Find the exact value of each expression.
tan 285°

Verified step by step guidance

1. Recognize that the angle 285° is in the fourth quadrant, where tangent values are negative.
2. Express 285° as a sum or difference of angles whose tangent values are known. For example, write 285° as 225° + 60°.
3. Use the tangent addition formula: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\), where \(A = 225^\circ\) and \(B = 60^\circ\).
4. Recall the exact values: \(\tan 225^\circ = 1\) and \(\tan 60^\circ = \sqrt{3}\), then substitute these into the formula.
5. Simplify the resulting expression step-by-step to find the exact value of \(\tan 285^\circ\).

Key Concepts

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.

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.

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.