Analyzing the Trigonometric Equation
We need to find the number of solutions for the equation $\sin^7x+\cos^7x=1$ in the interval $x \in [0, 4\pi]$.
Key Properties and Inequalities
Recall the fundamental trigonometric identity: $\sin^2x+\cos^2x=1$. For any real number $x$, we know that $-1 \le \sin x \le 1$ and $-1 \le \cos x \le 1$. Since the power is odd (7), $\sin^7x$ has the same sign as $\sin x$, and $\cos^7x$ has the same sign as $\cos x$. Consider the interval where $\sin x > 0$ and $\cos x > 0$, which is the first quadrant $(0, \pi/2)$. In this interval, $0 < \sin x < 1$ and $0 < \cos x < 1$. For $0 < a < 1$, we have $a^7 < a^2$. Therefore, for $x \in (0, \pi/2)$: $\sin^7x < \sin^2x$ $\cos^7x < \cos^2x$ Adding these inequalities, we get $\sin^7x+\cos^7x < \sin^2x+\cos^2x = 1$. Thus, there are no solutions in the open interval $(0, \pi/2)$. Similar analysis shows no solutions in the open intervals $(\pi/2, \pi)$, $(\pi, 3\pi/2)$, and $(3\pi/2, 2\pi)$ because in these intervals, either $\sin^7x < \sin^2x$ or $\cos^7x < \cos^2x$ (or both potentially negative), and the sum cannot reach 1.
Identifying Potential Solutions
The equation $\sin^7x+\cos^7x=1$ can only hold true when the terms are close to their maximum possible values, specifically when $\sin x$ or $\cos x$ are equal to $0$ or $1$. Let's check these cases:
- Case 1: $\sin x = 1$. This implies $\cos x = 0$. Substituting into the equation: $1^7 + 0^7 = 1$. This is true.
- Case 2: $\cos x = 1$. This implies $\sin x = 0$. Substituting into the equation: $0^7 + 1^7 = 1$. This is true.
- Case 3: $\sin x = 0$. This implies $\cos x = \pm 1$.
- If $\cos x = 1$, $0^7 + 1^7 = 1$. This is true (same as Case 2).
- If $\cos x = -1$, $0^7 + (-1)^7 = -1 \ne 1$. This is not a solution.
- Case 4: $\cos x = 0$. This implies $\sin x = \pm 1$.
- If $\sin x = 1$, $1^7 + 0^7 = 1$. This is true (same as Case 1).
- If $\sin x = -1$, $(-1)^7 + 0^7 = -1 \ne 1$. This is not a solution.
Therefore, the solutions occur only when $(\sin x, \cos x)$ is $(1, 0)$ or $(0, 1)$.
General Solutions
The conditions are met when:
- $\sin x = 1$ and $\cos x = 0$. This happens when $x = \frac{\pi}{2} + 2k\pi$ for any integer $k$.
- $\sin x = 0$ and $\cos x = 1$. This happens when $x = 2k\pi$ for any integer $k$.
Solutions in the Interval $[0, 4\pi]$
We need to find the values of $k$ that place $x$ within the interval $[0, 4\pi]$.
- For $x = 2k\pi$:
- If $k=0$, $x = 0$.
- If $k=1$, $x = 2\pi$.
- If $k=2$, $x = 4\pi$.
- For $x = \frac{\pi}{2} + 2k\pi$:
- If $k=0$, $x = \frac{\pi}{2}$.
- If $k=1$, $x = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2}$.
- If $k=2$, $x = \frac{\pi}{2} + 4\pi = \frac{9\pi}{2}$ (This is greater than $4\pi$, so it's outside the interval).
The distinct solutions in the interval $[0, 4\pi]$ are $0, \frac{\pi}{2}, 2\pi, \frac{5\pi}{2}, 4\pi$.
Counting the Solutions
There are 5 distinct solutions.
Similar Questions
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- Number of solutions of $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi]$ is :
- Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right), \theta \in \mathbf{R}$. Then $\alpha + 2\beta$ is equal to :
- The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$. The coordinates of A and B are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and C is at a distance of 10 units from B. The area (in sq. units) of $\Delta ABC$ is :
- If $\frac{\pi}{4} + \sum_{p=1}^{11} \tan^{-1} \left( \frac{2^{p-1}}{1 + 2^{2p-1}} \right) = \alpha$, then $\tan \alpha$ is equal to _________.
- If $\text{S} = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\frac{5\theta}{2} = \cos 7\theta \cos\frac{7\theta}{2}\right\}$, then $\text{n(S)}$ is equal to ___________.
- Let $S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a, a \in \mathbf{Z}\}$. Then $n(S)$ is equal to :
- Let $P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}$ and $S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec2\theta = a^2, \theta \in P\}$. Then $n(S)$ is :
- If $\text{A} = \frac{\sin 3^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 27^\circ} + \frac{\sin 27^\circ}{\cos 81^\circ}$ and $\text{B} = \tan 81^\circ - \tan 3^\circ$, then $\frac{\text{B}}{\text{A}}$ is equal to _____.
Important Questions from Trigonometry
Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.
- The number of elements in the set $\{x \in [0, 180^\circ] : \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ)\}$ is _______.
- Number of solutions of $\sqrt{3}\cos 2\theta + 8\cos \theta + 3\sqrt{3} = 0, \theta \in [-3\pi, 2\pi]$ is :
- Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta) = 4\left(\sin^4\left(\frac{7\pi}{2} - \theta\right) + \sin^4(11\pi + \theta)\right) - 2\left(\sin^6\left(\frac{3\pi}{2} - \theta\right) + \sin^6(9\pi - \theta)\right), \theta \in \mathbf{R}$. Then $\alpha + 2\beta$ is equal to :
- The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \frac{1 - y}{-2} = \frac{z - 2}{3}$. The coordinates of A and B are $(1, 6, 3)$ and $(4, 9, \alpha)$ respectively and C is at a distance of 10 units from B. The area (in sq. units) of $\Delta ABC$ is :
