21 Trignometry Formulas for Competitive Exams 

One of the most important concepts to understand in the quantitative aptitude section is trigonometry. The study of triangles is basically what trigonometry is, where “trigon” stands for triangle and “metric” for measurement. Trigonometry deals with how a right-angled triangle’s angles and sides relate to one another and how to use this information to calculate distances and heights. Read our blog to go through the list of the most important maths trigonometry formulas that a student preparing for competitive exams should not miss.

Firstly the trigonometry table can be used to answer the problems as it is a helpful and easy-to-understand resource. The trigonometry table for angles that are frequently used to solve problems is shown below:

Angles (In Degrees)	0°	30°	45°	60°	90°	180°	270°	360°
Angles (In Radians)	0	π/6	π/4	π/3	π/2	π	3π/2	2π
sin	0	1/2	1/√2	√3/2	1	0	-1	0
cos	1	√3/2	1/√2	1/2	0	-1	0	1
tan	0	1/√3	1	√3	∞	0	∞	0
cot	∞	√3	1	1/√3	0	∞	0	∞
cosec	∞	2	√2	2/√3	1	∞	-1	∞
sec	1	2/√3	√2	2	∞	-1	∞	1

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21 Most Important Formulas in Trigonometry

Since many students find it challenging to memorize all of the formulas. It is very important to understand the basic formulas first, as these can be used to derive all other formulas. So are you excited to know what these fundamental identities of trigonometry are? If you’re new to trigonometry, or if you just need to revise your concepts, here are the most important maths trigonometry formulas you need to know:

Basic Ratios sin θ = Opposite Side/Hypotenuse cos θ = Adjacent Side/Hypotenuse tan θ = Opposite Side/Adjacent Side sec θ = Hypotenuse/Adjacent Side cosec θ = Hypotenuse/Opposite Side cot θ = Adjacent Side/Opposite Side	Reciprocal Identities cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ sin θ = 1/cosec θ cos θ = 1/sec θ tan θ = 1/cot θ
Periodic Identities  sin (θ + 2π) = sin θ cos (θ + 2π) = cos θ tan (θ + 2π) = tan θ	Co-function Identities  sin (90° – θ) = cos θ cos (90° – θ) = sin θ tan (90° – θ) = cot θ
Sum and Difference Identities sin (A + B) = sin A cos B + cos A sin B cos (A + B) = cos A cos B – sin 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	Double Angle Identities sin 2θ = 2 sin θ cos θ cos 2θ = 2 cos² θ – 1 or 1 – 2 sin² θ

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Solved Examples

Q1. Solve for x in the equation 2sin(x) – 1 = 0.

Given equation: 2sin(x) – 1 = 0

Adding 1 to both sides: 2sin(x) = 1

Dividing both sides by 2: sin(x) = 1/2

This implies that x is in the first and second quadrants, where sin(x) is positive.

Using the fact that sin(30°) = 1/2, we can write: x = 30° or x = 180° – 30°

x = 30° and x = 150°

Q2. If cos(A) = -3/5 and A is in the second quadrant, find sin(A) and tan(A).

Given, cos(A) = -3/5

Since A is in the second quadrant, sin(A) is positive. 

Using the Pythagorean identity sin² (A) + cos² (A) = 1

sin² (A) + (-3/5)²  = 1

sin² (A) + 9/25 = 1

sin² (A) = 16/25

Taking the positive square root:

sin(A) = 4/5

tan(A) = sin(A)/cos(A) = (4/5)/(-3/5) 

tan (A)= -4/3

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