What are the double angle formulas for csc and sec?
Generally these are best calculated by changing them to sines and cosines
None of the first 4 pages I looked at after "googling" double angle formulas gave me a formula for csc 2x or sec 2x
We could however find them:
change csc 2x to 1/sin 2x
= 1/(2sinxcosx)
=1/2(cscx)(secx)
similarly
sec 2x = 1/cos2x
= 1/(cos^2 x - sin^2 x)
Thanks!
To find the double angle formulas for csc (cosecant) and sec (secant), we can start by recalling the definitions of these trigonometric functions:
- The cosecant (csc) of an angle is the reciprocal of the sine (sin) of that angle.
- The secant (sec) of an angle is the reciprocal of the cosine (cos) of that angle.
Let's use the double angle formula for sine to derive the double angle formulas for csc and sec:
1. Double angle formula for sine:
sin(2θ) = 2sin(θ)cos(θ)
2. Double angle formula for csc:
We know that csc(θ) is the reciprocal of sin(θ), so:
csc(2θ) = 1 / sin(2θ)
= 1 / (2sin(θ)cos(θ)) [substituting the double angle formula for sine]
= 1 / 2sin(θ) * 1/cos(θ)
= csc(θ) / 2cos(θ) [since csc(θ) = 1/sin(θ)]
Therefore, the double angle formula for csc is:
csc(2θ) = csc(θ) / 2cos(θ)
3. Double angle formula for sec:
We know that sec(θ) is the reciprocal of cos(θ), so:
sec(2θ) = 1 / cos(2θ)
= 1 / (2cos^2(θ) - 1) [using the double angle formula for cosine]
= 1 / (2(1 - sin^2(θ)) - 1) [since cos^2(θ) - sin^2(θ) = 1]
= 1 / (2 - 2sin^2(θ) - 1)
= 1 / (1 - 2sin^2(θ))
= sec^2(θ) / (1 - 2sin^2(θ)) [since sec^2(θ) = 1/(cos^2(θ))]
Therefore, the double angle formula for sec is:
sec(2θ) = sec^2(θ) / (1 - 2sin^2(θ))
These are the double angle formulas for csc and sec, which allow us to express csc(2θ) and sec(2θ) in terms of csc(θ), sec(θ), and trigonometric functions of θ.