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TRIGONOMETRY – APPLICATIONS, RATIOS AND IDENTITIES EXPLAINED

The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.


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The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.

Application of Trigonometry

Trigonometry has various applications in our daily life, it can be used to:

  1. Find height of tall building or tower without actually measuring it
  2. Find the height at which an object , airplane or balloon is flying
  3. Find the distance of a vehicle or ship from a tower.
  4. Used extensively in the field of construction.

Trigonometric Ratios

There are a total of six trigonometric ratios which can be represented in terms of the sides and angles of a triangle. The six trigonometric ratios – sine, cosine, tangent, secant, cosecant and cotangent along with their formulas are given below:

In right ?ABC

sine of ? A = Perpendicular/Hypotenuse = BC/AC

cosine of ? A = Base/ Hypotenuse  = AB/AC

tangent of ? A = Perpendicular /Base = BC/AB

cosecant of ? A = 1/ sine of A = AC/BC

secant of ? A = 1/cosine of A  = AC/AB

cotangent of ? A = 1/tangent of A = AB/BC

  • The value of sin A or cos A is always between +1 and – 1, whereas the value of sec A or cosec A is always greater than or equal to 1 or less than and equal to -1.
  • Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.
  • The values of different trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, given the angle remains the same.

Greek letter ? (theta) is also used to denote an angle

Trigonometric Identities

Some common formulas associated with trigonometric ratios which are useful in solving sums are :

sin2 A + cos2 A = 1,

secA – tan2 A = 1 for 0° ? A < 90°,

cosec2 A = 1 + cot2 A for 0° < A ? 90°.

 

Numerical:  Express the ratios cos A, tan A and sec A in terms of sin A.

Solution:     cos2 A + sin2 A = 1,

Or  cos2 A = 1 – sin2 A, i.e., cos A = ±? 1 ? sin2 A

or  cos A = ?1 ? sinA      (Ignoring the negative value , since sin A is positive, Cos A will also be positive.)

tan A = sin A/cos A   = sin A/?1 ? sinA

sec A = 1/cosA  = 1/?1 ? sinA