Answer to Question #228674 in Physical Chemistry for Nidg

Question

tan (22.5) can be expressed as

A.     (1+2^0.5)1/2

B.     (-1+2^0.5)

C.     (2+2^0.5)1/2

D.     None

Answer 1

tan 2x = 2 tanx / ( 1 – tan²x) (1)

Here, x = 22.5 °

Therefore, 2x = 2 × 22.5° = 45°

Also, we know that tan 45° = 1 (from trigonometric table values)

Let us consider tan 22.5° = y

Substituting x = 22.5 ° and tan 22.5° = y in (1) we get,

⇒1 = 2y / (1 – y²)

⇒1 – y² = 2y

⇒ y² + 2y – 1 = 0

⇒ y = ( – 2 ± 8^0.5 ) / 2 and y = ( – 2 ± 2$\times$2^0.5 ) / 2 

⇒ y = – 1 ± 2^0.5  [dividing numerator and denominator by 2]

⇒ y = 2^0.5 – 1 or  y = -1 – 2^0.5

Since the value of tan 22.5 degrees lies in the 1st quadrant, therefore, the required value should be positive.

Therefore, the value of tan 22.5 degree is 2^0.5 – 1.

Abswer: B. (-1+2^0.5)