Trigonometry in Maths: Table, Formulas, Identities and 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. The fundamental ratios used to study these relationships are known as trigonometric ratios, which include sine, cosine, tangent, cotangent, secant, and cosecant.
The term “trigonometry” is a 16th-century Latin derivative and the concept was given by the Greek mathematician Hipparchus. Trigonometry word is formed from ancient Greek words “trigonon” and “metron” which mean triangle and measure respectively, thus collectively called Trigonometry which means measures of a triangle.
The most important topics in trigonometry are trigonometry table, trigonometry formulas, trigonometric identities, and trigonometric ratios. In this article, we will see the basics of trigonometry, including its fundamental identities and formulas.
Table of Content
- Introduction to Trigonometry
- What is Trigonometry?
- Trigonometry Basics
- All Trigonometry Functions
- Even and Odd Trigonometric Functions
- Trigonometric Ratios
- Trigonometry Angles
- Trigonometry Chart
- Trigonometry Table
- Applications of Trigonometry
- Trigonometry Formulas
- 1. Pythagorean Trigonometric Identities
- 2. Sum and Difference Identities
- 3. Double angle Identities
- 4. Half Angle Identities
- 5. Product Sum Identities
- 6. Product Identities
- 7. Triple Angle Identities
- Unit Circle
- Trigonometry Identities
- Euler’s Formula for Trigonometry
- Trigonometry Real-Life Examples
- Trigonometry Solved Examples
- Practice Problems on Trigonometry
- Trigonometry Class 10 PDF
Introduction to Trigonometry
Trigonometry is the study of the relationships between the angles and sides of triangles. It primarily deals with right triangles, where one angle measures 90 degrees. The three main trigonometric functions—sine, cosine, and tangent—describe these relationships.
- Sine (sin) is the ratio of the length of the side opposite an angle to the length of the hypotenuse.
- Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side.
- These functions are fundamental in understanding periodic phenomena, such as waves and oscillations, and are widely used in fields like physics, engineering, and astronomy.
What is Trigonometry?
Trigonometry is the branch of mathematics that deals with the sides and angles of a triangle, especially the right-angle triangle.
Other than right-angle triangles, trigonometry is helpful in many different geometric figures, either 2-dimensional or 3-dimensional.
Studies of Trigonometry can be classified into three parts which are as follows:
- Core Trigonometry (deal with right angle triangles only)
- Plane Trigonometry (deals with all types of 2-dimensional geometry)
- Spherical Trigonometry (deals with all types of 3-dimension geometry)
Trigonometry Definition
Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles, especially right-angled triangles.
Trigonometry Basics
Trigonometry Basics are the core concepts of trigonometry without which it can’t be defined, some of these basics of trigonometry are as follows:
- Angles: The measure of space between two intersecting lines are known as angles.
- Right-angle Triangle: A triangle with one of its interior angles being the right angle i.e., 90°, is called a right angles triangle.
- Pythagoras Theorem: In right angles triangle, according to the Pythagoras theorem, the square of the hypotenuse is equal to the sum of squares of the other two sides,
- Trigonometric Ratios: Trigonometric Ratios are defined as the ratio of the sides of the right angle triangles. As there are 3 ways to choose two sides out of three and two ways for each chosen pair to arrange in ratio, thus there are 3×2 =6 trigonometric ratios which are defined for each possible pair of sides of the right angle triangle.
All Trigonometry Functions
The important trigonometric functions include sin and cos, as all the other trigonometric ratios can be defined in terms of sin and cos. The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas.
Functions | Abbreviation | Relationship to sides of a right triangle |
---|---|---|
Sine Function | sin | Opposite side/ Hypotenuse |
Cosine Function | cos | Adjacent side / Hypotenuse |
Tangent Function | tan | Opposite side / Adjacent side |
Cosecant Function | cosec | Hypotenuse / Opposite side |
Secant Function | sec | Hypotenuse / Adjacent side |
Cotangent Function | cot | Adjacent side / Opposite side |
Even and Odd Trigonometric Functions
Odd Trigonometric Functions
A trigonometric function is said to be an odd function if f(-x) = -f(x) and symmetric with respect to the origin.
Even Trigonometric Functions
A trigonometric function is said to be an even function, if f(-x) = f(x) and symmetric to the y-axis.
- Sin (-x) = – Sin x
- Cos (-x) = Cos x
- Tan (-x) = -Tan x
- Csc (-x) = – Csc x
- Sec (-x) = Sec x
- Cot (-x) = -Cot x
Trigonometric Ratios
In the study of Trigonometry, there are six trigonometric ratios:
- Sine
- Cosine
- Tangent
- Cosecant
- Secant
- Cotangent
These trigonometric ratios as the name suggests are defined as the ratios between two sides of a right-angle triangle.
For the sin trigonometric ratio, we defined it as the ratio of the perpendicular (opposite to the angle for which we are calculating sin,) to the hypotenuse of the right-angle triangle i.e., for triangle ABC right angles at B:
Similarly, cos is defined as the ratio of the base and hypotenuse of the right-angle triangle.
and trigonometric ratio tan is defined as the ratio of perpendicular to the base of the right-angle triangle.
Other, then sin, cos and tan, cot, sec, and cosec are also defined as the ratio of the sides of right-angle triangle as follows:
Note: sec, cosec, and cot are the reciprocals of the trigonometric ratios cos, sin, and tan respectively.
Trigonometry Table
In this table we summarize and briefly depict trigonometry ratios and how to calculate them:
Trigonometry Ratio | Short Form | Ratio of |
---|---|---|
Sine Function | sin | opposite side / Hypotenuse |
Cosine Function | cos | adjacent side / Hypotenuse |
Tangent Function | tan | opposite side / adjacent side |
Cosecant Function | cosec | 1 / sine |
Secant Function | sec | 1 / sec |
Cotangent Function | cot | 1 / tan |
As you can see in this table, calculating these trigonometry ratio is very simple, as they are related to each other.
- Also Read: Trigonometry Table
Trigonometry Angles
The angle for which the trigonometric ratio is defined is the trigonometric angle. Angles can either be measured in degrees (°) or can be measured in radians (rad).
Some standard angles for which we create the table of trigonometric ratios are 0°, 30°, 45°, 60°, and 90°. Other than these angles, we also sometimes need to deal with 15°, 18°, 75°, and 72°.
Trigonometry Chart
Trigonometric Chart is the table of values of trigonometric ratio at some specific angle values. The Trigonometry Chart for the value of trigonometric ratios at different angles is given below:
Trigonometry Table
Below is a full trigonometry table with all six trigonometric ratios:
Trigonometric Ratios | Degrees and Radians Trigonometric Table | ||||||
---|---|---|---|---|---|---|---|
0° | 30° | 45° | 60° | 90° | 180° | 270° | |
0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | |
Sin | 0 | 1/2 | 1/√2 | √(3)/2 | 1 | 0 | -1 |
Cos | 1 | √(3)/2 | 1/√2 | 1/2 | 0 | -1 | 0 |
Tan | 0 | 1/√3 | 1 | √3 | Not Defined | 0 | Not Defined |
Cosec | Not Defined | 2 | √2 | 2/√3 | 1 | Not Defined | -1 |
Sec | 1 | 2/√3 | √2 | 2 | Not Defined | -1 | Not Defined |
Cot | Not Defined | √3 | 1 | 1/√3 | 0 | Not Defined | -1 |
How to Remember Trigonometry Table?
The best way to remember the trigonometry table is to learn this trick to create one whenever needed. To create a table of different values of trigonometric ratios at different angles, we can use the following algorithm.
Using this trick we can complete the trigonometry table without remembering the exact values of the ratios for different angles. After learning the below mentioned steps ypu’ll never ask how to learn trigonometry table?
The steps required in this algorithm are as follows:
Step 1: Write first five whole numbers with some distance between them.
0 | 1 | 2 | 3 | 4
Step 2: Divide each number by 4.
0 | 1/4 | 2/4 | 3/4 | 4/4
OR
0 | 1/4 | 1/2 | 3/4 | 1
Step 3: Take Square Root for each resulting number in step 2..
√0 | √(1/4) | √(1/2) | √(3/4) | √1
OR
0 | 1/2 | 1/√2 | √(3)/2 | 1
Step 4: Resulting values for step 3, are the value of trigonometric ratio sine for angles 0°, 30°, 45°, 60°, 90°.
Step 5: Reverse the order of the resulting values in step 3, to get the value of cos for the same angles.
1 | √(3)/2 | 1/√2 | 1/2 | 0
Step 6: Find the ratio of the results in step 3 to results in step 5, to get the value of cos for angles 0°, 30°, 45°, 60°, 90°.
0÷1 | 1/2÷√(3)/2 | 1/√2÷1/√2 | √(3)/2÷1/2 | 1÷0
OR
0 | 1/√3 | 1 | √(3) | Not defined
Thus, using these steps, resulting table is formed.
Angle
0°
30°
45°
60°
90°
sin
0
1/2
1/√2
√(3)/2
1
cos
1
√(3)/2
1/√2
1/2
0
tan
0
1/√3
1
√3
Not Defined
Applications of Trigonometry
Trigonometry has so many applications in the real world, we can even say that is the most used mathematics concept throughout mathematics. Some of the applications of trigonometry are as follows:
- Trigonometry is very essential for modern-day navigation systems such as GPS or any other similar system.
- In most streams of engineering, trigonometry is used extensively for various kinds of analysis and calculations, which helps engineers to make more sound decisions for the construction of various kinds of structures.
- Various trigonometric concepts and formulas are used in the computer graphics of the modern age, as computer graphics are created in 3-D environments, so all the calculations are done by the graphics processing unit of the computer to deliver the computer graphics as output.
- Various astronomical calculations such as the radius of celestial bodies, the distance between objects, etc. involve the use of trigonometry and its different trigonometric ratios.
- In Physics, we use trigonometry to understand and evaluate many real-world systems such as the orbits of planets and artificial satellites, the reflection or refraction of light in various environments, etc.
Trigonometry Formulas
In trigonometry, there are a lot of formulas or identities that relate the different trigonometric ratios with each other for different values of angles. Some of these identities are:
1. Pythagorean Trigonometric Identities
- sin^{2} θ + cos^{2} θ = 1
- 1+tan^{2} θ = sec2 θ
- cosec^{2} θ = 1 + cot^{2} θ
2. Sum and Difference Identities
- sin (A+B) = sin A cos B + cos A sin B
- sin (A-B) = sin A cos B – cos A sin B
- cos (A+B) = cos A cos B – sin 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)
- tan (A-B) = (tan A – tan B)/(1 + tan A tan B)
3. Double angle Identities
- sin 2θ = 2 sinθ cosθ
- cos 2θ = cos^{2}θ – sin ^{2}θ = 2 cos ^{2 }θ – 1 = 1 – sin ^{2} θ
- tan 2θ = (2tanθ)/(1 – tan^{2}θ)
4. Half Angle Identities
- [Tex]\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos \theta}{2}} [/Tex]
- [Tex]\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos \theta}{2}} [/Tex]
- [Tex]\tan \frac{\theta}{2} = \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} =\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta} [/Tex]
5. Product Sum Identities
- [Tex]\sin A+\sin B=2 \sin \frac{A+B}{2} \cos \frac{A-B}{2} [/Tex]
- [Tex]\cos A+\cos B=2 \cos \frac{A+B}{2} \cos \frac{A-B}{2} [/Tex]
- [Tex]\sin A-\sin B=2 \cos \frac{A+B}{2} \sin \frac{A-B}{2} [/Tex]
- [Tex]\cos A-\cos B=-2 \sin \frac{A+B}{2} \sin \frac{A-B}{2} [/Tex]
6. Product Identities
- [Tex]\sin A \cos B=\frac{\sin (A+B)+\sin (A-B)}{2} [/Tex]
- [Tex]\cos A \cos B=\frac{\cos (A+B)+\cos (A-B)}{2} [/Tex]
- [Tex]\sin A \sin B=\frac{\cos ^2(A-B)-\cos (A+B)}{2} [/Tex]
7. Triple Angle Identities
- [Tex]\sin 3 \theta=3 \sin \theta-4 \sin ^3 \theta [/Tex]
- [Tex]\cos 3 \theta= 4 \cos^3 \theta-3 \cos \theta [/Tex]
- [Tex]\cos 3 \theta=\frac{3 \tan \theta-\tan ^3 \theta}{1-3 \tan ^2 \theta} [/Tex]
Unit Circle
The concept of Unit Circle was developed to simplify the process to find angles of sin, cos and tan.
Origin of unit circle is at (0,0) and the radius is 1 unit.
Suppose length of base is X and length of perpendicular is Y. This will give us:
Sin θ | y/1 = y |
Cos θ | x/1 = x |
Tan θ | y/x |
Trigonometry Identities
An equation, which deals with different trigonometric ratios is called Trigonometric Identity if it is true for all possible values of the angles.
Trigonometric identities relate different trigonometric ratios i.e., sin, cos, tan, cot, sec, and cosec, with each other for various different angles. Out of all one of the most basic as well as useful identities are Pythagorean trigonometric identities which are given as follows:
sin^{2} θ + cos^{2} θ = 1
1+tan^{2} θ = sec2 θ
cosec^{2} θ = 1 + cot^{2} θ
Euler’s Formula for Trigonometry
For the imaginary power of exponent e(Euler’s number), Euler gave an identity that relates the imaginary power of e to the trigonometric ratios sin and cos, the identity is given as follows:
[Tex]e^{i\phi} = \cos \phi + i\sin \phi [/Tex]
Where, i is the imaginary number which is defined as i = √(-1), and
Φ is the angle.
Putting, -Φ instead of Φ in the above identity, we get
[Tex]e^{-i\phi} = \cos \phi – i\sin \phi [/Tex]
Now, adding and subtracting these two values together we get, values of sin and cos in terms of imaginary power of Euler’s number,
[Tex]\frac{e^{i\phi}+e^{-i\phi}}{2} = \cos \phi [/Tex]
and
[Tex]\frac{e^{i\phi}-e^{-i\phi}}{2i} = \sin \phi [/Tex]
Trigonometry Real-Life Examples
As Trigonometry is the most widely used concept in mathematics, it has various applications and examples in the real world. One such example is height and distance.
In Height and distance, we can calculate various lengths and angles involving everyday scenarios. For example, we see a shadow of a tree on the ground and want to find the height of that tree (which is very hard to measure as we have to reach to the top of the tree to measure its height).
Using trigonometry, we can calculate the height of the tree without climbing the tree at all. We just need to measure the angle of the sun at the moment and the length of the shadow of the tree at the same moment and using this information we can use trigonometric ratio tan to calculate the height of the tree.
Trigonometry Solved Examples
Example 1: A ladder is leaning against a wall. The angle between the ladder and the ground is 45 degrees, and the length of the ladder is 10 meters. How far is the ladder from the wall?
Solution:
Let the distance between the ladder and the wall be x meters.
Here, ladder, wall and ground together makes a right angle triangle, where for given angle,
Length of ladder = hypotenous = 10 meter,
Distance between wall and laddar = base = x meterUsing trigonometric ratio cos, we get
⇒ cos(45°) = = x/10
⇒ cos(45°) = 1
⇒1/√2 = x/10
⇒ x = 10/√2 = 5√2 meters
Therefore, the ladder is 5√2 meters away from the wall.
Example 2: A right-angled triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30°. What are the lengths of the other two sides?
Solution:
Let’s call the side opposite to the 30° angle as ‘a’ and the side adjacent to it as ‘b’.
Now, sin (30°) = perpendicular/hypotenous = a/10
⇒ a = 10 × sin(30°) = 5 cm [sin(30°) = 1/2]
and cos(30°) = b/10
⇒ b = 10 × cos(30°) = 10 × √(3)/2 ≈ 8.66 cm
Therefore, the lengths of the other two sides are 5 cm and 8.66 cm (approx.).
Example 3: Prove that (cos x/sin x) + (sin x/cos x) = sec x × cosec x.
Solution:
LHS = (cos x/sin x) + (sin x/cos x)
⇒ LHS = [cos^{2}x + sin^{2}x]/(cos x sin x)
⇒ LHS = 1/(cosx sinx) [Using cos^{2}x + sin^{2}x = 1]
⇒ LHS = (1/cosx) × (1/sinx)
⇒ LHS = secx × cosecx = RHS [ 1/cosx = sec x and 1/sinx = cosec x]
Example 4: A person is standing at a distance of 10 meters from the base of a building. The person measures the angle of elevation to the top of the building as 60°. What is the height of the building?
Solution:
Let h be the height of the building.
and, all the distances here in the question make a right angle triangle, with a base of 10 meters and height h meter.
As tan θ = Perpendicular/Base
⇒ tan(60°) = h / 10
⇒ h = 10 tan(60°)
Using the values of tan(60°) = √3, we get:
h = 10√3 ≈ 17.32 m
Therefore, the height of the building is approximately 17.32 meters.
Example 5: Find the value of x in the equation cos^{-1}(x) + sin^{-1}(x) = π/4.
Solution:
For, cos^{-1}(x) + sin^{-1}(x) = π/4
As we know, cos^{-1}(x) + sin^{-1}(x) = π/2, above equation becomes
π/2 = π/4, which is not true.
Thus, the given equation has no such value of x, which can satisfy the equation.
Practice Problems on Trigonometry
1. Given a right triangle ABC where angle A is 90∘, angle B is 30∘, and the hypotenuse AC is 10 units long. Find the lengths of side AB (adjacent to angle B) and side BC (opposite to angle B).
2. Prove that sin^{2}θ+cos^{2}θ= 1 for any angle θ.
3. From the top of a 50-meter tall building, the angle of elevation to the top of a nearby building is 15∘. If the buildings are 100 meters apart on the ground, find the height of the second building.
4. Solve the equation tan(x) = 3 for x, where x is measured in degrees, and 0∘≤x<360∘.
Trigonometry Class 10 PDF
Download the Class 10 Chapter 8 Trigonometry in PDF format to revise concepts and learn concepts from the official NCERT book for CBSE exams.
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Trigonometry in Math’s – FAQs
What is trigonometry in math’s?
Trigonometry is the branch of mathematics that deals with the relationship between angles and lengths in geometric shapes.
What are the three types of trigonometry?
The three types of trigonometry are as follows:
- Core Trigonometry
- Plane Trigonometry
- Spherical Trigonometry
What are the 6 ratios of trigonometry?
The six trigonometric ratios are sin, cos, tan, cot, sec, and cosec.
What is the primary function of trigonometry?
The primary function of the trigonometry is to study and understand the relationships between sides and angles of triangles.
What are trigonometry identities?
An equation that holds true for all angles involving different trigonometric ratios is known as trigonometric identity.
Who invented trigonometry?
Trigonometry as a concept, was first introduced by an ancient Greek mathematician Hipparchus.
What are the basics of trigonometry?
The right-angle triangle is the most basic thing in trigonometry as all the trigonometric ratios are defined as the ratio of sides of right angle triangles.
Where is trigonometry used in real life?
Trigonometry and its functions find diverse applications in our everyday lives. It’s instrumental in geography for measuring distances between landmarks, in astronomy for gauging distances to nearby stars, and crucial in satellite navigation systems for accurate positioning and mapping.
What is trigonometry used for?
Trigonometry solves angle and distance problems in physics, engineering, astronomy, and navigation. It finds unknown angles/side lengths in triangles, analyzes periodic phenomena like waves, oscillations, and models real-world situations involving angles/distances.
How to do trigonometry?
Trigonometry uses angles, triangles, and trig functions (sine, cosine, tangent) to find unknown values. Identify known information, use trig ratios, and practice with identities and formulas for complex problems.