# Study Notes on Trigonometric Applications

By Gaurav Mohanty|Updated : December 27th, 2021

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 of this chapter. We will also discuss the concepts of Heights and Distances.

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

(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 (πc) = 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

1. 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.
2. 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.
3. 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:

1. Since angles 360° - θ and (- θ) are coterminal angles, the trigonometric ratios of (360° - θ) and (- θ) must be identical.
2. 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:

Example: -1

Find the Minimum value of 9 cos 2x + 2 sec 2x

sol - this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

• Minimum Value = 2√ 9 x  2= 2√ 18

Example:-2

Find the Minimum value of 8 tan 2x + 7 cot 2x
sol - this equation is a typical example of our type-3 so apply the formula 2√ ab   so,

• Minimum Value = 2√ 8 x  7= 2√ 56

Type -II:

Example -1

Find the Maximum and Minimum Value of 3 sin x + 4 cos y

Sol- If you find the question of this kind, apply the above formulae.

• Maximum Value = √ 9 + 16   = √ 25   = 5
• Minimum Value = - √ 9 + 16   = - √ 25   =  - 5

Example-2

Find the Maximum and Minimum Value of 3 sin x + 2 cos y

Sol- If you find the question of this kind, apply the above formulae.

• Maximum Value = √ 9 + 4   = √ 13
• Minimum Value = - √ 9 + 4   = - √ 13

Type III:

Example -1

Find the maximum and Minimum Value of 3 sin 2x + 4 cos 2x

Sol- Here the 4> 3 so

•  Maximum Value = 4
• Minimum Value = 3

Example –2

Find the maximum and Minimum Value of 5 sin 2x + 3 cos 2x

Sol - Here 5>3

• Maximum Value = 5
• Minimum Value = 3

Type-IV

Find the Minimum Value of  Sec 2x +  cosec 2x

Sol - 1 + tan 2x +  cosec 2x    --------------------------------------------(Sec 2x = 1 + tan 2x)

= 1+ tan 2x + 1 +  cot 2x ------------------------------------------------(cosec 2x = 1 + cot 2x )

=2 + tan 2x +  cot 2x---------------------------------------------------apply type-3 formula

=2 + 2 √ 1 x  1 = 2 + 2 =4

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