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



   

Trigonometric Ratios I and II

Trigonometric Ratios Part One 

If P(θ)=(x,y) is the point on the terminal arm of angle θ that intersects the uni circle, notice that... 

Where:                                                                                         Reciprocal functions:

cosθ= adjacent/hypotenuse   →  cosθ= Y        →   secθ= 1/X or hypotenuse/adjacent

sinθ= opposite/hypotenuse   →  sinθ= X        →   cscθ=1/Y or hypotenuse/adjacent

tanθ= opposite/adjacent        →  tanθ= Y/X   →   cotθ=X/Y or adjacent/hypotenuse

or SOH CAH TOA 

Recall that the unit circle is a circle with a radius of 1, centre at (0,0) and equation 
x^2+y^2=1. You can now describe the equation for the unit circle as cos^2θ+sin^2θ=1

Exact Values and Approximate Values of Trigonometric Ratios 

• Exact Values for the trigonometric ratios can be determined using special triangles 30°,60°,45°... and multiples of θ= 0, π/6,π/4,π/3... for points P(θ) on the unit circle. 
• Approximate Values can determine using scientific calculator. Set the mode first to Degree (D) or Radian (R) to correct the measure. 

Examples:

The point A(3/5,4/5) lies at the intersection of the unit circle and the terminal arm of an angle θ in standard position. 

a) Draw a diagram to model the situation.
b) Determine the values of the six trigonometric ratios for θ, in lowest form. 

*Recall the CAST Rule and SOH CAH TOA

a)

The diagram is in Quadrant I, where all the trigonometric ratios are positive. 

b) cosθ= adj./hyp.= 3/5                           secθ= hyp./adj.= 5/3
     sinθ= opp./hyp.= 4/5                          cscθ= hyp./opp.= 5/4
     tanθ= opp./adj.= 4/3                           cotθ= adj./opp.= 3/4 

Determine the exact value. 
a) cos 5π/6 = -√3/2 
because 5π/6 is a special triangle and it is negative because it lies in Quadrant II . 

Determine the approximate value.  
a) cos 260°=
Solution: 
 In degree mode, 
cos(260 = -0.1736481777 (answer) 

Trigonometric Ratios Part Two 

 Approximate Values of Angle
- To determine θ, use the inverse trigonometric function key on calculator. 

Example:

a) sinθ= 0.879 in the domain 0≤θ≤2π. 
Solution:
    *Recall, CAST Rule. Then, determine which Quadrant is going to lie. Since sinθ is positive it will lies in Quadrant I and II. Using calculator enter the inverse sine

sinθ= 0.879
         sin^{−1 (0.879}
            θR= 1.073760909
QI θ=θR                                                           QII θ=π-θR
       θ=1.073760909                                                  π-1.073760909
                                                                                    θ=2.067831744

Answer: θ= 1.073760909, 2.067831744 

THE END :)) 

and I choose.....

 Cathlene☺

Special Right Triangles and Quadrantal Triangles

First and foremost, I want to acknowledge John Paul Aguilar for choosing me to do the next blog.

Special Right Triangles

To determine the exact values of trigonometric ratios, we should first memorize the trigonometric functions and their second functions.

sinθ = Opposite                                  cscθ = Hypotenuse
        Hypotenuse                                           Opposite


cosθ = Adjacent                                 secθ = Hypotenuse
         Hypotenuse                                          Adjacent



tanθ = Opposite                                  cotθ = Adjacent

          Adjacent                                             Opposite



* The special angles on the unit circle are based on reference angles of either 30°, 45° or 60°.

** 30° is the same as π/6
45° is the same as π/4
60° is the same as π/3
90° is the same as π/2
π = 180°


45°, 45°, 90° Triangle





30°, 60°, 90° Triangle with a 30° reference angle




30°, 60°, 90° Triangle with a 60° reference angle




To get the exact value of the trigonometric ratios, just plug in the opposite, hypotenuse or adjacent angles in the SOH CAH TOA.

For example:

1. Determine the exact value of sinθ in the diagram below.



sinθ = Opposite
         Hypotenuse

sinθ = 1/√2


2. Determine the exact value of cotθ in the diagram below.


cotθ = √3




Quadrantal Angles


       The quadrantal angles are shown in degrees and they                                                                   are 0°, 90°, 180° and 270°.


     The quadrantal angles are shown in exact values. They                                                                          are π/2, π, 3π/2 and 2π



Note: Decimals can also be used as labels and they are called approximate values.





FIN

KONIEC


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RAIZZA BONDOC.


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