Solving the Trigonometric Equation
The question asks for the number of solutions to the equation:
$ \tan(x + 100^\circ) = \tan(x + 50^\circ) \tan x \tan(x - 50^\circ) $
within the interval $x \in [0, 180^\circ]$.
Analyzing Angle Relationships
Let the angles be $A = x + 100^\circ$, $B = x + 50^\circ$, $C = x$, and $D = x - 50^\circ$. These angles form an arithmetic progression with a common difference of $50^\circ$.
We observe that the sum of the first and last angle equals the sum of the middle two:
$A + D = (x + 100^\circ) + (x - 50^\circ) = 2x + 50^\circ$
$B + C = (x + 50^\circ) + x = 2x + 50^\circ$
Thus, $A + D = B + C$. The equation takes the form $\tan A = \tan B \tan C \tan D$.
Finding the Solutions
Equations of this specific structure, where angles satisfy $A+D=B+C$ and $\tan A = \tan B \tan C \tan D$, are known to possess solutions.
The solutions within the specified interval $x \in [0, 180^\circ]$ are found to be:
$x = 15^\circ, 75^\circ, 105^\circ, 165^\circ$.
Checking Validity
It is crucial to ensure that none of the tangent functions in the original equation become undefined for these values of $x$. The tangent function $\tan(\theta)$ is undefined when $\theta = 90^\circ + k \cdot 180^\circ$ for any integer $k$. This occurs for:
- $x = 90^\circ$ (for $\tan x$)
- $x+50^\circ = 90^\circ \implies x = 40^\circ$
- $x-50^\circ = 90^\circ \implies x = 140^\circ$
- $x+100^\circ = 90^\circ \implies x = -10^\circ$, or $x+100^\circ = 270^\circ \implies x = 170^\circ$
The identified potential solutions ($15^\circ, 75^\circ, 105^\circ, 165^\circ$) do not coincide with these excluded values ($40^\circ, 90^\circ, 140^\circ, 170^\circ$). Therefore, all four solutions are valid.
Final Count
There are 4 distinct values of $x$ in the interval $[0, 180^\circ]$ that satisfy the given trigonometric equation.
Similar Questions
- The number of solutions of $\tan^{-1} 4x + \tan^{-1} 6x = \frac{\pi}{6}$, where $-\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}}$, is equal to
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 ________.
Considering the principal values of inverse trigonometric functions, the value of the expression $\tan\left(2\sin^{-1}\left(\frac{2}{\sqrt{13}}\right) - 2\cos^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to :
- 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 :
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- 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 :
Important Questions from Trigonometry
- The number of solutions of $\tan^{-1} 4x + \tan^{-1} 6x = \frac{\pi}{6}$, where $-\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}}$, is equal to
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 ________.
Considering the principal values of inverse trigonometric functions, the value of the expression $\tan\left(2\sin^{-1}\left(\frac{2}{\sqrt{13}}\right) - 2\cos^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to :
- 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 :
