Trigonometry 10th-Standard
Trigonometry The word ‘Trigonometry’ means measurement of triangles.
Concept of Perpendicular, Base and Hypotenuse in a Right Triangle
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For any acute angle θ in a right angled triangle;
(i) For angle θ Perpendicular (P)= BC Base (B) = AB Hypotenuse (H)= AC |
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Trigonometry Ratios
The ratios of the sides of a right-angled triangle with respect to its acute angles are called Trigonometric ratios.
The three sides of a right-angled triangle give six trigonometrical ratios: sine, cosine, tangent, cotangent, secant and cosecant. In short, these ratios are written as sin, cos, tan, cot, sec, and cosec respectively.
- Trigonometric Ratios of θ in right angled △ABC are defined below
| sinθ | (Perpendicular/Hypotenuse) | BC/AC |
| cosθ | (Base/Hypotenuse) | AB/AC |
| tanθ | (Perpendicular/Base) | BC/AB |
| cosecθ | (Hypotenuse/Perpendicular) | AC/BC |
| secθ | (Hypotenuse/Base) | AC/AB |
| cotθ | (Base/Perpendicular) | AB/BC |
- Trigonometric Ratios of complementary Angles If the sum of two angles is 90°, then one angle is said to be complementary of the other. For Example in the below figure ∠B = 90°, ∠A + ∠C = 90°. So, ∠A & ∠C are complementary to each other.
| (i) sin (90° – θ) = (AB/AC) =cos θ
(ii) cos (90° – θ) = (BC/AC) = sin θ (iii) tan (90° – θ) = (AB/BC) =cot θ (iv) cosec (90° – θ) = (AC/AB) = sec θ (v) sec (90° – θ) = (AC/BC) = cosec θ (vi) cot (90° – θ) = (BC/AB) = tan θ |
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- Reciprocal Relation
(i) sin θ = (1/cosec θ) = (sin θ) . (cosec θ) = 1
(ii) cos θ = (1/sec θ) = (cos θ) . (sec θ) = 1
(iii) tan θ = (1/cot θ) = (tan θ) . (cot θ) = 1
- Quotient Relation
(i) tan θ = (sin θ/cos θ) (ii) cot θ = (cos θ/sin θ)
- Fundamental Identities
| sin2θ + cos2θ = 1 ⇒ sin2θ = 1 – cos2θ and cos2θ = 1 – sin2θ |
| sec2θ – tan2θ = 1 ⇒ 1+ tan2θ = sec2θ and sec2θ – 1 = tan2θ |
| cosec2θ – cot2θ = 1 ⇒ 1 + cot2θ = cosec2θ and cosec2θ – 1 = cot2θ |