1.How do you solve for sec(2pi/3) * tan(2pi/3)
2.2X=7
3.What is the smallest number (positive integer) when spelled out, the letters appear in alphabetical order? The word is explain it briefly
The Correct Answer and Explanation is :
1. Solving for (\sec\left(\frac{2\pi}{3}\right) \times \tan\left(\frac{2\pi}{3}\right)):
To solve this, we’ll break it down into two steps: first find (\sec\left(\frac{2\pi}{3}\right)) and (\tan\left(\frac{2\pi}{3}\right)), then multiply them together.
- Step 1: Find (\sec\left(\frac{2\pi}{3}\right)) (\sec(\theta) = \frac{1}{\cos(\theta)}) The cosine of (\frac{2\pi}{3}) is in the second quadrant, where cosine is negative. We know that (\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}). Therefore,
[
\sec\left(\frac{2\pi}{3}\right) = \frac{1}{-\frac{1}{2}} = -2.
] - Step 2: Find (\tan\left(\frac{2\pi}{3}\right)) (\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}) We know that (\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}) and (\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}). Therefore,
[
\tan\left(\frac{2\pi}{3}\right) = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}.
] - Step 3: Multiply (\sec\left(\frac{2\pi}{3}\right)) and (\tan\left(\frac{2\pi}{3}\right)) Now, multiply the results:
[
\sec\left(\frac{2\pi}{3}\right) \times \tan\left(\frac{2\pi}{3}\right) = (-2) \times (-\sqrt{3}) = 2\sqrt{3}.
]
So, the answer is (2\sqrt{3}).
2. Solving (2X = 7):
To solve for (X), divide both sides of the equation by 2:
[
X = \frac{7}{2} = 3.5.
]
So, (X = 3.5).
3. Smallest number whose spelled-out letters appear in alphabetical order:
The number you’re looking for is 40.
- The word for 40 is “forty.”
- When spelled out, the letters are: f, o, r, t, y.
- The letters are in alphabetical order: f < o < r < t < y.
Now, let’s explain why 40 is the smallest number with this property.
Explanation:
When spelled out, the letters of a number should appear in increasing alphabetical order. To determine the smallest number where this occurs, we start by examining numbers in increasing order and checking their spellings.
For instance:
- The word for 1 is “one”, which does not have letters in alphabetical order.
- The word for 10 is “ten”, which does not have letters in alphabetical order.
- As we move through numbers, we continue checking their spellings until we reach 40, which is “forty.”
In “forty”, the letters are arranged in alphabetical order: f, o, r, t, and y. None of the smaller numbers (like 1, 10, 12, 20, etc.) have their letters in alphabetical order, making 40 the smallest number with this property.
While numbers larger than 40 may also satisfy this condition, 40 is the first number where it happens. This pattern continues for larger numbers, but none are smaller than 40 with this specific alphabetical order.