Given that \( \sin x=\cos \left(2 x-48^{\circ}\right) \), find \( \tan x \), correct to 4 significant figures.

To solve the equation \( \sin x = \cos(2x - 48^\circ) \), we can use the identity that relates sine and cosine: \[ \sin x = \cos(90^\circ - x) \] This means we can rewrite the equation as: \[ \cos(90^\circ - x) = \cos(2x - 48^\circ) \] Using the property of cosine, we know that if \( \cos A = \cos B \), then: \[ A = B + 360^\circ n \quad \text{or} \quad A = -B + 360^\circ n \] for some integer \( n \). Applying this to our equation, we have two cases: ### Case 1: \[ 90^\circ - x = 2x - 48^\circ + 360^\circ n \] Rearranging gives: \[ 90^\circ + 48^\circ = 2x + x + 360^\circ n \] \[ 138^\circ = 3x + 360^\circ n \] \[ 3x = 138^\circ - 360^\circ n \] \[ x = \frac{138^\circ - 360^\circ n}{3} \] ### Case 2: \[ 90^\circ - x = -(2x - 48^\circ) + 360^\circ n \] Rearranging gives: \[ 90^\circ - x = -2x + 48^\circ + 360^\circ n \] \[ 90^\circ - 48^\circ = -2x + x + 360^\circ n \] \[ 42^\circ = -x + 360^\circ n \] \[ x = 360^\circ n - 42^\circ \] Now we will analyze the solutions from both cases. ### Case 1: For \( n = 0 \): \[ x = \frac{138^\circ}{3} = 46^\circ \] For \( n = 1 \): \[ x = \frac{138^\circ - 360^\circ}{3} = \frac{-222^\circ}{3} = -74^\circ \quad (\text{not in the range of } [0^\circ, 360^\circ]) \] For \( n = -1 \): \[ x = \frac{138^\circ + 360^\circ}{3} = \frac{498^\circ}{3} = 166^\circ \] ### Case 2: For \( n = 0 \): \[ x = -42^\circ \quad (\text{not in the range of } [0^\circ, 360^\circ]) \] For \( n = 1 \): \[ x = 360^\circ - 42^\circ = 318^\circ \] Now we have the valid solutions for \( x \): 1. \( x = 46^\circ \) 2. \( x = 166^\circ \) 3. \( x = 318^\circ \) Next, we will calculate \( \tan x \) for these angles. 1. For \( x = 46^\circ \): \[ \tan 46^\circ \approx 1.0355 \] 2. For \( x = 166^\circ \): \[ \tan 166^\circ = -\tan(180^\circ - 166^\circ) = -\tan 14^\circ \approx -0.2493 \] 3. For \( x = 318^\circ \): \[ \tan 318^\circ = \tan(-42^\circ) = -\tan 42^\circ \approx -0.9004 \] Among these, the value of \( \tan x \) that is positive and correct to 4 significant figures is: \[ \boxed{1.0355} \]