To solve this problem using the cofunction theorem, let’s break down the process step-by-step.
1. Understanding the Cofunction Theorem:
The cofunction theorem states that for complementary angles, the sine of one angle is equal to the cosine of its complement. Mathematically, this can be expressed as:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
where [tex]\(\theta\)[/tex] is an angle measured in degrees.
2. Identify the Given Angle:
In the given problem, we know that:
[tex]\[
\sin(41^\circ)
\][/tex]
3. Applying the Cofunction Theorem:
According to the theorem, we can find the cosine of the complementary angle. The complementary angle [tex]\( \phi \)[/tex] to [tex]\(\theta\)[/tex] is found by subtracting [tex]\(\theta\)[/tex] from 90 degrees:
[tex]\[
\phi = 90^\circ - \theta
\][/tex]
Substituting [tex]\(\theta = 41^\circ\)[/tex], we get:
[tex]\[
\phi = 90^\circ - 41^\circ
\][/tex]
4. Perform the Calculation:
[tex]\[
\phi = 90^\circ - 41^\circ = 49^\circ
\][/tex]
5. Conclusion:
Thus, according to the cofunction theorem, we have:
[tex]\[
\sin(41^\circ) = \cos(49^\circ)
\][/tex]
So, the expression becomes a true statement with the blank filled in as follows:
[tex]\[
\sin(41^\circ) = \cos(49^\circ)
\][/tex]
