what is the value of sec15°=?? & how it came

Question

Sec15°=??

Answer 1

We can take sec β = (1/cosβ)

and cos (θ/2)= ±√((1+cosθ)/2)

If we take θ=30 degrees we have

cos 15º = ±√((1+cos 30º)/2) = ±√(1±(√3)/2)/2

= √((1+0.866)/2) = √0.933 = 0.9659=cos 15

cos^-1 0.9659 = 15.00 degrees

so sec 15 = 1/0,9659 = 1.03530

 

Answer 2

Instead of the 1/2 angle formulas, I prefer using the difference formula, which is cos(A-B)=cos A cosB + sinAsinB. So using 45 and 30we get cos (45-30)=cos45cos30 + sin45sin30, which leads to 1/sqrt2 * sqrt3/2 + 1/sqrt2 * 1/2, which simplifies down to

(sqrt3 +1)/2sqrt2. now since secant is the reciprocal of cosine, flip over the fraction to get 2sqrt2/(sqrt3 + 1), This is the exact answer. When you put square roots in the calculator, you get an approximation. (these being irrational numbers). If you wanted an approximation, you could have just put 1/cos 15 in the calculator and be done with it. This is equivalent to the answer above though but this way, you dont have to worry about square roots inside other square roots.