Trigonometry warm up

Geometry Level 3

sin ⁡ ( a − b ) cos ⁡ ( a + b ) + sin ⁡ ( b − c ) cos ⁡ ( b + c ) + sin ⁡ ( c − a ) cos ⁡ ( a + c ) \sin(a-b)\cos(a+b)+\sin(b-c)\cos(b+c)+\sin(c-a)\cos(a+c)

Simplify the expression above.

Give your answer to 2 decimal places.


The answer is 0.00.

P C by P C , 20, Vietnam

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1 solution

Rishabh Jain
May 1, 2016

The expression can be written as: 1 2 ( ∑ cyc 2 cos ⁡ ( a + b ) sin ⁡ ( a − b ) ) \large\frac 12\left(\displaystyle\sum_{\text{cyc}}2\cos(a+b)\sin(a-b)\right)

( Use 2 cos ⁡ A sin ⁡ B = sin ⁡ ( A + B ) − sin ⁡ ( A − B ) ) (\small{\color{teal}{\text{Use } 2\cos A\sin B=\sin (A+B)-\sin (A-B)}})

= 1 2 ( ∑ cyc ( sin ⁡ 2 a − sin ⁡ 2 b ) ) = 0 \large=\dfrac 12\left(\displaystyle\sum_{\text{cyc}}(\sin 2a-\sin 2b)\right)=\boxed{0}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Wow, your solution is much shorter than mine

P C - 5 years, 1 month ago

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Thanks... :-)

Rishabh Jain - 5 years, 1 month ago

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