Trigonometry For High School

Trigonometry (from the Greek trigonon = three angles and metro = measure) is a part of elementary mathematics dealing with angles, triangles and trigonometric functions such as sine (abbreviated sin), cosine (abbreviated cos) and tangent (abbreviated tan). It has some connection to geometry, although there is disagreement on exactly what that connection is; for some, trigonometry is just a section of geometry.
Overview and definitions in Trigonometry
A standard right triangle.
Trigonometry uses a large amount of specific words to describe parts of a triangle. Some of the definitions in trigonometry are:
Right triangle - A right triangle is a triangle that has one angle that is equal to 90 degrees. (A triangle can not have more than one right angle.) The standard trigonometric ratios can only be used on right triangles.
Hypotenuse - The hypotenuse of a triangle is the longest side, and the side that is opposite the right angle. For example, for the triangle on the right, the hypotenuse is side c.
Opposite of an angle - The opposite side of an angle is the side that does not intersect with the vertex of the angle. For example, side a is the opposite of angle A in the triangle to the right.
Adjacent of an angle - The adjacent side of an angle is the side that intersects the vertex of the angle but is not the hypotenuse. For example, side b is adjacent to angle A in the triangle to the right.
There are three main trigonometric ratios for right triangles, and three reciprocals of those ratios. There are 6 total ratios. They are:
Sine (sin) - The sine of an angle is equal to the opposite/hypotenuse
Cosine (cos) - The cosine of an angle is equal to the adjacent/hypotenuse
Tangent (tan) - The tangent of an angle is equal to the opposite/adjacent
The reciprocals of these ratios are:
Cosecant (csc) - The cosecant of an angle is equal to the hypotenuse/opposite or 1/sin
Secant (sec) - The secant of an angle is equal to the or hypotenuse/adjacent
Cotangent (cot) - The cotangent of an angle is equal to the opposite/adjacent or 1/tangent
The sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters, such as SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
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Geometry For Elementary School
Geometry
Geometry is a kind of mathematics used to work with shapes.
Examples of Shapes
There are flat shapes and solid shapes in geometry. Squares, circles and triangles are some of the simplest shapes in flat geometry. Cubes, cylinders, cones and spheres are simple shapes in solid geometry.
Measuring in Geometry
Geometry can be used to measure a flat shape's area and perimeter.It can also be used to measure a solid shape's volume and surface area.
Many things have the shapes found in geometry. Geometry can be used to measure many things by seeing them as made of geometrical shapes. For example, geometry can help people find:
the surface area of a house, so they can buy the right amount of paint
the volume of a box, to see if it is big enough to hold a litre of food
the area of a farm, so it can be divided into equal parts
the distance around the edge of a pond, to know how much fencing to buy.

Some Simple Ideas in Geometry

In mathematics, geometry starts with a few simple ideas:
A point is shown on paper by touching it with a pencil or pen, without making any sideways movement. We know where the point is, but it has no size.
A straight line is the shortest distance between two points. For example, Sophie pulls a piece of string from one point to another point. A straight line between the two points will follow the path of the tight string.
A plane is flat surface that does not stop in any direction. A ball placed any place on this flat surface will not move if gravity on the surface is constant.

Triangle
A triangle is a shape. It has three straight sides and three points. The three angles of a triangle add to 180 degrees. It is the polygon with the least possible number of sides.

Triangles can be grouped according to how long their sides are
In an equilateral triangle all three sides have the same length
In an isosceles triangle two sides have the same length
In a scalene triangle all sides have different lengths

Triangles can also be grouped by their angles.
A right triangle has one angle that is 90 degrees (a right angle). The side opposite the right angle is the hypotenuse.
An obtuse triangle has one angle that is larger than 90 degrees (an obtuse angle)
An acute triangle has angles that are all less than 90 degrees (acute angles)

Perimeter
In geometry, perimeter is the distance around a flat object. For example, all four sides of a square rhombus have the same length, so a rhombus with side length 2 inches would have a perimeter of 8 inches (2+2+2+2=8).
Real-life objects have perimeters as well. A football field, including the end zones, is 360 feet long and 160 feet wide. So the perimeter of the field is 360+160+360+160=1040 feet.
The perimeter of a circle is usually called the circumference. It may be calculated by multiplying the diameter times "Pi". Pi is a constant which is equal to 3.14159; however, the places to the right of the decimal are endless. The number of places used depend on the accuracy required for the result.

A right triangle, (also called a right-angled triangle), has one angle that is 90 degrees. The other two angles always add up to 90 degrees but can be different sizes. The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.

Examples of shapes

2D shapes
circles
squares
triangles
ovals
kites

These are two dimensional shapes or flat plane geometry shapes. They can look like anything and can have any number of sides. The sides can be straight or curved. Triangles and squares are polygons.
The sides of these shapes are lines.

3D shapes
spheres
cubes
cones
pyramids
These are three dimensional shapes. The sides of these shapes are surfaces. Again, the sides can be straight or curved