If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

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Identify the given values: \(\cos x = -0.750\) and \(\sin x \approx 0.6614\).

Recall the half-angle formula for tangent: \(\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x}\) or \(\tan \frac{x}{2} = \frac{\sin x}{1 + \cos x}\). Choose the form that avoids division by zero or undefined expressions based on the quadrant of \(x\).

Since \(\cos x\) is negative and \(\sin x\) is positive, \(x\) lies in the second quadrant. For \(x\) in the second quadrant, use the formula \(\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x}\) to ensure the correct sign.

Substitute the given values into the chosen formula: \(\tan \frac{x}{2} = \frac{1 - (-0.750)}{0.6614} = \frac{1 + 0.750}{0.6614}\).

Simplify the expression to find the approximate value of \(\tan \frac{x}{2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relationship Between Sine, Cosine, and Tangent

Sine, cosine, and tangent are fundamental trigonometric functions related by the identity tan x = sin x / cos x. Understanding these relationships helps in converting between functions and solving for unknown values.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the original angle. For tangent, tan(x/2) can be found using formulas like tan(x/2) = ±√((1 - cos x)/(1 + cos x)) or tan(x/2) = sin x / (1 + cos x), depending on the quadrant.

Sign Determination Based on Quadrants

The sign of trigonometric functions depends on the angle's quadrant. Since cos x is negative and sin x is positive, x lies in the second quadrant, which affects the sign choice in half-angle formulas for tan(x/2).

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