determined real numbers \latex{\alpha ,\beta ,\gamma} depending only on \latex{\overrightarrow{v}} for which

\latex{\overrightarrow{v}=\alpha \times \overrightarrow{a}+\beta \times \overrightarrow{b}+\gamma \times \overrightarrow{c}.} (Figure 61)

IV. Vectors in the coordinate system

DEFINITION: The position vector of the point \latex{P(x; y)} in the Cartesian coordinate system is the vector pointing to \latex{ P } from the origin. (Figure 62)

DEFINITION: The coordinates of a vector in the Cartesian coordinate system equal the coordinates of the terminal point of its representation with initial point being the origin. Notation:\latex{\overrightarrow{a}(a_{1};a_{2} )}. (Figure 62)

It follows from the two definitions above that the coordinates of an arbitrary point of the coordinate system equal that of its position vector.

The basis vectors of the Cartesian coordinate system are the vectors
\latex{i(1;0)} and \latex{j(0;1)} with unit length. (Figure 63)

With the help of the vectors \latex{ i } and \latex{ j } we can define the coordinates of a vector in a different way.

DEFINITION: If in the Cartesian coordinate system we have that

\latex{\overrightarrow{a}=a_{1}\times i+a_{2}\times j,}

then the first coordinate of \latex{\overrightarrow{a}} is \latex{a_{1}}, and its second is \latex{a_{2}.}

The two definitions given for the coordinates of a vector are equivalent, that is, either one can be viewed as the definition, the other one follows as a property.

THE COORDINATES OF THE SUM OF VECTORS

If vectors \latex{\overrightarrow{a}(a_{1};a_{2} )} and \latex{\overrightarrow{b}(b_{1};b_{2} )} are given, then

\latex{\overrightarrow{a}+\overrightarrow{b}=(a_{1}i+a_{2}j )+(b_{1}i+b_{2}j )=(a_{1}+b_{1} )\times i+(a_{2}+b_{2} )\times j,}

that is, the first coordinate of the sum vector is the sum of the first coordinates of the two vectors, and the second coordinate of it is the sum of the second coordinates.

THE COORDINATES OF THE DIFFERENCE OF VECTORS

If vectors \latex{\overrightarrow{a}(a_{1};a_{2} )} and \latex{\overrightarrow{b}(b_{1};b_{2} )} are given, then

\latex{\overrightarrow{a}-\overrightarrow{b}=(a_{1}i+a_{2}j )-(b_{1}i+b_{2}j )=(a_{1}+b_{1} )\times i+(a_{2}+b_{2} )\times j,}

that is, the first coordinate of the difference vector is the difference of the corresponding first coordinates of the two vectors, and the second coordinate of it is the difference of the corresponding second coordinates.

THE COORDINATES OF THE MULTIPLE OF A VECTOR

Let the vector \latex{\overrightarrow{a}(a_{1};a_{2} )} and the real number \latex{\lambda} be given. Then

\latex{\lambda\times \overrightarrow{a}=\lambda\times (a_{1}i+a_{2}j )=(\lambda\times a_{1} )\times i+(\lambda\times a_{2} )\times j,}

that is, the coordinates of the \latex{\lambda}-times multiple of a vector are the \latex{\lambda}-times multiples of the coordinates.

THE SCALAR PRODUCT OF TWO VECTORS USING COORDINATES

The scalar product of the vectors \latex{\overrightarrow{a}(a_{1};a_{2} )} and \latex{\overrightarrow{b}(b_{1};b_{2} )} is

\latex{\overrightarrow{a}\times \overrightarrow{b}=a_{1}\times b_{1}+a_{2}\times b_{2}.}

Note: This formula follows easily using the distributivity of scalar product and using \latex{i\times i=i^{2}=1,j\times j=j^{2}=1} and \latex{i\times j=0.}

An easy application of the formula above for scalar product yields that the absolute value of the vector \latex{\overrightarrow{a}(a_{1};a_{2} )} is

\latex{|\overrightarrow{a}|=\sqrt{\overrightarrow{a^{2} } }=\sqrt{a_{1}^{2}+a_{2}^{2} } .}

Extension of the definitions of trigonometric functions

I. Definitions, basic properties

DEFINITION: The direction angle of the unit vector \latex{\overrightarrow{e}} in the coordinate system is the signed angle of the rotation which maps the base vector \latex{ i } to \latex{\overrightarrow{e}}. (Figure \latex{ 64 })

Let the coordinates of the unit vector \latex{\overrightarrow{e}} with direction angle \latex{\alpha} and \latex{e_{1}} and \latex{e_{2}}, that is \latex{\overrightarrow{e}=e_{1}\times i+e_{2}\times j.} (Figure 65)

DEFINITION: \latex{e_{1}=\cos \alpha} and \latex{e_{2}=\sin \alpha}, the cosine of the direction angle \latex{\alpha} is the first coordinate of the unit vector \latex{\overrightarrow{e}} with direction angle \latex{\alpha} while its sine is the second coordinate of said vector. (Figure 66)

\latex{\overrightarrow{e}(e_{1};e_{2} )}

\latex{e_{1} \times i}

\latex{\alpha}

\latex{e_{2} \times j}

\latex{i}

\latex{j}

\latex{ x }

\latex{ y }

Figure 65

\latex{\overrightarrow{e}(\cos \alpha ;\sin \alpha )}

\latex{\sin \alpha \times j}

\latex{\alpha}

\latex{\cos \alpha \times i}

\latex{i}

\latex{j}

\latex{ x }

\latex{ y }

Figure 66

Notes:

As it can be seen on Figure \latex{ 67 }, the definition above coincides with the one previously given for acute angles.
Direction angle can be measured in degrees or in radian. The measure corresponding to real numbers is radian, thus the definition above gives the cosine and sine of arbitrary real number, therefore it is also a definition of the functions cosine and sine.
It follows from the definition of direction angle that

\latex{\sin \alpha =\sin (\alpha +k\times 2\pi )} and \latex{\cos \alpha =\sin (\cos +k\times 2\pi )}
for arbitrary integer \latex{ k }. (Both the sine and cosine functions are \latex{2\pi}-periodic.)

DEFINITION:

\latex{\tan \alpha =\frac{\sin \alpha }{\cos \alpha };} \latex{(\alpha \neq \frac{\pi }{2}+k\times \pi ,k\in \Z ),}

\latex{\cot \alpha =\frac{\cos \alpha }{\sin \alpha }; } \latex{(\alpha \neq k\times \pi ,k\in \Z).}

Notes:

The general definition of tangent and cotangent also coincide with the one given for acute angles.
The tangent and cotangent of the real number a can be illustrated well using the tangent lines to the points \latex{(1; 0)} and \latex{(0; 1)} of the unit circle. (Figure \latex{ 68 })
It follows from the definitions that

\latex{\tan \alpha =\tan (\alpha +k\times \pi )} and \latex{\cot (\alpha +k\times \pi )}
for arbitrary integer \latex{ k } and any real number \latex{\alpha} for which the trigonometric
functions are defined. (Both the sine and cosine functions are \latex{\pi}-periodic.)

II. Identities

FOR SINE AND COSINE

\latex{\alpha} is an arbitrary real number (direction angle measured in radian):

\latex{\sin ^{2}\alpha +\cos ^{2}\alpha =1,}

\latex{\sin \left(\frac{\pi }{2}-\alpha \right)=\cos \alpha ,}

\latex{\cos \left(\frac{\pi }{2}-\alpha \right)=\sin \alpha ,}

\latex{\sin \left(\alpha +\frac{\pi }{2} \right)=\cos \alpha ,}

\latex{\cos \left(\alpha +\frac{\pi }{2} \right)=-\sin \alpha ,}

\latex{\sin (\pi -\alpha )=\sin \alpha,}

\latex{\cos (\pi -\alpha )=-\cos \alpha,}

\latex{\sin (\alpha +\pi )=-\sin \alpha ,}

\latex{\cos (\alpha +\pi )=-\cos \alpha ,}

\latex{\sin (-\alpha )=-\sin\alpha ,}

\latex{\cos (-\alpha )=\cos \alpha.}

FOR TANGENT AND COTANGENT

\latex{\alpha \neq k\times \frac{\pi }{2} (k\in \Z):}

\latex{\tan \alpha =\frac{1}{\cot \alpha },}

\latex{\tan \left(\frac{\pi }{2}-\alpha \right)=\cot \alpha ,}

\latex{\cot \left(\frac{\pi }{2}-\alpha \right)=\tan \alpha ,}

\latex{\tan \left(\alpha +\frac{\pi }{2} \right)=-\cot \alpha ,}

\latex{\cot \left(\alpha +\frac{\pi }{2} \right)=-\tan \alpha ,}

\latex{\tan (-\alpha )=-\tan \alpha ,}

\latex{\cot (-\alpha )=-\cot \alpha .}

ADDITION THEOREMS

For arbitrary real numbers \latex{\alpha} and \latex{\beta} the following addition formulae are satisfied by the functions sine and cosine:

\latex{\cos (\alpha -\beta )=\cos \alpha \times \cos \beta +\sin \alpha \times \sin \beta ,}

\latex{\cos (\alpha +\beta )=\cos \alpha \times \cos \beta -\sin \alpha \times \sin \beta ,}

\latex{\cos 2\alpha =\cos ^{2}\alpha -\sin ^{2}\alpha ,}

\latex{\sin (\alpha -\beta )=\sin \alpha \times \cos \beta -\cos \alpha \times \sin \beta ,}

\latex{\sin (\alpha +\beta )=\sin \alpha \times \cos \beta +\cos \alpha \times \sin \beta ,}

\latex{\sin 2\alpha =2\times \sin \alpha \times \cos \alpha .}

*The following factorization formulae are satisfied as well:

\latex{\sin \alpha +\sin \beta =2\times\sin \frac{\alpha +\beta }{2}\times \cos \frac{\alpha -\beta }{2},}

\latex{\sin \alpha -\sin \beta =2\times\cos \frac{\alpha +\beta }{2}\times \sin \frac{\alpha -\beta }{2},}

\latex{\cos \alpha +\cos \beta =2\times\cos \frac{\alpha +\beta }{2}\times \cos \frac{\alpha -\beta }{2},}

\latex{\cos \alpha -\cos \beta =-2\times\sin\frac{\alpha +\beta }{2}\times \sin \frac{\alpha -\beta }{2},}

For every real numbers \latex{\alpha} and \latex{\beta}, for which the expressions below are defined, the following addition formulae are satisfied by the tangent function:

\latex{\tan (\alpha -\beta )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \times \tan \beta },}

\latex{\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \times \tan \beta },}

\latex{\tan 2\alpha =\frac{2\times \tan \alpha }{1-\tan ^{2}\alpha }.}

Exercises

{{exercise_number}}. The angle between the leaning tower of Pisa and the ground is approximately \latex{ 85 }º. From how high had that item been dropped which landed \latex{ 3\,m } away from the bottom of the tower?

{{exercise_number}}. In a theater, the enclosed angle of the stairs leading to the top of a \latex{ 1.5\,m } high podium and the stage is \latex{ 10 }º. How far is the start of the stairs from the edge of the podium?

{{exercise_number}}. A ski elevator has a length of \latex{ 1,200\,m } and its elevation is \latex{ 510\,m }. Compute the angle of elevation.

{{exercise_number}}. Compute the trigonometric functions for the acute angle \latex{\alpha} without computing \latex{\alpha} itself using

\latex{\cos \alpha =0.8;}

\latex{\sin \alpha =\frac{3}{5};}

\latex{\tan \alpha =2.1;}

\latex{\cot \alpha =\sqrt{5}-2.}

{{exercise_number}}. Divide the side \latex{ BC } of the triangle \latex{ ABC } into \latex{ 6 } equal parts and let \latex{\overrightarrow{AB}=\overrightarrow{b},\overrightarrow{AC}=\overrightarrow{c}.} Express the vectors pointing from \latex{ A } to the dividing points as a linear combination of \latex{\overrightarrow{b}} and \latex{\overrightarrow{c}}.

{{exercise_number}}. The regular tetrehedron \latex{ OABC } is given. Let \latex{\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b}} and \latex{\overrightarrow{OC}=\overrightarrow{c}:} those with initial point being \latex{ O } and terminal point being

a side bisector of the triangle \latex{ ABC };

the centroid of the triangle \latex{ ABC }.

{{exercise_number}}. The position vectors of the vertices of a convex quadrilateral using a given point \latex{ O } in the plane of the quadrilateral as origin are \latex{\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}.} in a positive order. Using these vectors, express the following position vectors:

the intersection of the segments joining the midpoints of the opposite sides;

the midpoints of the diagonals;

the segment determined by the midpoints of the diagonals.

What observation can we make?

{{exercise_number}}. The points \latex{A(a_{1};a_{2} )} and \latex{B(b_{1};b_{2} )} are given in the coordinate system with position vectors being \latex{\overrightarrow{a}} and \latex{\overrightarrow{b}}, respectively. Let \latex{\overrightarrow{p}} be the position vector of an arbitrary point \latex{P(x;y)} of the segment \latex{ AB }. Prove that if \latex{\overrightarrow{p}=\alpha \times \overrightarrow{a}+\beta \times \overrightarrow{b,}} then

\latex{\alpha \geq 0} and \latex{\beta \geq 0;}

\latex{\alpha +\beta =1.}

{{exercise_number}}. Determine the value of \latex{ y } for which the vectors \latex{\overrightarrow{p}(3;4)} and \latex{\overrightarrow{q}(8;y)} are perpendicular to each other.

{{exercise_number}}. Prove that the following equalities are satisfied by any real number \latex{ x } in their corresponding domains:

\latex{(\sin x+\cos x)^{2}-\sin 2x=\cos 2x+2\times \sin ^{2}x;}

\latex{\frac{\cos 2x}{\cos x-\sin x}-\cos x=\sin x.}

{{exercise_number}}. Let \latex{\alpha ,\beta} and \latex{\gamma} be the angles of a triangle. Prove that

\latex{\sin \alpha \times \cos \beta +\cos \alpha \times \sin \beta =\sin \gamma .}

Mathematics 12.

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