1 / 4Simplify \tan 100^\circ + 4 \sin 100^\circ.tan100∘+4sin100∘.
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 \tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ and \sin 2\theta = 2 \sin \theta \cos \theta.sin2θ=2sinθcosθ.
3
I can use these identities to rewrite the expression as \frac{\sin 100^\circ}{\cos 100^\circ} + 4 \sin 100^\circ.cos100∘sin100∘+4sin100∘.
4
I can combine the terms by finding a common denominator of \cos 100^\circ.cos100∘.
5
I get \frac{\sin 100^\circ + 4 \sin 100^\circ \cos 100^\circ}{\cos 100^\circ}.cos100∘sin100∘+4sin100∘cos100∘.
6
I can simplify the numerator by using the double angle identity again, but in reverse.
7
I get \frac{\sin 100^\circ + 2 \sin 200^\circ}{\cos 100^\circ}.cos100∘sin100∘+2sin200∘.
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 \frac{\sin 100^\circ + 2 (\sin 180^\circ \cos 20^\circ + \cos 180^\circ \sin 20^\circ)}{\cos 100^\circ}.cos100∘sin100∘+2(sin180∘cos20∘+cos180∘sin20∘).
10
I recall that \sin 180^\circ = 0sin180∘=0 and \cos 180^\circ = -1,cos180∘=−1, so I can simplify further.
11
I get \frac{\sin 100^\circ - 2 \sin 20^\circ}{\cos 100^\circ}.cos100∘sin100∘−2sin20∘.
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 \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)}.(cos90∘cos10∘−sin90∘sin10∘)(sin90∘cos10∘+cos90∘sin10∘)−2sin20∘.
14
I recall that \sin 90^\circ = 1sin90∘=1 and \cos 90^\circ = 0,cos90∘=0, so I can simplify further.
15
I get \frac{\cos 10^\circ - 2 \sin 20^\circ}{-\sin 10^\circ}.−sin10∘cos10∘−2sin20∘.
16
I can simplify the sign by multiplying the numerator and denominator by -1.
17
I get \frac{2 \sin 20^\circ - \cos 10^\circ}{\sin 10^\circ}.sin10∘2sin20∘−cos10∘.
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 \frac{2 (\sin 30^\circ \cos 10^\circ - \cos 30^\circ \sin 10^\circ) - \cos 10^\circ}{\sin 10^\circ}.sin10∘2(sin30∘cos10∘−cos30∘sin10∘)−cos10∘.
20
I recall that \sin 30^\circ = \frac{1}{2}sin30∘=21 and \cos 30^\circ = \frac{\sqrt{3}}{2},cos30∘=23, so I can simplify further.
21
I get \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ - \cos 10^\circ}{\sin 10^\circ}.sin10∘cos10∘−3sin10∘−cos10∘.
22
I can cancel out the \cos 10^\circcos10∘ terms in the numerator.
23
I get \frac{-\sqrt{3} \sin 10^\circ}{\sin 10^\circ}.sin10∘−3sin10∘.
24
I can cancel out the \sin 10^\circsin10∘ terms in the numerator and denominator.
25
I get -\sqrt{3}.−3.
26
Answer: -\sqrt{3}−3
It is unknown how broadly these results will generalize beyond the domain of math, and we consider it important for future work to explore the impact of process supervision in other domains. If these results generalize, we may find that process supervision gives us the best of both worlds – a method that is both more performant and more aligned than outcome supervision.
Large language models (LLMs) have shown impressive success in various applications. However, these models are often not well aligned with human intents, which calls for additional treatments on them, that is, the alignment problem. To make LLMs better follow user instructions, existing alignment methods mostly focus on further training them. However, the extra training of LLMs are usually expensive in terms of GPU compute; worse still, LLMs of interest are oftentimes not accessible for user-demanded training, such as GPTs. In this work, we take a different perspective -- Black-Box Prompt Optimization (BPO) -- to perform alignments. The idea is to optimize user prompts to suit LLMs' input understanding, so as to best realize users' intents without updating LLMs' parameters. BPO is model-agnostic and the empirical results demonstrate that the BPO-aligned ChatGPT yields a 22% increase in the win rate against its original version, and 10% for GPT-4. Importantly, the BPO-aligned LLMs can outperform the same models aligned by PPO and DPO, and it also brings additional performance gains when combining BPO with PPO or DPO. Code and datasets are released at this https URL.
Comments: | work in progress |
Subjects: | Computation and Language (cs.CL) |
Cite as: | arXiv:2311.04155 [cs.CL] |
(or arXiv:2311.04155v2 [cs.CL] for this version) | |
https://doi.org/10.48550/arXiv.2311.04155 Focus to learn more |
We introduce GAIA, a benchmark for General AI Assistants that, if solved, would represent a milestone in AI research. GAIA proposes real-world questions that require a set of fundamental abilities such as reasoning, multi-modality handling, web browsing, and generally tool-use proficiency. GAIA questions are conceptually simple for humans yet challenging for most advanced AIs: we show that human respondents obtain 92\% vs. 15\% for GPT-4 equipped with plugins. This notable performance disparity contrasts with the recent trend of LLMs outperforming humans on tasks requiring professional skills in e.g. law or chemistry. GAIA's philosophy departs from the current trend in AI benchmarks suggesting to target tasks that are ever more difficult for humans. We posit that the advent of Artificial General Intelligence (AGI) hinges on a system's capability to exhibit similar robustness as the average human does on such questions. Using GAIA's methodology, we devise 466 questions and their answer. We release our questions while retaining answers to 300 of them to power a leader-board available at this https URL.
Subjects: | Computation and Language (cs.CL); Artificial Intelligence (cs.AI) |
Cite as: | arXiv:2311.12983 [cs.CL] |
(or arXiv:2311.12983v1 [cs.CL] for this version) | |
[2311.12983] GAIA: a benchmark for General AI Assistants Focus to learn more |