Simple Trigonometry

Geometry Level 3

If sin ⁡ x + cos ⁡ x = 7 5 \sin x + \cos x = \dfrac75 , and 1 sin ⁡ x + 1 cos ⁡ x = a b \dfrac1{\sin x} + \dfrac1{\cos x} = \dfrac ab , where a a and b b are coprime positive integers, find a + b a+b .


The answer is 47.

Ashish Gupta by Ashish Gupta , 22, India

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1 solution

Ashish Gupta
Apr 11, 2016

1 sin ⁡ x + 1 cos ⁡ x = sin ⁡ x + cos ⁡ x sin ⁡ x cos ⁡ x \frac{1}{\sin x}+\frac{1}{\cos x} =\frac{\sin x+\cos x}{\sin x\cos x} = 2 ⋅ sin ⁡ x + cos ⁡ x 2 sin ⁡ x cos ⁡ x = 2 ⋅ sin ⁡ x + cos ⁡ x ( sin ⁡ x + cos ⁡ x ) 2 − 1 =2\cdot \frac{\sin x+\cos x}{2\sin x\cos x} =2\cdot \frac{\sin x+\cos x}{(\sin x+\cos x)^2-1}

We know sin ⁡ x + cos ⁡ x = 7 5 \sin x+\cos x=\frac{7}{5} , so:

1 sin ⁡ x + 1 cos ⁡ x = 2 ⋅ 7 5 ( 7 5 ) 2 − 1 = 35 12 \frac{1}{\sin x}+\frac{1}{\cos x} = 2\cdot \frac{\frac 75}{\left(\frac{7}{5}\right)^2-1} =\frac{35}{12}

So, a = 35 a =35 and b = 12 b=12 . a + b = 47 a+b=47 .

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