Support the reading website called English Academy
Defenition: Trigonometry is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths.
Trigonometry is the branch of mathematics that studies the relationships between the sides and angles of triangles. It uses six trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—to find unknown side lengths or angles, and has applications in fields like astronomy, navigation, and engineering.
Key concepts:
Right triangle: A triangle with one 90-degree angle. Trigonometry is fundamentally based on the relationships within these triangles.
Angles: Can be measured in degrees or radians.
Applications: Used in various fields to calculate distances and angles, including astronomy, surveying, engineering, and even in physics and complex analysis.
How it works:
In a right triangle, the trigonometric functions are ratios of the lengths of the sides. For example, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
Because the ratios of the sides remain the same for a given angle, trigonometry allows you to calculate a missing side or angle if you know some others.
This video explains the trigonometric ratios (SOH CAH TOA) using an example:
representation of the basic trigonometric ratios in a right triangle, such as SOH CAH TOA. SOH (
sin(θ)=Opposite/Hypotenuses i n open paren theta close paren equals Opposite / Hypotenuse
𝑠𝑖𝑛(𝜃)=Opposite/Hypotenuse
), CAH (
cos(θ)=Adjacent/Hypotenusec o s open paren theta close paren equals Adjacent / Hypotenuse
𝑐𝑜𝑠(𝜃)=Adjacent/Hypotenuse
), and TOA (
tan(θ)=Opposite/Adjacentt a n open paren theta close paren equals Opposite / Adjacent
𝑡𝑎𝑛(𝜃)=Opposite/Adjacent
) are fundamental and can be visualized with a simple right-angled triangle diagram. Other key formulas include the Pythagorean identity (
sin2θ+cos2θ=1s i n squared theta plus c o s squared theta equals 1
𝑠𝑖𝑛2𝜃+𝑐𝑜𝑠2𝜃=1
) and reciprocal identities like
cosec(θ)=1/sin(θ)c o s e c open paren theta close paren equals 1 / s i n open paren theta close paren
𝑐𝑜𝑠𝑒𝑐(𝜃)=1/𝑠𝑖𝑛(𝜃)
Key concepts
Right-angled triangles: Trigonometry is often introduced using right-angled triangles, where the relationships between the angles and the ratios of the sides are defined.
Trigonometric functions: These functions are ratios of the sides of a right-angled triangle relative to its angles.
Sine (sin): The ratio of the opposite side to the hypotenuse.
Cosine (cos): The ratio of the adjacent side to the hypotenuse.
Tangent (tan): The ratio of the opposite side to the adjacent side.
Applications: Trigonometry is used for calculations in fields such as astronomy, mapmaking, surveying, and engineering to find distances and angles.
Identities: These are equations that are true for all values of the variables involved and are fundamental to solving complex problems.
How it works
Trigonometry allows you to find an unknown side or angle in a triangle if you know enough other information.
The ratios of the sides of a triangle remain constant for a given angle, even if the triangle is scaled up or down.
Related topics
Unit circle: A circle with a radius of one used in more advanced trigonometry to extend the concepts beyond right triangles to all angles.
Plane vs. Spherical Trigonometry: Plane trigonometry deals with triangles on a flat surface, while spherical trigonometry deals with triangles on a sphere.
If you want to learn about Trigonometry and watch videos and exerices, click this link below
https://www.khanacademy.org/math/trigonometry
