Trigonometry Examples

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Trigonometry
Verify the Identity sin(2pi-x)=-sin(x)
sin(2π-x)=-sin(x)
Step 1
Start on the left side.
sin(2π-x)
Step 2
Apply the difference of angles identity.
sin(2π)cos(x)-cos(2π)sin(x)
Step 3
Simplify the expression.
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Step 3.1
Simplify each term.
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Step 3.1.1
Subtract full rotations of 2π until the angle is greater than or equal to 0 and less than 2π.
sin(0)cos(x)-cos(2π)sin(x)
Step 3.1.2
The exact value of sin(0) is 0.
0cos(x)-cos(2π)sin(x)
Step 3.1.3
Multiply 0 by cos(x).
0-cos(2π)sin(x)
Step 3.1.4
Subtract full rotations of 2π until the angle is greater than or equal to 0 and less than 2π.
0-cos(0)sin(x)
Step 3.1.5
The exact value of cos(0) is 1.
0-1⋅1sin(x)
Step 3.1.6
Multiply -1 by 1.
0-1sin(x)
Step 3.1.7
Rewrite -1sin(x) as -sin(x).
0-sin(x)
0-sin(x)
Step 3.2
Subtract sin(x) from 0.
-sin(x)
-sin(x)
Step 4
Because the two sides have been shown to be equivalent, the equation is an identity.
sin(2π-x)=-sin(x) is an identity
sin⁡(2π-x)=-sin⁡(x)
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