Basics Concepts of Trigonometry

By Ashwini Shivhare|Updated : April 5th, 2023

Trigonometry is one of the important topics for competitive exams. In order to learn it well, one needs to start with the basics. We will cover all the important concepts in this chapter. We will also discuss the concepts of Heights and Distances. These concepts are very important from exam point of view for upcoming SSC exams 2023.

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Here are the important concepts and formulas in Trigonometry. Let's start with the values of different angles in different quadrants:

Let's read about some very important properties of the

(1) In the hexadecimal system, the angles are measured in degrees, minutes and seconds.

One complete rotation = 360 degrees (360⁰) 1⁰ = 60 minutes (60’); 1’ = 60 seconds (60”)

In the circular system, the angles are measured in radians. π radians (π^c) = 180⁰

(2) Let S = length of arc AB

θ = angle AOB expressed in radians and r = radius of the circle

Then, S = r × θ

(3) And the area of sector AOB is A = (1/2) × r² × θ

(4) Consider the right angle triangle ABC as shown

Sine θ = Opposite Side/Hypotenuse = BC/AC

Cosine θ = Adjacent Side/Hypotenuse = AB/AC

Tangent θ = Opposite Side/Adjacent Side = BC /AB

Cosec θ = 1/sinθ

Sec θ = 1/cos θ

Cot θ = 1/tan θ

(5) Limiting Value of the Trigonometrical Ratios

Since sin² θ + cos² θ = 1, hence each of sin θ and cos θ is numerically less than 1; or their maximum value is 1.
Since sec θ and cosec θ are respectively reciprocals of cos θ and sin θ, therefore the values of sec θ and cosec θ are always greater than 1; or their minimum value is 1.
tan θ and cot θ can have any value between - infinity and + infinity.

(6) Signs Of Trigonometrical Ratios

1. Quadrant: 0 to 90, All + ve

2. Quadrant: 90° to 180°, only sin θ and cosec θ + ve, rest - ve

3. Quadrant: 180° to 270°, only tan θ and cot θ + ve, rest - ve

4. Quadrant: 270° to 360°, only cos θ and sec θ + ve, rest - ve

(7) Trigonometrical Ratios Of Allied Angles

1. sin ( - θ) = - sin θ; cosec (- θ) = - cosec θ

cos ( - θ) = cos θ; sec (- θ) = sec θ;

tan ( - θ) = - tan θ; cot (- θ) = - cot θ

2. Trigonometric ratios of (90° - θ) in terms of those of θ, for all values of θ:

sin (90° - θ) = cos θ; cosec (90° - θ) = sec θ

cos (90° - θ) = sin θ; cosec (90° - θ) = cosec θ

tan(90° - θ) = cot θ; cot (90° - θ) = tan θ

3. sin (90° + θ) = cos θ; cosec (90° + θ) = sec θ

cos (90° + θ) = - sin θ; cosec (90° + θ) = - cosec θ

tan(90° + θ) = - cot θ; cot (90° + θ) = tan θ

4. sin (180° - θ) = cos θ; cosec (180° - θ) = cosec θ

cos (180° - θ) = - cos θ; cosec (180° - θ) = sec θ

tan(180° - θ) = - tan θ; cot (90° -θ) = - cot θ

Important Points:

Since angles 360° - θ and (- θ) are coterminal angles, the trigonometric ratios of (360° - θ) and (- θ) must be identical.
Since angles 360° - θ and θ are coterminal angles, the trigonometric ratios of 360° + θ and θ must be identical.

Maximum and Minimum Value Trigonometric Identity:

Type-I:

In case of sec²x, cosec²x, cot²x and tan²x, we cannot find the maximum value because they can have infinity as their maximum value. So in question containing these trigonometric identities, you will be asked to find the minimum values only. The typical question forms are listed below: