Question

Given that tan(x − π/3) = 1, solve x for x ∈(0°, 180°)

Answer 1

$\tan \left(x - \dfrac{\pi}{3} \right) = 1$

Replacement: $x - \dfrac{\pi}{3} = t$

$\tan (t) = 1$

$t = \tan^{-1}(1) + \pi k, \ k \in \mathbb{Z}$

$t = \dfrac{\pi}{4} + \pi k, \ k \in \mathbb{Z}$

Reverse replacement:

$x - \dfrac{\pi}{3} = \dfrac{\pi}{4} + \pi k, \ k \in \mathbb{Z}$

$x - \dfrac{\pi}{3} + \dfrac{\pi}{3} = \dfrac{\pi}{4} + \dfrac{\pi}{3} + \pi k, \ k \in \mathbb{Z}$

$x = \dfrac{7 \pi}{12} + \pi k, \ k \in \mathbb{Z}$

Since $x \in (0^{\circ}; \ 180^{\circ})$ i. e. $x \in (0; \ \pi)$ then:

$0 < \dfrac{7 \pi}{12} + \pi k < \pi, \ k \in \mathbb{Z}$

$0 - \dfrac{7 \pi}{12} < \dfrac{7 \pi}{12} - \dfrac{7 \pi}{12} + \pi k < \pi - \dfrac{7 \pi}{12}, \ k \in \mathbb{Z}$

$-\dfrac{7 \pi}{12} < \pi k < \dfrac{5 \pi}{12}, \ k \in \mathbb{Z}$

$-\dfrac{7 \pi}{12} \cdot \dfrac{1}{\pi} < \pi k \cdot \dfrac{1}{\pi} < \dfrac{5 \pi}{12} \cdot \dfrac{1}{\pi}, \ k \in \mathbb{Z}$

$-\dfrac{7}{12} < k < \dfrac{5}{12}, \ k \in \mathbb{Z}$

Thus $k = 0$

Therefore, $x = \dfrac{7 \pi}{12} + \pi \cdot 0 = \dfrac{7 \pi}{12}$

Answer: $x = \dfrac{7 \pi}{12}$

Answer 2

To solve the equation tan(x - π/3) = 1 for x ∈ (0°, 180°), we can use the properties of the tangent function and some trigonometric identities. Here's the step-by-step solution:

1. Start with the given equation: tan(x - π/3) = 1.

2. Use the identity tan(π/4) = 1. By comparing this with the given equation, we can see that x - π/3 must be equal to π/4.

   x - π/3 = π/4.

3. Add π/3 to both sides of the equation:

   x = π/4 + π/3.

4. Find a common denominator for π/4 and π/3, which is 12:

   x = (3π + 4π) / 12.

   Simplifying the numerator:

   x = 7π / 12.

5. The value of x is in radians. To convert it to degrees, multiply by 180/π:

   x = (7π / 12) * (180/π).

   Simplifying:

   x = 105°.

6. Check if the solution satisfies the given condition that x is in the interval (0°, 180°). In this case, 105° is within the specified range.

Therefore, the solution to the equation tan(x - π/3) = 1 for x ∈ (0°, 180°) is x = 105°.