Finding Range or Expressions Using Trigonometric Substitution

Trigonometric identities are essential for finding range or expressions using trigonometric substitution. Trigonometric identities are based on trigonometric ratios. We have 6 trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. The trigonometric ratios are structured based on the right angle triangle. 

Trigonometric identities are mainly helpful in simplifying trigonometric functions and equations that are complex. 

Trigonometric Ratios 

Trigonometric ratios are formed using the sides of a right-angle triangle by employing the Pythagoras theorem. We have six trigonometric ratios:

1. Sine: The ratio of the opposite side to the hypotenuse is equal to sin.

sinθ= opposite side/ hypotenuse 

2. Cosecant: The hypotenuse ratio to the opposite is equal to cosecθ.

cosecθ= hypotenuse/ opposite side = 1/sinθ 

3. Cosine: The ratio of the adjacent side to the hypotenuse is equal to cosθ.

cosθ= adjacent side/ hypotenuse 

4. Secant: The hypotenuse ratio to the adjacent side is equal to secθ.

secθ= hypotenuse/ adjacent side = 1/cosθ 

5. Tangent: The ratio of the opposite side to the adjacent side is equal to tanθ . 

tanθ= opposite side/ adjacent side = sinθ/cosθ

6. Cotangent: The ratio of the adjacent side to the opposite side is equal to cotθ.

cotθ = adjacent side /opposite side = 1/tanθ

By Pythagoras Theorem:

(side)2+(side)2=(hypotenuse)2 

From the above right angle theorem:

(a)2+(b)2=(h)2

Trigonometric Identities 

We have a few identities:

Sin2 θ +  cos2θ = 1

1 + tan2θ = sec2θ

1+ cot2θ = cosec2θ

The above identities will stand true for all the values.

For example: let us assume =900

For Sin2 θ +  cos2θ = 1

         1 + 0 = 1   

LHS = RHS which stands true.

Any equation that involves trigonometric ratios that is valid for all the values for which we have ratios is called trigonometric identities.

Some more trigonometric identities are :

1. secθ + tanθ = 1/secθ – tanθ

2.  secθ – tanθ = 1/sec θ+ tanθ

      a,b are formed using 1 + tan2θ=sec2θ 

    sec2θ – tan2θ = 1/ ( (a2– b2) = (a+b)(a-b))

     (secθ+tanθ)(secθ-tanθ) = 1

     (secθ+tanθ) = 1 secθ – tanθ (or) (secθ – tanθ) = 1 secθ + tanθ

1. cosecθ + cotθ = 1 cosecθ-cotθ

2. cosecθ – cotθ = 1 cosecθ+cotθ

From 1+ cot2θ = cosec2θ

cosec2θ– cot2θ = 1 

3. sinθ.cosecθ=1 

        Since, cosecθ=1/sinθ            

4. cosθ.secθ=1

Since, secθ=1/cosθ

5. tanθ.cotθ = 1

  Since, cotθ = 1/tanθ

Trigonometric Equations : 

The equations that involve trigonometric identities and trigonometric functions for finding range or expressions using trigonometric substitution are called trigonometric equations.

 Trigonometric equations contain one or more than one trigonometric function.

The values that will satisfy the unknown values are called solutions.

The set of the values that form a solution of the equations is called solution set or general solution.

Principal solution of an equation or principal values of trigonometric functions 

• For the Trigonometric equation sinθ = k where k[1,-1], then the principal value of the angle has a unique value where it is in the range of [-2,2 ].

Example: Find the principal solution of for equation sinθ = 12

The principal value of the above trigonometric equation is 6.

1. There is a principal value for trigonometric function cosine for trigonometric equation cos = k where k[1,-1]. The principal value of the angle has a unique value in the range of [0, ].

2. For trigonometric function tangent, there is a principal value for the trigonometric equation tan=k where k (-,). The principal value of the angle has a unique value in the range of (-2,2 ).

3. For trigonometric function cotangent, there exists a principal value for trigonometric equation cot=k where k (-,) then the principal value of the angle has a unique value which is in the range of [-2,0)(0,2 ].

4. For trigonometric function cotangent, there exists a principal value for trigonometric equation cosec=k where k (-,-1][1,) then the principal value of the angle has a unique value which is in the range of [-2,0)(0,2 ].

5. For trigonometric function cotangent, there exists a principal value for trigonometric equation sec=k where k (-,-1][1,) then the principal value of the angle has a unique value which is in the range of [0,2)(2, ].

General Solution for simple trigonometric equations

The general solution that combines the entire values of the angles in a simple equation that is in common for all the values is the equation’s general solution.

1. The general solution of trigonometric equation sinθ = k where k [-1,1] is 

n+(-1)n where nz and a principal solution [-2,2 ].

2. The general solution of trigonometric equation cosecθ = k where

 k (-,-1][1,) is  

n+(-1)n where nz and a principal solution [-2,0)(0,2 ].

3. The general solution of trigonometric equation cosθ = k where k [-1,1] is 

2n where NZ and principal solution [0, ].

4. The general solution of the trigonometric equation secθ = k where k (-,-1][1,) is 2n where nz and principal solution [0,2)(2, ].

5. The general solution of the trigonometric equation tanθ = k where k R is n+ where NZ and a principal solution (-2,2 ). 

6. The general solution of the trigonometric equation cotθ = k where k R is n+ where nz and a principal solution [-2,0)(0,2 ]. 

Domain and Range of Trigonometric functions:

• Function
• Domain
• Range

Conclusion

Trigonometric identities and equations used in finding range or expressions using trigonometric substitution are most prominently used in finding the limits and range to which they belong and finding the solution of the equation. Important points to remember while solving the trigonometric equations: Squaring of the equation should be avoided, check the denominator that if it has zero, undefined values, check if the angles at any point are undefined or infinite, see that the equation satisfies all the values of the domain. Remembering the identities, principal solutions, values and general solutions and substituting them in the equations will help find the range of trigonometric equations.