1
I notice that the angle 100 degrees is not a special angle, so I will need to use some trigonometric identities to simplify this expression.
2
I recall that tanθ=sinθcosθtan theta = frac{sin theta}{cos theta} and sin2θ=2sinθcosθ.sin 2theta = 2 sin theta cos theta.
3
I can use these identities to rewrite the expression as sin100∘cos100∘+4sin100∘.frac{sin 100^circ}{cos 100^circ} + 4 sin 100^circ.
4
I can combine the terms by finding a common denominator of cos100∘.cos 100^circ.
5
I get sin100∘+4sin100∘cos100∘cos100∘.frac{sin 100^circ + 4 sin 100^circ cos 100^circ}{cos 100^circ}.
6
I can simplify the numerator by using the double angle identity again, but in reverse.
7
I get sin100∘+2sin200∘cos100∘.frac{sin 100^circ + 2 sin 200^circ}{cos 100^circ}.
8
I notice that 200 degrees is the same as 180 degrees plus 20 degrees, so I can use the sum angle identity for sine.
9
I get sin100∘+2(sin180∘cos20∘+cos180∘sin20∘)cos100∘.frac{sin 100^circ + 2 (sin 180^circ cos 20^circ + cos 180^circ sin 20^circ)}{cos 100^circ}.
10
I recall that sin180∘=0sin 180^circ = 0 and cos180∘=−1,cos 180^circ = -1, so I can simplify further.
11
I get sin100∘−2sin20∘cos100∘.frac{sin 100^circ – 2 sin 20^circ}{cos 100^circ}.
12
I notice that 100 degrees is the same as 90 degrees plus 10 degrees, so I can use the sum angle identity for sine and cosine.
13
I get (sin90∘cos10∘+cos90∘sin10∘)−2sin20∘(cos90∘cos10∘−sin90∘sin10∘).frac{(sin 90^circ cos 10^circ + cos 90^circ sin 10^circ) – 2 sin 20^circ}{(cos 90^circ cos 10^circ – sin 90^circ sin 10^circ)}.
14
I recall that sin90∘=1sin 90^circ = 1 and cos90∘=0,cos 90^circ = 0, so I can simplify further.
15
I get cos10∘−2sin20∘−sin10∘.frac{cos 10^circ – 2 sin 20^circ}{-sin 10^circ}.
16
I can simplify the sign by multiplying the numerator and denominator by -1.
17
I get 2sin20∘−cos10∘sin10∘.frac{2 sin 20^circ – cos 10^circ}{sin 10^circ}.
18
I notice that 20 degrees is the same as 30 degrees minus 10 degrees, so I can use the difference angle identity for sine.
19
I get 2(sin30∘cos10∘−cos30∘sin10∘)−cos10∘sin10∘.frac{2 (sin 30^circ cos 10^circ – cos 30^circ sin 10^circ) – cos 10^circ}{sin 10^circ}.
20
I recall that sin30∘=12sin 30^circ = frac{1}{2} and cos30∘=32,cos 30^circ = frac{sqrt{3}}{2}, so I can simplify further.
21
I get cos10∘−3sin10∘−cos10∘sin10∘.frac{cos 10^circ – sqrt{3} sin 10^circ – cos 10^circ}{sin 10^circ}.
22
I can cancel out the cos10∘cos 10^circ terms in the numerator.
23
I get −3sin10∘sin10∘.frac{-sqrt{3} sin 10^circ}{sin 10^circ}.
24
I can cancel out the sin10∘sin 10^circ terms in the numerator and denominator.
25
I get −3.-sqrt{3}.
26
Answer: −3-sqrt{3}
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- Source: https://openai.com/research/improving-mathematical-reasoning-with-process-supervision
