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Trigonometry Part-1 || Chapter – 8 || Class 10th (For Hindi Medium)

Trigonometry Part-2 || Chapter – 8 || Class 10th (For Hindi Medium)

Trigonometric Ratios

Opposite & Adjacent Sides in a Right Angled Triangle

In the ΔABC right-angled at B, BC is the side opposite to ∠A, AC is the hypotenuse and AB is the side adjacent to ∠A.

Introduction To Trigonometry

Trigonometric Ratios

For the right ΔABC, right-angled at ∠B, the trigonometric ratios of the ∠A are as follows:

  • sin A=opposite side/hypotenuse=BC/AC
  • cos A=adjacent side/hypotenuse=AB/AC
  • tan A=opposite side/adjacent side=BC/AB
  • cosec A=hypotenuse/opposite side=AC/BC
  • sec A=hypotenuse/adjacent side=AC/AB
  • cot A=adjacent side/opposite side=AB/BC

To know more about Trigonometric Ratios, visit here.

Visualization of Trigonometric Ratios Using a Unit Circle

Draw a circle of the unit radius with the origin as the centre. Consider a line segment OP joining a point P on the circle to the centre which makes an angle θ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

  • sinθ=PQ/OP=PQ/1=PQ
  • cosθ=OQ/OP=OQ/1=OQ
  • tanθ=PQ/OQ=sinθ/cosθ
  • cosecθ=OP/PQ=1/PQ
  • secθ=OP/OQ=1/OQ
  • cotθ=OQ/PQ=cosθ/sinθ
Introduction To Trigonometry

Visualisation of Trigonometric Ratios Using a Unit Circle

Relation between Trigonometric Ratios

  • cosec θ =1/sin θ
  • sec θ = 1/cos θ
  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ=1/tan θ

Trigonometric Ratios of Specific Angles

Range of Trigonometric Ratios from 0 to 90 degrees

For 0∘≤θ≤90∘,

  • 0≤sinθ≤1
  • 0≤cosθ≤1
  • 0≤tanθ<∞
  • 1≤secθ<∞
  • 0≤cotθ<∞
  • 1≤cosecθ<∞

tanθ and secθ are not defined at  90∘.
cotθ and cosecθ are not defined at 0∘.

Variation of trigonometric ratios from 0 to 90 degrees

As θ increases from 0∘ to 90∘

  • sin θ increases from 0 to 1
  • cos θ decreases from 1 to 0
  • tan θ increases from 0 to ∞
  • cosec θ decreases from ∞ to 1
  • sec θ increases from 1 to ∞
  • cot θ decreases from ∞ to 0

Standard values of Trigonometric ratios

∠A 0o 30o 45o 60o 90o
sin A 0  1/2  1/√2  √3/2  1
cos A 1  √3/2 1/√2  1/2 0
tan A 0  1/√3  1 √3  not defined
cosec A not defined  2  √2  2/√3  1
sec A 1 2/√3 √2  2  not defined
cot A not defined  √3  1 1/√3 0
Introduction To Trigonometry

To know more about Trigonometric Ratios of Standard Angles, visit here.

Trigonometric Ratios of Complementary Angles

Complementary Trigonometric ratios

If θ is an acute angle, its complementary angle is 90∘−θ. The following relations hold true for trigonometric ratios of complementary angles.

  • sin (90∘− θ) = cos θ
  • cos (90∘− θ) = sin θ
  • tan (90∘− θ) = cot θ
  • cot (90∘− θ) = tan θ
  • cosec (90∘− θ) = sec θ
  • sec (90∘− θ) = cosec θ

To know more about Trigonometric Ratios of Complementary Angles, visit here.

Trigonometric Identities

  • sin2θ+cos2θ=1
  • 1+cot2θ=coesc2θ
  • 1+tan2θ=sec2θ
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