Which of the following counterexamples proves that sinxtanx=cosx is not a trigonometric identity? Select all that apply.
-2π
-3π
-3π/4
-π/4
For x = - 2 π
sin ( - 2 π ) = 0 , cos ( - 2 π ) = 1 , tan ( - 2 π ) = 0
sin ( - 2 π ) ∙ tan ( - 2 π ) = 0 ∙ 0 = 0 ≠ cos ( - 2 π )
0 ≠ 1
For x = - 3 π
sin ( - 3 π ) = 0 , cos ( - 3 π ) = - 1 , tan ( - 3 π ) = 0
sin ( - 3 π ) ∙ tan ( - 3 π ) = 0 ∙ 0 = 0 ≠ cos ( - 3 π )
0 ≠ - 1
For x = - 3 π / 4
sin ( - 3 π / 4 ) = -√ 2 / 2 , cos ( - 3 π / 4 ) = - √ 2 / 2 , tan ( - 3 π / 4 ) = 1
sin ( - 3 π / 4 ) ∙ tan ( - 3 π / 4 ) = -√ 2 / 2 ∙ 1 = - √ 2 / 2 = cos ( - 3 π / 4 )
- √ 2 / 2 = - √ 2 / 2
For x = - π / 4
sin ( - π / 4 ) = -√ 2 / 2 , cos ( - π / 4 ) = √ 2 / 2 , tan ( - π / 4 ) = - 1
sin ( - π / 4 ) ∙ tan ( - π / 4 ) = -√ 2 / 2 ∙ ( - 1 ) = √ 2 / 2 = cos ( - π / 4 )
√ 2 / 2 = √ 2 / 2
So x = - 2 π and x = - 3 π proves that
sin x ∙ tan x = cos x is not a trigonometric identity,
Seems like Reiny already helped you with this, but here goes again.
If it is an identity, then all of the values suggested should work.
So, does
sin(-2π)*tan(-2π) = cos(-2π) ?
0*0 = 1 ?
Nope. So, using -2π proves it is not an identity.
Now you try the others. If you get stuck, come on back and show what you got.
Also you can do this:
sin x ∙ tan x = cos x
sin x ∙ sin x / cos = cos x
sin² x / cos = cos x
Multiply both sides by cos x
sin² x = cos² x
For x = - 2 π
sin ( - 2 π ) = 0 , cos ( - 2 π ) = 1
sin² ( - 2 π ) = 0 ≠ cos² ( - 2 π )
0 ≠ 1²
0 ≠ 1
For x = - 3 π
sin ( - 3 π ) = 0 , cos ( - 3 π ) = - 1 ,
sin² ( - 3 π ) = 0 ≠ cos² ( - 3 π )
0 ≠ ( - 1 )²
0 ≠ 1
For x = - 3 π / 4
sin ( - 3 π / 4 ) = -√ 2 / 2 , cos ( - 3 π / 4 ) = - √ 2 / 2
sin² ( - 3 π / 4 ) = ( -√ 2 / 2 )² = 2 / 4 = 1 / 2 =
cos² ( - 3 π / 4 ) = ( - √ 2 / 2 )² = 2 / 4 = 1 / 2
1 / 2 = 1 / 2
For x = - π / 4
sin ( - π / 4 ) = -√ 2 / 2 , cos ( - π / 4 ) = √ 2 / 2
sin² ( - π / 4 ) = ( -√ 2 / 2 )² = 2 / 4 = 1 / 2 =
cos² ( - π / 4 ) = ( - √ 2 / 2 )² = 2 / 4 = 1 / 2
1 / 2 = 1 / 2
So x = - 2 π and x = - 3 π proves that
sin x ∙ tan x = cos x is not a trigonometric identity.
Or, having arrived at
sin² x = cos² x
then
tan² x = 1
which is only true for odd multiples of π/4
so, for -2π and -3π it won't work
To prove that sin(x)tan(x) = cos(x) is not a trigonometric identity, we need to find counterexamples. A counterexample is a value or values that satisfy the equation for one side but not for the other side.
First, let's simplify the equation sin(x)tan(x) = cos(x). We know that tan(x) = sin(x)/cos(x). So, substituting this in the equation, we get sin(x) * (sin(x)/cos(x)) = cos(x).
Simplifying further, we have (sin^2(x))/cos(x) = cos(x).
Now, let's go through the given options and plug them into the equation to see if they satisfy both sides:
-2π:
sin^2(-2π)/cos(-2π) = cos(-2π)
This becomes (0)/1 = 1, which is not true. Therefore, -2π is a counterexample.
-3π:
sin^2(-3π)/cos(-3π) = cos(-3π)
This becomes (0)/1 = 1, which is not true. Therefore, -3π is a counterexample.
-3π/4:
sin^2(-3π/4)/cos(-3π/4) = cos(-3π/4)
This becomes (1)/(-√2/2) = -√2/2, which is not true. Therefore, -3π/4 is a counterexample.
-π/4:
sin^2(-π/4)/cos(-π/4) = cos(-π/4)
This becomes (1)/(√2/2) = √2/2, which is true. Therefore, -π/4 is not a counterexample.
Therefore, the counterexamples that prove sin(x)tan(x) = cos(x) is not a trigonometric identity are -2π, -3π, and -3π/4.