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TRIGONOMERTRY: Basic Concepts, Formulae & Solved Examples (Part 1)

Trigonometry

Angles: The amount of turn between two straight lines that have a common point is called angles.

Generally, angles are measured in degree or radian

1 degree = 60 minutes

1 minutes = 60 seconds

Suppose an angle is given as 45°23’53’’, we read it as 45 degrees 23 minutes and 53 seconds.

The basics of trigonometry start with right angle triangle. It is given as below :

Trigonometric ratios

sinθ = P/H                   cosecθ = H/P

cosθ = B/H                   secθ = H-P

tanθ = P/B                    cotθ = B/P

 

Let’s see the sign of trigonometric ratios in various quadrants

  • In the first quadrant (0<θ<90), all trigonometric ratios are positive
  • In the 2nd quadrant (90<θ<180), only sinθ and cosecθ are positive
  • In the 3rd quadrant (180<θ<270), only tanθ and cotθ are positive
  • In the 4th quadrant (270<θ<360), only cosθ and secθ are positive

 

Below table shows the values of various trigonometric ratios at defined angles

 

Some common Trigonometric Identitites are :

  • 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
  • sinθ × cosecθ = 1
  • cosθ × secθ = 1
  • tanθ × cotθ = 1

 

 

  • sinθ = 1/ cosec θ
  • cosθ = 1/ secθ
  • tanθ = sinθ/cosθ

 

 

  • sin2 θ + cos2 θ = 1
  • sec2 θ – tan2 θ = 1
  • cosec2 θ – cot2 θ = 1

 

 

  • sin (- θ) = – sin θ
  • cos (- θ) = cos θ
  • tan (- θ) = – tan θ
  • cot (- θ) = – cot θ
  • cosec (- θ) = – cosec θ
  • sec (- θ) = sec θ

 

 

 

Some practice questions are given below:

1. For all real A, the simplified value of the expression sin6A + cos6A + 3sin2A cos2A is
  • -2
  • 2
  • 0
  • 1

 

 

2. If cotθ =√3, find the value of tan2θ– sin2θ + cos2θ
  • 3/5
  • 3√3 – (1/2)
  • √3 +(1/2)
  • 5/3

 

 

3.  If x = rcosi .cosi, y= rcosi .sini and z = r sini, then the value of x2 + y2 + z2 is-
  • r^2
  • 1/r
  • 1/r^2
  • r

 

 

4. If sinθ + cosecθ = 2, then the value of sin9θ + cosec9θ is –
  • 4
  • 0
  • 1
  • 2

 

 

5. If 2y cosA -x sinA = 0 and 2x secA -y cosec A = 3, then the value of x2 +4y2 is-
  • 2
  • 4
  • 1
  • 0

 

 

6. What is the value of [(1-sin2θ)sec2θ+tan2θ] (cos2θ+1) when 0o∠θ∠90o?
  • <2
  • 2
  • >2
  • > or = 2

 

 

7. 0o < x < 90o and 2sinx+15cos2x=7, then what is the value of tan x ?
  • 3/5
  • 3/4
  • 4/3
  • 4/5

 

 

Answer

1.  1

2. √3 +(1/2)

3.  r^2

4.  2

5. 4

6. > 2

7. 4/3

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