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Trigonometry Examples
sin(x)sin(x)
Step 1
Interchange the variables.
x=sin(y)x=sin(y)
Step 2
Step 2.1
Rewrite the equation as sin(y)=xsin(y)=x.
sin(y)=xsin(y)=x
Step 2.2
Take the inverse sine of both sides of the equation to extract yy from inside the sine.
y=arcsin(x)y=arcsin(x)
Step 2.3
Remove parentheses.
y=arcsin(x)y=arcsin(x)
y=arcsin(x)y=arcsin(x)
Step 3
Replace yy with f-1(x)f−1(x) to show the final answer.
f-1(x)=arcsin(x)f−1(x)=arcsin(x)
Step 4
Step 4.1
To verify the inverse, check if f-1(f(x))=xf−1(f(x))=x and f(f-1(x))=xf(f−1(x))=x.
Step 4.2
Evaluate f-1(f(x))f−1(f(x)).
Step 4.2.1
Set up the composite result function.
f-1(f(x))f−1(f(x))
Step 4.2.2
Evaluate f-1(sin(x))f−1(sin(x)) by substituting in the value of ff into f-1f−1.
f-1(sin(x))=arcsin(sin(x))f−1(sin(x))=arcsin(sin(x))
f-1(sin(x))=arcsin(sin(x))f−1(sin(x))=arcsin(sin(x))
Step 4.3
Evaluate f(f-1(x))f(f−1(x)).
Step 4.3.1
Set up the composite result function.
f(f-1(x))f(f−1(x))
Step 4.3.2
Evaluate f(arcsin(x))f(arcsin(x)) by substituting in the value of f-1f−1 into ff.
f(arcsin(x))=sin(arcsin(x))f(arcsin(x))=sin(arcsin(x))
Step 4.3.3
The functions sine and arcsine are inverses.
f(arcsin(x))=xf(arcsin(x))=x
f(arcsin(x))=x
Step 4.4
Since f-1(f(x))=x and f(f-1(x))=x, then f-1(x)=arcsin(x) is the inverse of f(x)=sin(x).
f-1(x)=arcsin(x)
f-1(x)=arcsin(x)