Trigonometric Ratios. Method of Coordinates. Transformation of Figures.

Right triangle and trigonometric ratios ☰

A right triangle is a triangle that has one interior angle measuring 90 degrees, which is also known as the "right angle". The side opposite to the right angle is called the hypotenuse, while the other two sides are called the legs.
Trigonometric ratios are mathematical functions that relate the angles and sides of a right triangle. There are six trigonometric ratios, which are commonly abbreviated as "sin", "cos", "tan", "csc", "sec", and "cot". Each ratio represents the ratio of two sides of the triangle, and the ratio depends on the angle that is being considered.

The three primary trigonometric ratios are sine, cosine, and tangent. Here's how to define each one:

• Sine: The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
  In other words, \(sin(\theta )=\frac{ \text{opposite}}{ \text{hypotenuse}} \) .
• Cosine: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
  In other words, \(cos(\theta )=\frac{ \text{adjacent}}{ \text{hypotenuse}} \) .
• Tangent: The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the adjacent side.
  In other words, \(tan(\theta )=\frac{ \text{opposite}}{ \text{adjacent}} \) .

The other three trigonometric ratios are reciprocal functions of the primary trigonometric ratios. The reciprocal of sine is cosecant (csc), the reciprocal of cosine is secant (sec), and the reciprocal of tangent is cotangent (cot).

• Cosecant: \( csc(\theta )=\frac{ \text{hypotenuse}}{ \text{opposite}} \) .
• Secant: \( sec(\theta )=\frac{ \text{hypotenuse}}{ \text{adjacent}} \) .
• Cotangent: \( cot(\theta )=\frac{ \text{adjacent}}{ \text{opposite}} \).

Trigonometric ratios are used in many different fields, including physics, engineering, and mathematics. They are particularly useful for solving problems involving right triangles, such as finding missing angles or sides.

Trigonometric identities ☰

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables in the equation. These identities can be used to simplify expressions, prove other mathematical results, and solve various problems in mathematics, science, and engineering.
There are many trigonometric identities, and they can be classified into several different categories based on their form and the functions involved. Some of the most commonly used trigonometric identities include:

• Pythagorean identities: These identities involve the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean identities are: $$ sin^2 ( \theta )+cos^2 ( \theta)=1$$ $$ tan^2 ( \theta )+1=sec^2 ( \theta ) $$ $$ cot^2 ( \theta )+1=csc^2 (\theta) $$
• Reciprocal identities: These identities involve the reciprocal functions of the primary trigonometric functions (sine, cosine, and tangent). The reciprocal identities are: $$csc( \theta )= \frac{1}{sin( \theta )} $$ $$ sec( \theta )=\frac{1}{cos( \theta )} $$ $$ cot( \theta )=\frac{1}{tan( \theta )} $$
• Quotient identities: These identities involve the quotient of two trigonometric functions. The quotient identities are: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$ and $$ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $$
• Even/odd identities: These identities involve the symmetry properties of the trigonometric functions. The even/odd identities are: $$ \sin(-\theta) = -\sin(\theta)$$ $$\cos(-\theta) = \cos(\theta) $$ $$\tan(-\theta) = -\tan(\theta) $$
• Angle sum and difference identities: These identities involve the sum or difference of two angles. The angle sum and difference identities are: $$\sin(a \pm b) = \sin(a) \cos(b) \pm \cos(a) \sin(b) $$ $$\cos(a \pm b) = \cos(a) \cos(b) \mp \sin(a) \sin(b) $$ $$\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)} $$

Trigonometric identities are used in a variety of applications, including physics, engineering, and mathematics. They are particularly useful for simplifying expressions, evaluating integrals, and solving equations involving trigonometric functions. By memorizing these identities and understanding how to apply them, one can become proficient in working with trigonometric functions and solving a wide range of problems.

Coordinates of the midpoint of a piece ☰

The coordinates of the midpoint of a line segment in a coordinate plane can be found using the midpoint formula. If the two endpoints of the line segment have coordinates \( (x_1,y_1 ) \) and \( (x_2,y_2 ) \), then the coordinates of the midpoint are:

\(\left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right) \).

This formula works because the midpoint of a line segment is the point that is exactly halfway between the two endpoints. To find the midpoint, we add the \(x\)-coordinates of the endpoints and divide by 2 to get the \(x\)-coordinate of the midpoint. Similarly, we add the \(y\)-coordinates of the endpoints and divide by 2 to get the \(y\)-coordinate of the midpoint.

For example, if the endpoints of a line segment are \((3,2) \) and \((9,8) \), the coordinates of the midpoint are:

\(\left(\frac{3 + 9}{2}, \frac{2 + 8}{2}\right) = (6, 5) \).
This means that the midpoint of the line segment is the point \( (6,5) \).

Equation of the straight line through two points. ☰

The equation of a straight line through two points \( (x_1,y_1 ) \) and \( (x_2,y_2 ) \) can be found using the point-slope formula:
\( y-y_1=\frac{y_2-y_1}{x_2-x_1} \cdot (x-x_1 ) \) , where \(m\) is the slope of the line, and \((x,y)\) is any point on the line.
To use the point-slope formula, we first find the slope of the line using the slope formula: \( m = \frac{y_2-y_1}{x_2-x_1} \).
Once we have the slope, we can plug in the coordinates of one of the points, say \( (x_1,y_1 ) \) and simplify the equation to get the final form: \( y-y_1=m(x-x_1 ) \) .
This form is known as the point-slope form of the equation of a line. Alternatively, we can simplify this equation further by expanding and rearranging it into the slope-intercept form: \(y=mx-mx_1+y_1\)
This form is often preferred because it makes it easier to see the \(y\)-intercept of the line, which is the value of \(y\) when \(x=0\).

For example, if we have two points \((2,4)\) and \((5,8)\), we can find the equation of the line passing through these points as follows:
First, we find the slope of the line: \( m = \frac{8-4}{5-2} = \frac{4}{3} \)
Then, we can use either of the two point-slope forms of the equation of a line to find the equation of the line. Let's use the first form: \( y-4 = \frac{4}{3} (x-2) \)
Expanding and simplifying, we get:
\( y = \frac{4}{3} x - \frac{8}{3} + 4 = \frac{4}{3} x + \frac{4}{3} \)
This is the slope-intercept form of the equation of the line passing through the points \((2,4)\) and \((5,8)\) .

The point-slope formula can be expressed as:
\( y-y_1 = \frac{y_2-y_1}{x_2-x_1} (x-x_1 ) \)
And the slope-intercept form can be expressed as: \(y=mx+b\) where \(m\) is the slope and \(b\) is the \(y\)-intercept.

Transformation of figures, rotation. ☰

Transforming figures involves moving a given figure to a new position without changing its shape. There are four main types of shape transformations:

• Translation: A translation involves moving a figure from one position to another, without changing its size or shape. This transformation is also known as a slide or shift. In translation, each point of the figure moves the same distance and in the same direction.
• Rotation: A rotation involves turning a figure around a fixed point by a certain angle. In this transformation, the shape and size of the figure remain the same. A rotation can be clockwise or counterclockwise.
• Reflection: A reflection involves flipping a figure across a line called the line of reflection or mirror line. Each point of the figure is reflected across the line, creating a mirror image of the original figure.
• Dilation: A dilation involves changing the size of a figure by stretching or shrinking it. In this transformation, the shape of the figure remains the same, but its size is either increased or decreased.

Transformations of figures can be performed on the coordinate plane using algebraic equations. For example, a translation of a figure can be performed by adding or subtracting a fixed amount from the x and y coordinates of each point of the figure. A rotation can be performed by using the rotation formulas to find the new coordinates of each point after the rotation. A reflection can be performed by using the reflection formulas to find the new coordinates of each point after the reflection. A dilation can be performed by multiplying the \(x\) and \(y\) coordinates of each point by a scaling factor.
Transformations of figures are used in many areas of mathematics and science, such as computer graphics, engineering, and physics. They are also used in art and design to create interesting and aesthetically pleasing patterns and designs.

Rotation is one of the basic transformations that can be applied to 2D figures. A rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. The figure is rotated by a certain angle \( \theta \) in either a clockwise or counterclockwise direction.
The rotation of a figure is performed by rotating each point of the figure by the same angle \( \theta \) around the center of rotation. The distance between the center of rotation and each point of the figure remains constant after the rotation. The resulting figure is called the image of the original figure.
To perform a rotation of a figure, we need to know the center of rotation and the angle of rotation. The center of rotation can be any point in the plane. The angle of rotation is measured in degrees or radians and can be positive (counterclockwise) or negative (clockwise).

If a figure is rotated by an angle \( \theta \) around the origin \((0,0) \), we can use the following formulas to find the coordinates of the rotated point \((x',y' ) \) given the coordinates of the original point \((x,y)\):
\( x' = x\cos(\theta) - y\sin(\theta) \)
\( y' = x\sin(\theta) + y\cos(\theta) \)

If a figure is rotated by an angle \( \theta \) around a point \((h,k)\), we can use the following formulas to find the coordinates of the rotated point \((x',y' ) \) given the coordinates of the original point \((x,y)\):
\( x' = (x - h)\cos(\theta) - (y - k)\sin(\theta) + h \)
\( y' = (x - h)\sin(\theta) + (y - k)\cos(\theta) + k \)
These formulas can be used to rotate any point in the plane around any center of rotation.

To visualize a rotation, consider the following example:
Suppose we want to rotate the point \((2,3)\) by an angle of \(90^\circ \) counterclockwise around the origin \( (0,0) \). Using the rotation formulas, we get:
\( x^{\prime} = 2\cos 90^\circ - 3\sin 90^\circ = -3 \)
\( y^{\prime} = 2\sin 90^\circ + 3\cos 90^\circ = 2\)
Thus, the rotated point is \( (-3,2) \). We can verify this by plotting the point \( (2,3) \) and the point \( (-3,2) \) on a coordinate plane and visually verifying that the rotated point is obtained by rotating the original point by \( 90^\circ \) counterclockwise around the origin.

The rotation formulas can be expressed as:
\( x' = x\cos(\theta) - y\sin(\theta) \)
\( y' = x\sin(\theta) + y\cos(\theta) \)
\( x' = (x - h)\cos(\theta) - (y - k)\sin(\theta) + h \)
\( y' = (x - h)\sin(\theta) + (y - k)\cos(\theta) + k \)

Similarity transformation. Homothety. ☰

Similarity transformation and homothety are two mathematical concepts used to describe geometric transformations in the plane. Both transformations preserve the shape of objects, but they differ in how they change their size and orientation.

Similarity Transformation:
A similarity transformation is a transformation that preserves the shape of an object while changing its size and orientation. This means that if two objects are similar, then one can be transformed into the other through a similarity transformation. A similarity transformation consists of a combination of a dilation (scaling) and a rotation.
More specifically, a similarity transformation of a point in the plane consists of multiplying its coordinates by a scaling factor \(k\) and rotating it by an angle \( \theta \). The transformation can be written as:
\( (x',y' )= k(R( \theta )(x,y)) \) where \((x,y) \) are the coordinates of the original point, \((x',y') \) are the coordinates of the transformed point, \(k\) is the scaling factor, and \( R( \theta ) \) is the rotation matrix with angle \( \theta \).

Some important properties of similarity transformations include:

• They preserve angles between lines.
• They preserve the ratio of distances between any two points on the object.
• They do not change the orientation (clockwise or counterclockwise) of the object.

A homothety is a type of similarity transformation that only involves a dilation (scaling) of an object. In other words, a homothety is a transformation that preserves the shape of an object while changing its size. A homothety can be described by a single parameter, the scaling factor \(k\).
Mathematically, a homothety of a point in the plane consists of multiplying its coordinates by a scaling factor \(k\). The transformation can be written as: \( (x',y' )= k(x,y) \), where \((x,y)\) are the coordinates of the original point, and \( (x',y') \) are the coordinates of the transformed point.

Some important properties of homotheties include:

• They preserve the shape of an object.
• They change the size of the object, but not its orientation.
• They preserve the ratio of distances between any two points on the object.

In summary, similarity transformations and homotheties are both important concepts in geometry that describe how objects can be transformed while preserving their shape. Similarity transformations involve a combination of scaling and rotation, while homotheties only involve scaling.