TRIGONOMETRIC IDENTITIES

About "Trigonometric identities"

On this web page "Trigonometric identities" we are going to see the formulas which are being used to solve all kinds of problems on Trigonometry. 

Trigonometric Identities

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

Now let us see the reciprocal trigonometric identities

sin θ = 1/Cosec θ

csc θ = 1/sin θ

cos θ = 1/sec θ

sec θ = 1/cos θ

tan θ = 1/cot θ

cot θ = 1/tan θ

Let us see some other important trigonometric identities

sin² θ  + cos² θ = 1

sin² θ  = 1 - cos² θ

cos² θ = 1 - sin² θ

sec² θ - tan² θ = 1

sec² θ  = 1 +  tan² θ

tan² θ  =  sec² θ - 1

csc² θ - cot² θ = 1

csc² θ = 1 + cot² θ

cot² θ =  csc² θ - 1

Double Angle Identities

Now, let us see double angle trigonometric formulas

1. sin 2A = 2 sin A cos A

2. cos 2A = cos² A - sin² A

3. tan 2A = 2 tan A / (1-tan² A)

4. cos 2A = 1 - 2sin² A

5. cos 2A = 2cos² A - 1

6. sin 2A = 2 tan A / (1+tan² A)

7. cos 2A = (1 - tan² A) / (1+tan² A)

8. sin²A = (1 - cos 2A) / 2

9. cos²A = (1 + cos 2A) / 2

These identities are applied in both the ways ,left to right and  right to left.

Half Angle Identities

Now, let us see half angle trigonometric-formulas

sin A  =  2sin(A/2)cos(A/2)

cos A  =  cos²(A/2) - sin²(A/2)

tan A  =  2 tan(A/2) / [1 - tan²(A/2)]

cos A  =  1 - 2sin²(A/2)

cos A  =  2cos²(A/2) - 1

sin A  =  2tan(A/2) / [1 + tan²(A/2)]

cos A  =  [1 - tan²(A/2)] / [1 + tan²(A/2)]

sin²A/2  =  (1 - cos A) / 2

Cos²A/2  =  (1 + cos A) / 2

tan²(A/2)  =  (1 - cos A) / (1 + cos A)

Compound Angles

Now, let us see compound angle trigonometric-formulas

1. Sin (A+B) = Sin A Cos B + Cos A Sin B

2. Sin (A+B) = Sin A Cos B + Cos A Sin B

3. Cos (A+B) = Cos A Cos B - Sin A Sin B

4. Cos (A-B) = Cos A Cos B + Sin A Sin B

5.Tan (A+B) = [Tan A + Tan B] /(1- Tan A Tan B)

6.Tan (A-B) = [Tan A - Tan B] /(1+ Tan A Tan B)

Sum to product Identities

Now, let us see sum to producty trigonometric-formulas

1.Sin C + Sin D = 2 Sin [(C+D)/2] cos [(C-D)/2]

2.Sin C - Sin D = 2 Cos [(C+D)/2] Sin [(C-D)/2]

3.Cos C + Cos D = 2 Cos [(C+D)/2] Cos [(C-D)/2]

4.Cos C - Cos D = 2 Sin [(C+D)/2] Sin [(C-D)/2]

sin3A cos3A tan3A formulas

Sin 3A = 3 Sin A - 4 sin³A

Cos 3A = 4 Cos³A - 3 Cos A

tan 3A = (3 tan A - tan³ A)/(1-3tan²A)

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.

After having gone through the stuff given above, we hope that the students would have understood "Trigonometric identities"

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