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Trigonometric Identities

Trigonometric identities are the correspondences including geometrical capacities and remain constant for each worth of the factors in question, with the end goal that the two sides of the fairness are characterised. In this small example, we will investigate mathematical characters. There are three essential geometrical proportions: sin, cos, and tan. The three other geometrical proportions sec, cosec, and cot in geometry are the reciprocals of sin, cos, and tan separately. How are these geometrical proportions (sin, cos, tan, sec, cosec, and bed) associated with one another? They are associated through Trigonometric identities(or in short trig identities).

Let’s understand some basics of Trigonometric Identities

Trigonometric identities are conditions that connect with various mathematical capacities and are valid for any value of the variable that is there in the space. Essentially, identity is a condition that remains constant for every one of the upsides of the variable(s) present in it.

Now let’s focus on some of the algebraic identities

(a + b)2 = a2 + 2ab + b2

(a – b)2 = a2 – 2ab+ b2

(a + b)(a-b)= a2 – b2

The algebraic identity is an interrelation between the variables whereas the trig identities relate the 6 trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent.

Basics of Reciprocal Trigonometric Identities

We already observed that the reciprocals of sin, cosine, and tangent are cosecant, secant, and cotangent respectively.

Due to that, reciprocal trigonometric identities are:

  • sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ
  • cos θ = 1/secθ (OR) sec θ = 1/cosθ
  • tan θ = 1/cotθ (OR) cot θ = 1/tanθ

Now let’s learn about Pythagorean Trigonometric Identities:

Pythagoras Theorem is used in Trigonometry to derive Pythagorean trigonometric identities. If we apply Pythagoras theorem to a Right Angled Triangle then the given below formulae are used:

Opposite2 + Adjacent2 = Hypotenuse2

Dividing both sides by Hypotenuse2

Opposite2/Hypotenuse2 + Adjacent2/Hypotenuse2 = Hypotenuse2/Hypotenuse2

  • sin2θ + cos2θ = 1

This is one of the Pythagorean identities. Similarly, we can derive 2 other Pythagorean trigonometric identities as follows:

  • 1 + tan2θ = sec2θ
  • 1 + cot2θ = cosec2θ

Now let’s understand the other part of trigonometric identities – Complementary and Supplementary Trigonometric Ideas:A pair of two angles such that their sum is equal to 90° is called a Complementary Angle. θ is the complement angle of (90 – θ). Now let’s learn the ratios of Complementary Angles:

  • sin (90°- θ) = cos θ
  • cos (90°- θ) = sin θ
  • cosec (90°- θ) = sec θ
  • sec (90°- θ) = cosec θ
  • tan (90°- θ) = cot θ
  • cot (90°- θ) = tan θ

A pair of two angles such that their sum is equal to 180° is called a Supplementary Angle. θ is the supplement angle of  (180 – θ). Now let’s learn the ratios of Supplementary Angles:

  • sin (180°- θ) = sinθ
  • cos (180°- θ) = -cos θ
  • cosec (180°- θ) = cosec θ
  • sec (180°- θ)= -sec θ
  • tan (180°- θ) = -tan θ
  • cot (180°- θ) = -cot θ

Now let’s understand The Sum And Difference of various Trigonometric Ideas:

The sum and difference identities consist of the formulas of sin(A+B), cos(A-B), cot(A+B), etc.

  • sin (A+B) = sin A cos B + cos A sin B
  • sin (A-B) = sin A cos B – cos A sin B
  • cos (A+B) = cos A cos B – sin A sin B
  • cos (A-B) = cos A cos B + sin A sin B
  • tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
  • tan (A-B) = (tan A – tan B)/(1 + tan A tan B)

Now let’s move on to the another interesting concept-

Double and Half Angle Trigonometric Ideas:

The sum and difference formulae can be applied to obtain the double angle trigonometric identities.

For example, from the above formulas:

sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, we get:

sin (θ + θ) = sinθ cosθ + cosθ sinθ

sin 2θ = 2 sinθ cosθ

Similarly, we can apply the other double-angle identities.

  • sin 2θ = 2 sinθ cosθ
  • cos 2θ = cos2θ – sin 2θ
    = 2 cos2θ – 1
    = 1 – sin 2 θ
  • tan 2θ = (2tanθ)/(1 – tan2θ)

Half Angle Formulas

Using one of the stated above double angle formulas,

cos 2θ = 1 – 2 sin2θ

2 sin2θ = 1- cos 2θ

sin2θ = (1 – cos2θ)/(2)

sin θ = ±√[(1 – cos 2θ)/2]

Replacing θ by θ/2 on both sides,

sin (θ/2) = ±√[(1 – cos θ)/2]

This is the half-angle formula of sin.

Similarly, we can apply the other half-angle formulas.

sin (θ/2) = ±√[(1 – cosθ)/2]

cos (θ/2) = ±√(1 + cosθ)/2

tan (θ/2) = ±√[(1 – cosθ)(1 + cosθ)]

Now let’s learn the Cosine and Sine Rule Trigonometric Identities:

The relation between the angles and the corresponding sides of a triangle is called the Sine Rule.

 We use the sine rule and the cosine rule for non-right angled triangles. Sine rule can be given as, 

  •       a/sinA = b/sinB = c/sinC
  • sinA/a = sinB/b = sinC/c
  • a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

The relation between the angles and the sides of a triangle which is usually used when two sides and the included angle of a triangle are given.In this case we can say that this is Cosine Rule.

  • a2 = b2 + c2 – 2bc·cosA
  • b2 = c2 + a2 – 2ca·cosB
  • c2 = a2 + b2 – 2ab·cosC

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