TRIGONOMETRIC IDENTITIES

Trigonometric Identities :

In mathematics, trigonometric identities are equalities where we would have trigonometric functions and they would be true for every value of the occurring variables. Geometrically, these are identities involving certain functions of one or more angles.

Trigonometric Identities - SOHCAHTOA

Sin θ = Opposite side / Hypotenuse side

Cos θ = Adjacent side / Hypotenuse side

Tan θ = Opposite side / Adjacent side

Csc θ = Hypotenuse side / Opposite side

Sec θ = Hypotenuse side / Adjacent side

Cot θ = Adjacent side / Opposite side

Reciprocal Trigonometric Identities

Sinθ = 1 / Cosecθ

Cscθ = 1 / Sinθ

Cosθ = 1 / Secθ

Secθ = 1 / Cosθ

Tanθ = 1 / Cotθ

Cotθ = 1 / Tanθ

Other Trigonometric Identities

Sin²θ + Cos²θ = 1

Sin²θ = 1 - Cos²θ

Cos²θ = 1 - Sin²θ

Sec²θ - Tan²θ = 1

Sec²θ = 1 + Tan²θ

Tan²θ = Sec²θ - 1

Csc²θ - Cot²θ = 1

Csc²θ = 1 + Cot²θ

Cot²θ = Csc²θ - 1

Double Angle Identities

Sin2A = 2 ⋅ SinA ⋅ CosA

Cos2A = Cos²A - Sin²A

Tan2A = 2 ⋅ TanA / (1 - Tan²A)

Cos2A = 1 - 2 ⋅ Sin²A

Cos2A = 2 ⋅ Cos²A - 1

Sin2A = 2 ⋅ TanA / (1 + Tan²A)

Cos2A = (1 - Tan²A) / (1 + Tan²A)

Sin²A = (1 - Cos2A) / 2

Cos²A = (1 + Cos2A) / 2

Half Angle Identities

SinA = 2 ⋅ Sin(A/2) ⋅ Cos(A/2)

CosA = Cos²(A/2) - Sin²(A/2)

TanA = 2 ⋅ Tan(A/2) / [1 - Tan²(A/2)]

CosA = 1 - 2 ⋅ Sin²(A/2)

CosA = 2 ⋅ Cos²(A/2) - 1

SinA = 2 ⋅ Tan(A/2) / [1 + Tan²(A/2)]

CosA = [1 - Tan²(A/2)] / [1 + Tan²(A/2)]

Sin²A/2 = (1 - Cos A) / 2

Cos²A/2 = (1 + Cos A) / 2

Tan²(A/2) = (1 - CosA) / (1 + CosA)

Compound Angles Identities

Sin(A + B) = SinA ⋅ CosB + CosA ⋅ SinB

Cos(A + B) = CosA ⋅ CosB - SinA ⋅ SinB

Cos(A - B) = CosA ⋅ CosB + SinA ⋅ SinB

Tan(A + B) = [TanA + TanB] / [1- TanA ⋅ TanB]

Tan(A - B) = [TanA - TanB] / [1 + TanA ⋅ TanB]

Sum to Product Identities

SinC + SinD = 2 ⋅ Sin[(C+D) / 2] ⋅ cos [(C-D) / 2]

SinC - SinD = 2 ⋅ Cos [(C+D) / 2] ⋅ Sin [(C-D) / 2]

CosC + CosD = 2 ⋅ Cos [(C+D) / 2] ⋅ Cos [(C-D) / 2]

CosC - CosD = 2 ⋅ Sin [(C+D) / 2] ⋅ Sin [(C-D) / 2]

Triple Angle Identities

Sin3A = 3 ⋅ SinA - 4 ⋅ sin³A

Cos3A = 4 ⋅ Cos³A - 3 ⋅ Cos A

Tan3A = [3 ⋅ TanA - Tan³A] / [1 - 3 ⋅ Tan²A]

Values of Certain Angles

Solving Word Problems Using Trigonometric Identities

Step 1 :

Understanding the question and drawing the appropriate diagram are the two most important things to be done in solving word problems in trigonometry.

Step 2 :

If it is possible, we have to split the given information. Because, when we split the given information in to parts, we can understand them easily.

Step 3 :

We have to draw diagram almost for all of the word problems in trigonometry. The diagram we draw for the given information must be correct. Drawing diagram for the given information will give us a clear understanding about the question.

Step 4 :

Once we understand the given information clearly and correct diagram is drawn, solving word problems in trigonometry would not be a challenging work.

Step 5 :

After having drawn the appropriate diagram based on the given information, we have to give name for each position of the diagram using English alphabets (it is clearly shown in the word problem given below). Giving name for the positions would be easier for us to identify the parts of the diagram.

Step 6 :

Now we have to use one of the three trigonometric ratios (sin, cos and tan) to find the unknown side or angle.

Once the diagram is drawn and we have translated the English Statement (information) given in the question as mathematical equation using trigonometric ratios correctly, 90% of the work will be over. The remaining 10% is just getting the answer. That is solving for the unknown.

These are the most commonly steps involved in solving word problems in trigonometry.