Trigonometric Identity Trigonometric Identity If two expressions are equal for all values of the variables, then the relation is called an identity. For example x2 – 9 = (x – 3) (x + 3) and (x – a) (x – b) / (c – a) (c – b) + (x – b) (x – c) / (a – b) (a – c) + (x – c) (x – a) / (b – c) (b – a) = 1. are algebraic identities as they are satisfied by every value of the variable x. In other words an equation that includes trigonometric ratios of an angle θ is said to be a trigonometric identity if it is satisfied for all values of θ for which the given trigonometric ratios are defined. For example cos2 θ - 1/2 = cos θ (cos θ - ½ ) is a trigonometric identity, whereas cos θ (cos θ - ½) = 0 is an equation. Also sec θ = 1/ cos θ is a trigonometric identity, because it holds for all values of q except for which cos θ = 0. For cos θ = 0. sec θ is not defined. Know More About Trig Function
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The following steps should be kept in mind while verifying trigonometric identities worksheet. a) Start with more complicated side of the identity and prove it equal to the other side. b) If the identity contains sine, cosine and other trigonometric ratios, then express all the ratios in terms of sine and cosine. c) If one side of an identity cannot be easily reduced to the other side then simplifies both sides and proves them identical equal. d) While proving identities, never transfer terms from one side to another. Lets prove some identities: 1. sin2 θ + cos2 θ = 1 In a right a right triangle ABC, let ∠C = θ, then by Pythagoras theorem, we have AB2 + BC2 = AC2 Dividing both sides by AC2 , we get AB2 / AC2 + BC2 / AC2 = AC2 / AC2 AB2 / AC2 + BC2 / AC2 = 1 (sin θ)2 + (cos θ )2 = 1 (sin θ = AB / AC, cos θ = BC / AC) Read More About LSA of Right Circular Cones Worksheet
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sin2 θ + cos2 θ = 1 As a sequence of the above identity we can use 1 – sin2 θ = cos2 θ and 1 - cos2 θ = sin2 θ 2. 1 + tan 2 θ = sec2 θ In a right a right triangle ABC, let ∠C = θ, then by Pythagoras theorem, we have AB2 + BC2 = AC2 Dividing both sides byBC2 , we get AB2 / BC2 + BC2 / BC2 = AC2 / BC2 AB2 / AC2 + BC2 / AC2 = 1 (AB / BC)2 + 1 = ( AC / BC) 2 (tan θ)2 + 1 = (sec θ )2 (AB / BC = tan θ , AC / BC = sec θ) 1 + tan 2 θ = sec2 θ As a sequence of the above identity we can use sec2 θ - tan2 = 1 and sec 2 - 1= tan2 θ Another identity is 1 + cot2 θ = cosec2 θ, we can also prove this by the same way.
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