sec(pi/2-u)= ?
The Correct Answer and Explanation is:
To find the value of:sec(π2−u)\sec\left(\frac{\pi}{2} – u\right)sec(2π−u)
we use a trigonometric identity involving co-function relationships.
Co-function Identity:
sec(π2−u)=csc(u)\sec\left(\frac{\pi}{2} – u\right) = \csc(u)sec(2π−u)=csc(u)
This identity comes from the general co-function rules in trigonometry, which relate the trigonometric function of an angle to the co-function of its complement. In this case, the complement of angle uuu is π2−u\frac{\pi}{2} – u2π−u. The secant and cosecant functions are co-functions of each other.
Here are some of the main co-function identities:
- sin(π2−u)=cos(u)\sin\left(\frac{\pi}{2} – u\right) = \cos(u)sin(2π−u)=cos(u)
- cos(π2−u)=sin(u)\cos\left(\frac{\pi}{2} – u\right) = \sin(u)cos(2π−u)=sin(u)
- tan(π2−u)=cot(u)\tan\left(\frac{\pi}{2} – u\right) = \cot(u)tan(2π−u)=cot(u)
- cot(π2−u)=tan(u)\cot\left(\frac{\pi}{2} – u\right) = \tan(u)cot(2π−u)=tan(u)
- sec(π2−u)=csc(u)\sec\left(\frac{\pi}{2} – u\right) = \csc(u)sec(2π−u)=csc(u)
- csc(π2−u)=sec(u)\csc\left(\frac{\pi}{2} – u\right) = \sec(u)csc(2π−u)=sec(u)
These identities are especially useful in simplifying expressions and solving trigonometric equations, especially when the angle is a complement of another.
Conclusion:
sec(π2−u)=csc(u)\sec\left(\frac{\pi}{2} – u\right) = \csc(u)sec(2π−u)=csc(u)
This means that the secant of the complement of angle uuu is equal to the cosecant of angle uuu. This relationship is rooted in the geometry of right triangles and the unit circle definition of trigonometric functions. Understanding these co-function identities enhances flexibility in solving trigonometric problems.
