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When $x=\theta$, the function $f(x)=\sin x-3\cos x$ reaches its maximum value. What is the value of $\cos \theta$? | The function $f(x)=\sin x-3\cos x$ can be rewritten as $\sqrt {10}( \frac { \sqrt {10}}{10}\sin x- \frac {3 \sqrt {10}}{10}\cos x)$, which simplifies to $\sqrt {10}\sin (x+\alpha)$, where $\cos \alpha= \frac { \sqrt {10}}{10}$ and $\sin \alpha=- \frac {3 \sqrt {10}}{10}$.
The function $f(x)$ reaches its maximum value ... | -\frac {3 \sqrt {10}}{10} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s... | [
"<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s... | <think>
Okay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this.
First, I remember that functions of the form a sin x + b cos x can be rewritten using a single si... | [
{
"from": "human",
"value": "When $x=\\theta$, the function $f(x)=\\sin x-3\\cos x$ reaches its maximum value. What is the value of $\\cos \\theta$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum val... |
When $x=\theta$, the function $f(x)=\sin x-3\cos x$ reaches its maximum value. What is the value of $\cos \theta$? | The function $f(x)=\sin x-3\cos x$ can be rewritten as $\sqrt {10}( \frac { \sqrt {10}}{10}\sin x- \frac {3 \sqrt {10}}{10}\cos x)$, which simplifies to $\sqrt {10}\sin (x+\alpha)$, where $\cos \alpha= \frac { \sqrt {10}}{10}$ and $\sin \alpha=- \frac {3 \sqrt {10}}{10}$.
The function $f(x)$ reaches its maximum value ... | -\frac {3 \sqrt {10}}{10} | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s... | [
"<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's start by recalling how to find the maximum of a trigonometric function like this. \n\nFirst, I remember that functions of the form a sin x + b cos x can be rewritten using a s... | <think>
Okay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum value at x = θ. Hmm, let's see. I remember that for functions of the form a sin x + b cos x, the maximum value is sqrt(a² + b²). But wait, here the function is sin x - 3 cos x. So maybe that formula still app... | [
{
"from": "human",
"value": "When $x=\\theta$, the function $f(x)=\\sin x-3\\cos x$ reaches its maximum value. What is the value of $\\cos \\theta$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of cos θ when the function f(x) = sin x - 3 cos x reaches its maximum val... |
Among the following items, which can form a set?
A: Very small numbers
B: Interesting books
C: Real numbers greater than 8
D: Tall people | Option A is incorrect. The standard for "very small numbers" is not fixed, so it cannot form a set.
Option B is incorrect. "Interesting books" do not satisfy the definiteness of a set, so they cannot form a set.
Option C is correct. Real numbers greater than 8 can form a set.
Option D is incorrect. "Tall people" ... | \text{C} | Logic and Puzzles | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel... | [
"<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel... | <think>
Okay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is "well-defined," which means that for any given object, we can definitively say wh... | [
{
"from": "human",
"value": "Among the following items, which can form a set? \nA: Very small numbers \nB: Interesting books \nC: Real numbers greater than 8 \nD: Tall people"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out which of these options can form a set. Let's st... |
Among the following items, which can form a set?
A: Very small numbers
B: Interesting books
C: Real numbers greater than 8
D: Tall people | Option A is incorrect. The standard for "very small numbers" is not fixed, so it cannot form a set.
Option B is incorrect. "Interesting books" do not satisfy the definiteness of a set, so they cannot form a set.
Option C is correct. Real numbers greater than 8 can form a set.
Option D is incorrect. "Tall people" ... | \text{C} | Logic and Puzzles | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | [
"<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel... | [
"<think>\nOkay, so I need to figure out which of these options can form a set. Let's start by recalling what a set actually is. From what I remember in math class, a set is a well-defined collection of distinct objects. The key term here is \"well-defined,\" which means that for any given object, we can definitivel... | <think>
Okay, so I need to figure out which of these options can form a set. The options are A: Very small numbers, B: Interesting books, C: Real numbers greater than 8, and D: Tall people. Hmm, let me recall what defines a set in mathematics. A set is a well-defined collection of distinct objects. The key term here is... | [
{
"from": "human",
"value": "Among the following items, which can form a set? \nA: Very small numbers \nB: Interesting books \nC: Real numbers greater than 8 \nD: Tall people"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out which of these options can form a set. The opti... |
"Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of(...TRUNCATED) | "1. **Given** $10^{m}=12$ **and** $10^{n}=3$, **find the value of** $10^{m-n}$.\n\n We start by ex(...TRUNCATED) | -1 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. (...TRUNCATED) | ["<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. (...TRUNCATED) | "<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. *(...TRUNCATED) | [{"from":"human","value":"Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^(...TRUNCATED) |
"Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of(...TRUNCATED) | "1. **Given** $10^{m}=12$ **and** $10^{n}=3$, **find the value of** $10^{m-n}$.\n\n We start by ex(...TRUNCATED) | -1 | Algebra | math-word-problem | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. (...TRUNCATED) | ["<think>\nOkay, let's try to solve these problems one by one. I'll start with the first one.\n\n1. (...TRUNCATED) | "<think>\nOkay, let's tackle these three problems one by one. I need to make sure I understand each (...TRUNCATED) | [{"from":"human","value":"Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^(...TRUNCATED) |
"Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\\ (n\\in(...TRUNCATED) | "**Analysis**\n\nThis question examines the application of the general formula of an arithmetic sequ(...TRUNCATED) | A | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a(...TRUNCATED) | ["<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a(...TRUNCATED) | "<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a_(...TRUNCATED) | [{"from":"human","value":"Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n(...TRUNCATED) |
"Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n+1}+a_{n+2}=3n+5\\ (n\\in(...TRUNCATED) | "**Analysis**\n\nThis question examines the application of the general formula of an arithmetic sequ(...TRUNCATED) | A | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a(...TRUNCATED) | ["<think>\nOkay, let's try to solve this problem step by step. So, we have an arithmetic sequence {a(...TRUNCATED) | "<think>\nOkay, let's see. The problem says that we have an arithmetic sequence {a_n}, and for any n(...TRUNCATED) | [{"from":"human","value":"Given that $\\left\\{ a_n \\right\\}$ is an arithmetic sequence, and $a_{n(...TRUNCATED) |
"$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （ ）\n\nA: $-\\frac{(...TRUNCATED) | "Solution: $\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ$\n\n$= \\sin(18^\\circ(...TRUNCATED) | D | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And (...TRUNCATED) | ["<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And (...TRUNCATED) | "<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And I(...TRUNCATED) | [{"from":"human","value":"$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （(...TRUNCATED) |
"$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （ ）\n\nA: $-\\frac{(...TRUNCATED) | "Solution: $\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ$\n\n$= \\sin(18^\\circ(...TRUNCATED) | D | Algebra | MCQ | Yes | Yes | cn_k12 | false | 2 | [
true,
true
] | 2 | ["<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And (...TRUNCATED) | ["<think>Okay, so I have this trigonometry problem here: sin 18° cos 12° + cos 18° sin 12°. And (...TRUNCATED) | "<think>\nAlright, let's see. I need to figure out the value of sin 18° cos 12° + cos 18° sin 12(...TRUNCATED) | [{"from":"human","value":"$\\sin 18^\\circ \\cos 12^\\circ + \\cos 18^\\circ \\sin 12^\\circ = $ （(...TRUNCATED) |
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