Trigonometric Identities

Hi guys! We've just started the trigonometric identities unit. So, I'll be talking about the basic trigonometric identities, sum and difference identities, and the double angle trigonometric identities

An equation is called a trigonometric identity if it is true for all permissible values of the variable for which both sides of the equation are defined. An equation is an identity when the LHS is identical to RHS

Reciprocal Identities:
csc θ= 1/sin θ
sec θ= 1/cos θ
cot θ= 1/tan 

Quotient identities:
tan θ= sin θ/cos θ
cot θ=cos θ/sin θ

Pythagorean Identity:
cos^2θ + sin^2θ=1

Example: Determine the non-permissible values, in degrees, for the equation  sec θ=tanθ/sinθ

LHS: sec θ=1/cos θ => cos θ ≠ 0   
                                     θ ≠ 90°, 270°...
                                     θ ≠ 90° + 180°n, where nEI
RHS: tan θ/sin θ => sin θ ≠ 0
                                     θ ≠ 0°, 180°, 360°...
                                      θ ≠ 180°n, where nEI
          tan θ=sin θ/cos θ=> cos θ ≠ 0   
                                                θ ≠ 90°, 270°...
                                                θ ≠ 90° + 180°n, where nEI
θ ≠ 90°n, where nEI

Example: Express the Pythagorean identity cos^2θ + sin^2θ = 1 as the equivalent identity 1+tan^2θ=sec^2θ.

   cos^2θ+sin^2θ=1 
           cos^2θ
   cos^2θ + sin^2θ  =     1        
   cos^2θ + cos^2θ     cos^2θ
  
1+tan^2θ=sec^2θ

Sum and Difference Identities

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) 
cos(α + β) = cos(α)cos(β) – sin(α)sin(β) 

tan(α + β) = tanα - tanβ 

                       1- tanαtanβ

sin(α – β) = sin(α)cos(β) – cos(α)sin(β) 

cos(α – β) = cos(α)cos(β) + sin(α)sin(β) 

tan(α - β) =  tanα - tanβ 

                       1+ tanαtanβ

Example: Exact Trigonometric Values for Angles
cos 5π/12  = cos (2π/12 + 3π/12)
                  = cos (π/6 + π/4)
                  = (cosπ/6)(cosπ/4)-(sinπ/6)(sinπ/4)
                  = (√3/2)(1/√2) - (1/2)(1/√2)
                  = √3/2√2 - 1/2√2

cos 5π/12 = -1 + √3
                        2√2    

Double Angle Trigonometric Identities
sin2α  = 2sinαcosα

cos2α = cos²α – sin²α 
cos2α = 1 – 2sin²α 
cos2α = 2cos²α – 1
tan2α =     2tanα      
               1 - tan^2α

Example: Write the expression as a single trigonometric function.
      2tan76°       =     2tanα     = tan2α
1 - tan^2(76°)    1- tan^2α         
                                                  =tan 2(76°) 
                                                  = tan152°      

Example: Simplify the expression 1-cos2x/sin2x to one of the three primary trigonometric functions.   
       1 - cos2x
          sin2x
    = 1 - (1 - 2sin^2x)
           2sinxcosx
    =1-1+2sin^2x
        2sinxcosx
    = sinx
       cosx
    = tanx

I hope you guys enjoyed my blog post. Have a great day!

I choose…

       Katrina Mojas