Just Cos B

Geometry Level 2

If tan ⁡ θ \tan\theta is equal to n th n^{\text{th}} smallest prime number where n n is the largest two digit prime, find the digit sum of B B when cos ⁡ θ \cos\theta is written as 1 B \frac{1}{\sqrt{B}} .


The answer is 26.

David Holcer by David Holcer , 20, Canada

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1 solution

Guilherme Niedu
Feb 22, 2017

n = 97 \large \displaystyle n = 97 (largest two-digit prime)

t a n ( θ ) = 509 \large \displaystyle tan(\theta) = 509 (509 is the 97th prime number)

sin ⁡ ( θ ) = 509 ⋅ c o s ( θ ) \large \displaystyle \sin(\theta) = 509\cdot cos(\theta)

c o s ( θ ) 2 ⋅ ( 1 + 50 9 2 ) = 1 \large \displaystyle cos(\theta)^2 \cdot(1 + 509^2) = 1

c o s ( θ ) = 1 259082 \large \displaystyle cos(\theta) = \frac{1}{\sqrt{259082}}

Then B = 259082 B = 259082 and the digit sum is 26 \fbox{26}

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