Rationalizing Trigonometry

Geometry Level 3

If 4 cos ⁡ 9 ∘ = a + b + b − b 4 \cos 9^\circ = \sqrt{a + \sqrt b} + \sqrt{b - \sqrt b} , find the value of a + b a + b .


The answer is 8.

Shubhrajit Sadhukhan by Shubhrajit Sadhukhan , 16, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Iliya Hristov
May 17, 2021

0 Helpful 0 Interesting 1 Brilliant 0 Confused

Alternatively: 0 = cos ⁡ 9 0 ∘ = cos ⁡ ( 5 ∗ 1 8 ∘ ) = cos ⁡ ( 1 8 ∘ + 4 ∗ 1 8 ∘ ) = . . . . . = 16 cos ⁡ 5 1 8 ∘ − 20 cos ⁡ 3 1 8 ∘ + 5 cos ⁡ 1 8 ∘ ⇒ cos ⁡ 2 1 8 ∘ = 5 ± 5 8 = ( 2 cos ⁡ 2 9 ∘ − 1 ) 2 ⇒ ( 4 cos ⁡ 9 ∘ ) 2 = 8 ± 40 ± 8 5 = . . . . . 0=\cos90^\circ=\cos(5*18^\circ)=\cos(18^\circ+4*18^\circ)=.....=16\cos^518^\circ-20\cos^318^\circ+5\cos18^\circ\;\;\;\Rightarrow\\ \cos^218^\circ=\frac{5\pm\sqrt{5}}{8}=(2\cos^29^\circ-1)^2\;\;\;\Rightarrow\;\;\;(4\cos9^\circ)^2=8\pm\sqrt{40\pm8\sqrt{5}}=.....

Iliya Hristov - 2 weeks, 5 days ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...