First off the definition of trigonometric function is an equation that involves at least one trigonometric function of a variable. This equation is referred to trigonometric identity when all permissible values of the variable for which both sides of the equation are shown.
To verify an identity: you show that the left hand side (LHS) is identical to the right hand side (RHS)
Reciprocal Identities:
cscΘ = 1÷sinΘ secΘ= 1÷cosΘ cotΘ=1÷tanΘ
Quotient Identities:
tanΘ=sinΘ÷cosΘ cotΘ=cosΘ÷sinΘ
Pythagorean Identity:
cos² Θ + sin² Θ =1
Example 1. We can verify a potential Identity by..
a) Determining the non-permissible values, in degrees, for the equation secΘ = tanΘ÷sinΘ
We can start on the LHS, follow the steps!
secΘ = tanΘ÷sinΘ
1.
secΘ = 1÷cosΘ • cos≠ 0
Θ≠ 0°, 90°, 270°
Θ≠ 180°n. where n E I
2.
tanΘ÷sinΘ • sin≠ 0
Θ≠ 0°, 180°, 360°
Θ≠ 180°n, where n E I
3.
tanΘ = sinΘ÷cosΘ • cos≠ 0
Θ≠ 0°, 90°, 270°
Θ≠ 180°n. where n E I
4.
90°n, where n E I
b) Numerically verify that Θ =60° and Θ = π÷4 are solutions of the equation.
1. Θ = 60°
secΘ = tanΘ÷sinΘ
sec60° = tan60°÷sin60°
2. Input perfect right triangles.
2 = √3÷√3÷2
2 = √3 • 2÷√3
2 = 2
this shows that Θ = 60° is a solution to the equation.
Example 2: Use identities to simplify expressions.
a) Determine he non-permissible values, in radians, of the variable in the expression
sec x÷tan x.
b) Simplify the expression.
Example 3. Pythagorean Identity.
a) Verify that the equation 1 + tan² Θ = sec² Θ is true when Θ = 3π÷4
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b) Express the Pythagorean identity cos² Θ + sin² Θ = 1 as the equivalent identity 1 + tan² Θ = sec² Θ.
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That is all happy precal-ing!
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