What is the exact value of tan 195?

Steps to Solve

1. Convert Degrees to Radians (if needed)

To evaluate the tangent function, we typically work with radians, but it's not necessary for this calculation because most calculators can handle degrees.

2. Find Reference Angle

The reference angle for 195 degrees is calculated by subtracting 180 degrees: $$ 195^\circ - 180^\circ = 15^\circ $$

3. Determine the Sign of the Tangent

The angle 195 degrees is in the third quadrant, where the tangent function is positive. Thus, $tan(195^\circ) = tan(15^\circ)$.

4. Use Tangent Angle Formula for Exact Value

To find $tan(15^\circ)$, we can use the tangent subtraction formula: $$ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $$ Letting $A = 45^\circ$ and $B = 30^\circ$, we have: $$ \tan(15^\circ) = \tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ} $$

5. Insert Known Values

We know that: $$ \tan 45^\circ = 1 \quad \text{and} \quad \tan 30^\circ = \frac{1}{\sqrt{3}} $$ Substituting these values into the formula: $$ \tan(15^\circ) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} $$

6. Simplify the Expression

Now simplify the numerator and denominator: $$ = \frac{\frac{\sqrt{3}}{\sqrt{3}} - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} $$ This simplifies to: $$ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} $$