# TRIGONOMETRIC IDENTITIES

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.

## SOHCAHTOA

Sinθ = Opposite side/Hypotenuse side

Cscθ = Hypotenuse side/Opposite side

sinθ = 1/cscθ

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

## Compound Angles Identities

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

cos(A + B) = cosAcosB - sinAsinB

cos(A - B) = cosAcosB + sinAsinB

tan(A + B) =  (tanA + tanB)/(1 - tanAtanB)

tan(A - B) =  (tanA - tanB)/(1 + tanAtanB)

## Double Angle Identities

sin2A = 2sinAcosA

cos2A = cos2A - sin2A

tan2A =  2tanA/(1 - tan2A)

cos2A = 1 - 2sin2A

cos2A = 2cos2A - 1

sin2A = 2tanA/(1 + tan2A)

cos2A = (1 - tan2A)/(1 + tan2A)

sin2A = (1 - cos2A)/2

cos2A = (1 + cos2A)/2

## Half Angle Identities

sinA = 2sin(A/2)cos(A/2)

cosA = cos2(A/2) - sin)2(A/2)

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

cosA = 1 - 2sin2(A/2)

cosA = 2cos2(A/2) - 1

sinA = 2tan(A/2)/[1 + tan2(A/2)]

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

sin2(A/2) = (1 - cosA)/2

cos2(A/2) = (1 + cosA)/2

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

## Triple Angle Identities

sin3A = 3sinA - 4sin3A

cos3A = 4cos3A - 3cosA

tan3A = (3tanA - tan3A)/(1 - 3tan2A)

## Sum to Product Identities

sinC + sinD  =  2sin[(C + D)/2]cos[(C - D)/2]

sinC - sinD = 2cos[(C + D)/2]sin[(C - D)/2]

cosC + cosD = 2cos[(C + D)/2]cos[(C - D)/2]

cosC - cosD = 2sin[(C + D)/2]sin[(D - C)/2]

## Values of Trigonometric Ratios for Standard 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.

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