Basic Trigonometry Formulas List: Trigonometry Table & Identities

Trigonometric formulas are first introduced to students in NCERT Class 10 Maths textbook. Trigonometric formulas and identities are helpful tools for students to study the applications of trigonometry. Besides, scientists, geologists, and astronomers apply these formulas when carrying out certain experiments. 

Here, you can learn basic trigonometric formulas that will come into use in higher classes as well. As you scroll, you can also explore trigonometric formulas and identities that are part of Class 11 and 12 Maths syllabus. Before you proceed with formulas, know the definition and applications of trigonometry. 

What is Trigonometry?

In Greek, the word ‘trigonometry’ is broken down into three parts; ‘tri’ means three, ‘gon’ means sides, and ‘metron’ means to measure. In other words, trigonometry is a branch of Mathematics through which you can study the relationship between the sides and angles of a triangle. 

Trigonometry Formulas for Class 10

This chapter is first introduced in CBSE Class 10 Maths Book so that students learn to calculate the distance between the stars and planets from Earth or other celestial objects. Students who wish to pursue engineering, architecture or astronomy shall learn the basic concepts of trigonometry in NCERT Class 10 Maths. If you have started practising this chapter, our trigonometry formulas for Class 10 will be helpful for you. 

1. Trigonometric Ratios

In trigonometric ratios, we will define the ratios for angles of measure 0° and 90°. With the help of formulas, we can calculate trigonometric ratios for some specific angles. 

Taking a right-angled triangle as a reference, let us know the trigonometric ratios of angle 𝚹. 

• sine of ∠𝚹 = Side opposite to angle 𝚹/ Hypotenuse
• cosine of ∠𝚹 = Side adjacent to angle 𝚹/ Hypotenuse
• tangent of ∠𝚹 = Side opposite to angle 𝚹/ Side adjacent to angle 𝚹
• cosecant of ∠𝚹 = 1/ sin 𝚹 =  Hypotenuse/ Side opposite to angle 𝚹
• secant of ∠𝚹 = 1/ cos 𝚹 = Hypotenuse/ Side adjacent to angle 𝚹
• cotangent of ∠𝚹 = 1/ tan 𝚹 = Side adjacent to angle 𝚹/ Side opposite to angle 𝚹

2. Trigonometry Table

With the help of trigonometry table, you can know the values of all the trigonometric ratios of measure 0°, 30°, 45°, 60° and 90°. 

3. Trigonometric identities

In earlier classes, you have learnt that an equation is called an identity when its left hand is equal to right hand side. Similarly, a trigonometry identity is an equation which involves the use of trigonometric ratios. Go through the list of trigonometric identities of Class 10 below. 

• cos2𝚹 + sin2𝚹 = 1
• 1 + tan2𝚹 = sec2𝚹
• 1 + cot2𝚹 = cosec2𝚹

4. Trigonometric ratios of complementary angles

Recall from earlier classes, two angles are said to be complementary if their sum is equal to 90°. 

• sin (90° - 𝚹) = cos 𝚹
• cos (90° - 𝚹) = sin 𝚹
• tan (90° - 𝚹) = cot 𝚹
• cot (90° - 𝚹) = tan 𝚹
• sec (90° - 𝚹) = cosec 𝚹
• cosec (90° - 𝚹) = sec 𝚹

Trigonometric Formulas for Class 11

After you have brushed up the basic trigonometry formulas, you can proceed with attempting trigonometry questions of NCERT Class 11 Maths Book by applying relevant formulas given here. 

1. Relation between degree and radian

A complete angle measure 360° which forms a circle. An angle can also be measured in degrees or in radians. 

• Radian measure = (π/180) x Degree measure
• Degree measure = (180/π) x Radian measure

2. Signs of trigonometric functions in different quadrants

You can find the signs of trigonometric functions in different quadrants by referring to the table below. 

3. Negative angles (Even-odd identities) 

• sin (-x) = - sin x
• cos (-x) = cos x
• tan (-x) = - tan x
• sec (-x) - sec x
• cosec (-x) = - cosec x
• cot (-x) = -cot x

4. Shifting angle by π/2, π and 3π/2

5. Trigonometric Functions of Sum and Difference of Two Angles

Below is the list of derived trigonometry functions having sum and difference of two angles. 

• cos (x + y) = cos x cos y - sin x sin y
• cos (x - y) = cos x cos y + sin x sin y
• sin (x + y) = sin x cos y + cos x sin y
• sin (x - y) = sin x cos y - cos x sin y
• tan (x + y) = (tan x + tan y)/ (1 - tan x tan y)
• tan (x - y) = (tan x - tan y)/ (1 + tan x tan y)

6. Trigonometry Formulas involving Double Angle

• sin 2x = 2 sin x cos x = 2 tan x/ (1 + tan2 x) 
• cos 2x = cos2x - sin2 x = 1 - 2sin2x = 2cos2x - 1 = (1 - tan2x)/ (1 + tan2x)
• tan 2x = 2 tan x/ (1 - tan2x)

7. Trigonometry Formulas involving Triple Angle 

• sin 3x = 3 sin x - 4 sin3x
• cos 3x = 4 cos3x - 3 cos x
• tan 3x = (3 tan x - tan3 x)/ (1 - 3 tan2x)

8. Trigonometry Formulas involving Half-Angle Identities

• sin x = 2 sin (x/2) cos (x/2) = 2 tan (x/2)/ {1 + tan2(x/2)}
• cos x = cos2(x/2) - sin2(x/2) = 1 - 2sin2(x/2) = 2cos2(x/2) - 1 = {1 - tan2(x/2)}/ {1 + tan2(x/2)}
• tan x = 2 tan (x/2)/ {1 - tan2 (x/2)}
• cos2 (x/2) = ½ (1 + cos x)
• sin2 (x/2) = ½ (1 - cos x)

9. Trigonometry Formulas involving Sum Identities (Sum to Product Identities)

• cos x + cos y = 2 cos {(x + y)/2} cos {(x - y)/2}
• cos x - cos y = -2 sin {(x + y)/2} sin {(x - y)/2}
• sin x + sin y = 2 sin {(x + y)/2} cos {(x - y)/2}
• sin x - sin y = 2 cos {(x + y)/2} sin {(x - y)/2}

10. Trigonometry Formulas (Product Identities)

• 2 cos x cos y = cos (x + y) + cos (x - y)
• -2 sin x sin y = cos (x + y) - cos (x - y)
• 2 sin x cos y = sin (x + y) + sin (x - y)
• 2 cos x sin y = sin (x + y) - sin (x - y)

Note down the trigonometry formulas list in your notebook for future reference. Develop a habit of revising all trigonometry formulas on a daily basis so that you can remember these for the long term.