Chapter 8 – Introduction To Trigonometry

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 ratio 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 ∞  1
• sec θ increases from 1 to ∞
• cot θ decreases from ∞  0

Standard values of Trigonometric Ratios

Complementary angles are those sets of two angles whose sum is equal to 90 o. Let us understand it through an example. Let say there are two angles ∠A and ∠B. They will be called complementary if,

∠A + ∠B = 90 o

Here,  ∠A is complementary to ∠B and vice versa. 

The lengths of sides of a right-angle triangle’s relationship with the acute angle are shown by trigonometric ratios.  The following relation is established 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

So it is concluded that:-

• Sin of an angle = Cos of its complementary angle
• Cos of an angle = Sin of its complementary angle
• Tan of an angle = Cot of its complementary angle

Trigonometric identities are helpful and useful when trigonometric functions are included in an expression or an equation. However, these identities have certain functions of one or more angles. 

Trigonometric Identities List

Reciprocal Identities 

• Sin θ = 1/Csc θ or Csc θ = 1/Sin θ
• Cos θ = 1/Sec θ or Sec θ = 1/Cos θ
• Tan θ = 1/Cot θ or Cot θ = 1/Tan θ

Pythagorean Identities

• sin2 a + cos2 a = 1
• 1+tan2 a = sec2 a
• Cosec2 a = 1 + cot2 a

Ratio Identities

• Tan θ = Sin θ/Cos θ
• Cot θ = Cos θ/Sin θ

Opposite Angle Identities

• Sin (-θ) = – Sin θ
• Cos (-θ) = Cos θ
• Tan (-θ) = – Tan θ
• Cot (-θ) = – Cot θ
• Sec (-θ) = Sec θ
• Csc (-θ) = – Csc θ