What is Plane Trigonometry? A Comprehensive Guide

What is Plane Trigonometry?

Have you ever wondered how to calculate the height of a building or the distance to the horizon? If so, then you have used trigonometry! Trigonometry is a branch of mathematics that deals with the relationships between angles and sides of triangles. It is used in a wide variety of applications, from navigation to surveying to construction.

In this article, we will explore the basics of plane trigonometry. We will learn how to define angles and sides of triangles, and how to use trigonometric ratios to solve problems. We will also see how trigonometry is used in real-world applications.

By the end of this article, you will have a solid understanding of the basics of plane trigonometry and how it can be used to solve problems. So let’s get started!

Term	Definition	Example
Angle	The measure of the rotation of a line around a fixed point	A right angle is 90 degrees
Side	A line segment that connects two vertices of a triangle	The hypotenuse of a right triangle is the side opposite the right angle
Ratio	The relationship between two sides of a triangle	The sine of an angle is the ratio of the opposite side to the hypotenuse

Trigonometry is a branch of mathematics that deals with the relationships between angles and side lengths of triangles. It is one of the oldest and most important branches of mathematics, with applications in navigation, surveying, astronomy, and many other fields.

Plane trigonometry is the study of trigonometry in two dimensions. It deals with triangles that lie in a plane, such as right triangles and oblique triangles. Plane trigonometry is the foundation for many other branches of mathematics, including calculus and analytical geometry.

This article provides an to plane trigonometry. We will cover the history of trigonometry, the fundamentals of plane trigonometry, and some applications of trigonometry.

History of Plane Trigonometry

The earliest known use of trigonometry dates back to the Babylonians in the 2nd millennium BC. The Babylonians used trigonometry to solve problems in astronomy and surveying. The Egyptians also used trigonometry in their construction of pyramids.

The Greek mathematician Euclid wrote a treatise on trigonometry in the 3rd century BC. Euclid’s work was based on the work of the Babylonians and Egyptians, but he also made some important contributions of his own. Euclid’s treatise on trigonometry was the standard text on the subject for centuries.

In the 1st century AD, the Greek astronomer Hipparchus made important contributions to trigonometry. Hipparchus developed tables of trigonometric functions, which were used by astronomers and mathematicians for centuries.

The Indian mathematician Aryabhata made important contributions to trigonometry in the 5th century AD. Aryabhata developed new methods for calculating trigonometric functions, and he also wrote a treatise on trigonometry.

The Chinese mathematician Zu Chongzhi made important contributions to trigonometry in the 5th century AD. Zu Chongzhi developed new methods for calculating trigonometric functions, and he also wrote a treatise on trigonometry.

The Islamic mathematician Al-Khwarizmi made important contributions to trigonometry in the 9th century AD. Al-Khwarizmi developed new methods for calculating trigonometric functions, and he also wrote a treatise on trigonometry.

The European mathematician Regiomontanus made important contributions to trigonometry in the 15th century AD. Regiomontanus developed new methods for calculating trigonometric functions, and he also wrote a treatise on trigonometry.

The development of trigonometry continued in the 16th, 17th, and 18th centuries. During this time, mathematicians developed new methods for calculating trigonometric functions, and they also used trigonometry to solve problems in physics, astronomy, and navigation.

In the 19th century, mathematicians began to study trigonometry from a more abstract perspective. They developed new theories of trigonometry, and they also used trigonometry to solve problems in calculus and analysis.

Today, trigonometry is a fundamental branch of mathematics. It is used in a wide variety of fields, including navigation, surveying, astronomy, physics, and engineering.

Fundamentals of Plane Trigonometry

The fundamentals of plane trigonometry include the following topics:

• Angles and their measurement
• Right triangles and trigonometric ratios
• The unit circle and trigonometric functions
• Graphs of trigonometric functions

Angles and their measurement

An angle is formed by two rays that share a common endpoint. The common endpoint is called the vertex of the angle, and the rays are called the sides of the angle.

Angles are measured in degrees. A right angle is an angle that measures 90 degrees. A straight angle is an angle that measures 180 degrees.

There are two ways to measure an angle:

• Inscribed angle: An inscribed angle is an angle that is formed by two chords that intersect inside a circle. The measure of an inscribed angle is half the measure of the intercepted arc.
• Central angle: A central angle is an angle that is formed by two radii that intersect at the center of a circle. The measure of a central angle is equal to the measure of the intercepted arc.

Right triangles and trigonometric ratios

A right triangle is a triangle that has one angle that measures 90 degrees. The sides of a right triangle are called the hypotenuse, the opposite side, and the adjacent side. The hypotenuse is the side opposite the right angle. The opposite side is the side opposite the angle that is not the right angle. The adjacent side is the side adjacent to the angle that is not the right angle.

The trigonometric ratios of a right triangle are the ratios of the lengths of the sides to the angles. The trigonometric ratios are:

• Sine: The sine of an angle is the ratio of the opposite side to the hypotenuse.
• Cosine: The cosine of an angle is

3. Applications of Plane Trigonometry

Plane trigonometry is used in a wide variety of applications, from solving triangles to navigation to astronomy. Some of the most common applications include:

• Solving triangles

One of the most basic applications of plane trigonometry is solving triangles. A triangle is a closed, three-sided figure. The sides of a triangle are called legs, and the angles opposite the legs are called angles. The sum of the interior angles of a triangle is always 180 degrees.

To solve a triangle, you need to know at least three of the following pieces of information:

• Two sides and the included angle
• Two angles and a side
• Three sides

Once you know these pieces of information, you can use the sine, cosine, and tangent functions to find the other sides and angles of the triangle.

• Navigation

Plane trigonometry is also used in navigation. Sailors use trigonometry to determine their position on the Earth’s surface. They do this by measuring the angle between the horizon and the North Star. This angle, along with the sailor’s latitude and longitude, can be used to determine the sailor’s position on a map.

• Astronomy

Plane trigonometry is also used in astronomy. Astronomers use trigonometry to measure the distances to stars and planets. They do this by observing the angles between the stars and planets and the Earth.

• Physics

Plane trigonometry is also used in physics. Physicists use trigonometry to study the motion of objects. They do this by measuring the angles between the objects and the forces acting on them.

• Engineering

Plane trigonometry is also used in engineering. Engineers use trigonometry to design bridges, buildings, and other structures. They do this by calculating the angles and forces that the structures will be subjected to.

4. Challenge Problems in Plane Trigonometry

Plane trigonometry is full of challenging problems. Some of the most common challenge problems include:

• The law of sines

The law of sines states that in a triangle, the ratio of the sine of an angle to the length of the opposite side is the same for all angles. This can be written as:

“`
sin(A)/a = sin(B)/b = sin(C)/c
“`

where A, B, and C are the angles of the triangle and a, b, and c are the lengths of the sides opposite those angles.

• The law of cosines

The law of cosines states that in a triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them. This can be written as:

“`
a^2 = b^2 + c^2 – 2bc*cos(A)
“`

where a, b, and c are the lengths of the sides of the triangle and A is the angle opposite side a.

• The Pythagorean theorem

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This can be written as:

“`
a^2 + b^2 = c^2
“`

where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse.

• The sum and difference identities

The sum and difference identities relate the sine, cosine, and tangent of the sum or difference of two angles to the sine, cosine, and tangent of the individual angles. These identities can be written as:

“`
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
tan(A + B) = (tan(A) + tan(B))/(1 – tan(A)tan(B))
“`

• The product-to-sum identities

The product-to-sum identities relate the product of two trigonometric functions to the sum or difference of those functions. These identities can be written as:

“`
sin(A)sin(B) = (1 – cos(A – B))/2
cos(A)cos(B) = (1 + cos(A – B))/2
tan(A)tan(B) = (tan(A) – tan(B))/(1 +

What is Plane Trigonometry?

Plane trigonometry is the study of triangles in a plane, specifically the relationships between the sides and angles of a triangle. It is a branch of mathematics that has been used for centuries to solve problems in navigation, surveying, and construction.

What are the main trigonometric functions?

The main trigonometric functions are sine, cosine, and tangent. These functions are defined in terms of the angles of a right triangle, and they can be used to find the lengths of the sides of a triangle given the angles or vice versa.

How do I use trigonometry to solve problems?

Trigonometry can be used to solve a variety of problems, such as finding the distance between two points, the height of a building, or the area of a triangle. To use trigonometry to solve a problem, you first need to draw a diagram of the situation and identify the angles and sides that are known. Then, you can use the trigonometric functions to find the unknown angles or sides.

What are some applications of trigonometry?

Trigonometry has a wide variety of applications, including:

• Navigation: Triangulation is used to determine the position of a ship or aircraft by measuring the angles to known landmarks.
• Surveying: Trigonometry is used to measure the distance between two points, the height of a building, or the area of a field.
• Construction: Trigonometry is used to design and build structures such as bridges, buildings, and towers.
• Physics: Trigonometry is used to study the motion of objects in space, such as planets, stars, and comets.
• Engineering: Trigonometry is used to design and build machines such as engines, turbines, and robots.

What are the different types of trigonometric identities?

There are a number of different types of trigonometric identities, including:

• Pythagorean identities: These identities relate the sine, cosine, and tangent of an angle to each other.
• Sum and difference identities: These identities relate the sine, cosine, and tangent of the sum or difference of two angles to the sine, cosine, and tangent of each angle individually.
• Double-angle identities: These identities relate the sine, cosine, and tangent of twice an angle to the sine, cosine, and tangent of the angle itself.
• Half-angle identities: These identities relate the sine, cosine, and tangent of half an angle to the sine, cosine, and tangent of the angle itself.

How can I learn more about trigonometry?

There are a number of resources available to help you learn more about trigonometry, including:

• Textbooks: There are a number of textbooks available on trigonometry, both at the high school and college level.
• Online resources: There are a number of online resources available that can help you learn about trigonometry, such as Khan Academy and YouTube.
• Classes: You can also take classes on trigonometry at your local community college or university.

  plane trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a useful tool for solving problems in a variety of fields, such as surveying, navigation, and engineering. The basic concepts of plane trigonometry are the sine, cosine, and tangent functions, which are used to find the measures of angles and sides of triangles. Trigonometry is a fascinating and challenging subject that can be used to solve a wide variety of problems. I hope this overview has given you a better understanding of what plane trigonometry is and how it can be used.