Certainly! Let's determine the given function [tex]\( \tan(78^\circ) \)[/tex] using its cofunction.
1. Understanding Cofunctions:
The cofunction identity for tangent and cotangent tells us that:
[tex]\[
\tan(90^\circ - \theta) = \cot(\theta)
\][/tex]
2. Apply the Cofunction Identity to Our Angle:
Here, we are given [tex]\( 78^\circ \)[/tex]. We will rewrite this in terms of its cofunction:
[tex]\[
\tan(78^\circ)
\][/tex]
We recognize that:
[tex]\[
\tan(78^\circ) = \tan(90^\circ - 12^\circ)
\][/tex]
3. Express in Terms of Cotangent:
According to the cofunction identity, we can rewrite [tex]\( \tan(90^\circ - 12^\circ) \)[/tex] as:
[tex]\[
\tan(90^\circ - 12^\circ) = \cot(12^\circ)
\][/tex]
4. Conclusion:
Therefore:
[tex]\[
\tan(78^\circ) = \cot(12^\circ)
\][/tex]
As a result of these steps and converting accordingly:
- [tex]\( \tan(78^\circ) \)[/tex] is calculated to be approximately [tex]\( 4.704630109478451 \)[/tex]
- [tex]\( \cot(12^\circ) \)[/tex] is also calculated to be approximately [tex]\( 4.704630109478455 \)[/tex]
Both values are nearly identical, substantiating that:
[tex]\[
\tan(78^\circ) = \cot(12^\circ)
\][/tex]
Thus, we have successfully expressed [tex]\( \tan(78^\circ) \)[/tex] in terms of its cofunction, as [tex]\( \cot(12^\circ) \)[/tex].
