Trigonometric Symbols are the symbols that are used in trigonometry and help to solve various problems. The trigonometric symbols are used to represent various trigonometric ratios. Trigonometry is a branch of mathematics that explores the relationships between the ratios of the sides of a right-angled triangle and its angles.
In this article, we will learn about trigonometric symbols their representation, and others in detail.
Table of Content
What are Trigonometric Symbols
Trigonometric Symbols we use mostly in mathematics are sine (sin), cosine (cos), tangent (tan), and their reciprocals are the cotangent (cot), secant (sec), and cosecant (csc), which are less used. Again, each of these six trigonometric functions has a corresponding inverse function, and these have an analog among the hyperbolic functions.
Representation of Trigonometric Symbols
In trigonometry, there are many functions are used to relate angles within a right triangle to its various lengths or ratios. The below table explains some of the most common functions.
|
Trigonometric Symbol |
Trigonometric Symbol Name |
Explanation |
|---|---|---|
|
θ |
Theta |
It is used to represent an angle in trigonometric equations. |
|
sin |
Sine |
It is the ratio of the length of the side opposite the angle to the length of the hypotenuse. It is represented as “sin(θ).” |
|
cos |
Cosine |
It is the ratio of the length of the adjacent side to the length of the hypotenuse. It is represented as “cos(θ).” |
|
tan |
Tangent |
The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle. It is represented as “tan(θ).” |
|
csc |
Cosecant |
The angle of cosecant is the reciprocal of the sine, csc(θ) = 1/sin(θ). |
|
sec |
Secant |
The angle of secant is the reciprocal of the cosine, sec(θ) = 1/cos(θ). |
|
cot |
Cotangent |
The angle of cotangent is the reciprocal of the tangent, cot(θ) = 1/tan(θ). |
|
arcsin x or sin⁻¹(x) |
Arcsine function Inverse sine |
It is used to find the angle for a given trigonometric ratio |
|
arccos x or cos⁻¹(x) |
Arcsine function Inverse cosine |
It is used to find the angle for a given trigonometric ratio |
|
arctan x or tan⁻¹(x) |
Arcsine function Inverse tan |
It is used to find the angle for a given trigonometric ratio |
|
π |
Pi |
It is used in trigonometry to express angles in radians. |
|
r |
Radius |
In polar coordinates, r represents the distance from the origin to a point, it is used in trigonometry while dealing with polar coordinates. |
|
Δ |
Delta |
It is used to represent the change in an angle, such as Δθ |
These Trigonometric Symbols are used to calculate and describe the relationship between angles and sides in geometric figures.
Trigonometric Symbols in Right-Angled Triangle
These symbols and notations are fundamental in trigonometry and are used to describe and calculate relationships between angles and sides in triangles and other geometric figures. They are also used in various applications, including physics, engineering, and many other fields where angles and periodic phenomena are involved.
As given in the above image in a right-angle triangle:
Side opposite the right angle is called the hypotenuse
Side opposite to other angles is called the opposite side
- For angle R opposite side is PQ
- For angle P opposite side is RQ
The side adjacent to an angle is called the adjacent side
- For angle R adjacent side is RQ
- For angle P adjacent side is PQ
There are 6 basic trigonometric relations that form the basics of trigonometry. These 6 trigonometric relations are ratios of all the different possible combinations in a right-angled triangle.
These trigonometric symbols are known as:
- Sine
- Cosine
- Tangent
- Cosecant
- Secant
- Cotangent
The mathematical symbol θ is used to denote the angle.
Sine (sin)
Sine of an angle is defined by the ratio of lengths of sides which is opposite to the angle and the hypotenuse. It is represented as sin θ.
Cosine (cos)
Cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. It is represented as cos θ
Tangent (tan)
Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle. It is represented as tan θ
Cosecant (cosec)
Cosecant of an angle is defined by the ratio of the length of the hypotenuse and the side opposite the angle. It is represented as cosec θ
Secant (sec)
Secant of an angle is defined by the ratio of the length of the hypotenuse and the side and the side adjacent to the angle. It is represented as sec θ
Cotangent (cot)
Cotangent of an angle is defined by the ratio of the length of sides that is adjacent to the angle and the side which is opposite to the angle. It is represented as cot θ.
|
Trigonometric Symbols of θ |
Trigonometric Function in Right-Angled Triangle |
|---|---|
|
Sine (sin θ) |
sin θ = opposite/hypotenuse |
|
Cosine (cos θ) |
cos θ = adjacent/hypotenuse |
|
Tangent (tan θ) |
tan θ = opposite/adjacent |
|
Cosecant (csc θ) |
csc θ = adjacent/opposite |
|
secant (sec θ) |
sec θ = hypotenuse/adjacent |
|
cotangent (cot θ) |
cot θ = hypotenuse/opposite |
Applications of Trigonometric Symbols
- Trigonometry is used in geography to measure the distance between landmarks.
- Trigonometry is used in astronomy to measure the distance of nearby stars and also in the satellite navigation system.
- The calculus is based on trigonometry and algebra.
- The fundamental trigonometric functions like sine and cosine are used to describe the sound and light waves
- Trigonometry is used in oceanography to calculate the heights of waves and tides in oceans.
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Examples on Trigonometric Symbols
Example 1: If θ is an acute angle in a right triangle, and sin θ = 3/5, find the value of cos θ.
Solution:
Given,
- sin θ = 3/5
Suppose the side opposite θ is 3, and the hypotenuse is 5. According to the Pythagorean theorem, the adjacent side can be found using
cos θ =
cos θ =
cos θ =
cos θ =
cos θ =
∴ cos θ = 4/5
Example 2: Given that cot α = -4/3, calculate the value of tan α.
Solution:
Given,
- cot α = -4/3
Lets assume that the adjacent side is -4, and the opposite side is 3.
Now, the tangent of an angle tan α is the ratio of the opposite side to the adjacent side.
tan α =
tan α = 3/ (-4)
∴ tan α = -3/4
Practice Question on Trigonometric Symbols
Q1. Express cosθ in terms of sin θ using a Pythagorean trigonometric identity.
Q2. If tan α = 3/4, find the values of cos α and csc α.
Q3. Given that cot β = -2, calculate the values of sin β and sec β.
Q4. If csc β = 5/2, determine the values of sin β and tan β.
Q5. Express sin2θ in terms of cos2θ using a Pythagorean trigonometric identity.
Frequently Asked Questions on Trigonometric Symbols
What are Symbols used in Trigonometry?
Trigonometric Symbols we use mostly in mathematics are sine (sin), cosine (cos), tangent (tan),And their reciprocals are the cotangent (cot), secant (sec), and cosecant (csc) which are less used.
What is θ in Math?
Theta (θ) is a Greek Alphabet that is used to represent angular position of a vector, or angles in Mathematics.
What is the symbol for sine in trigonometry?
The symbol for sine in trigonometry is “sin”.
What does the cosine symbol look like?
The symbol for cosine is “cos”.
How is tangent represented in trigonometry?
Tangent is represented by the symbol “tan”.
Is there a symbol for secant in trigonometry?
Yes, the symbol for secant in trigonometry is “sec”.