Find the exact value of each expression.
tan 285°

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Recognize that the angle 285° is in the fourth quadrant, where tangent values are negative.

Express 285° as a sum or difference of angles whose tangent values are known. For example, write 285° as 225° + 60°.

Use the tangent addition formula: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\), where \(A = 225^\circ\) and \(B = 60^\circ\).

Recall the exact values: \(\tan 225^\circ = 1\) and \(\tan 60^\circ = \sqrt{3}\), then substitute these into the formula.

Simplify the resulting expression step-by-step to find the exact value of \(\tan 285^\circ\).

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Reference Angles and Angle Reduction

To find the exact value of trigonometric functions for angles greater than 360° or in different quadrants, we use reference angles by subtracting or adding full rotations (360°) or known angles. For 285°, recognizing its position in the fourth quadrant helps simplify the calculation.

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Reference Angles on the Unit Circle

Tangent Function and Its Sign in Quadrants

The tangent function is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Its sign depends on the quadrant: tangent is positive in the first and third quadrants and negative in the second and fourth. Since 285° lies in the fourth quadrant, tan 285° is negative.

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Quadratic Formula

Using Angle Sum or Difference Identities

When the angle is not standard, express it as a sum or difference of known angles (e.g., 285° = 270° + 15°) and apply the tangent addition or subtraction formulas: tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b). This allows exact evaluation using known values.

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