Question

Find the value of \( \tan(105^\circ) \) using compound angle identities.

• A. \(2+\sqrt3\)
• B. \(-(2+\sqrt3)\)
• C. \(1+\sqrt3\)
• D. \(\sqrt3-2\)

Hint:

Angles such as \(15^\circ, 75^\circ, 105^\circ\) are usually solved using compound angle identities like: \[ \sin(A\pm B),\quad \cos(A\pm B),\quad \tan(A\pm B) \] Break the angle into known angles such as \(30^\circ, 45^\circ, 60^\circ\).

Answer 1

The Correct Option is B

Concept: To evaluate trigonometric functions of angles like \(105^\circ\), we use compound angle identities. Since \[ 105^\circ = 60^\circ + 45^\circ \] we apply the identity \[ \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

Step 1: Substitute the angles. \[ \tan(105^\circ) = \tan(60^\circ + 45^\circ) \] \[ = \frac{\tan60^\circ + \tan45^\circ}{1 - \tan60^\circ \tan45^\circ} \]

Step 2: Use known trigonometric values. \[ \tan60^\circ = \sqrt3 \] \[ \tan45^\circ = 1 \] Substitute: \[ \tan(105^\circ) = \frac{\sqrt3 + 1}{1 - \sqrt3} \]

Step 3: Rationalize the denominator. Multiply numerator and denominator by \(1+\sqrt3\): \[ \frac{(\sqrt3+1)(1+\sqrt3)}{(1-\sqrt3)(1+\sqrt3)} \] Denominator: \[ 1-3 = -2 \] Numerator: \[ (\sqrt3+1)^2 = 3 + 2\sqrt3 +1 = 4 + 2\sqrt3 \] Thus, \[ \tan(105^\circ) = \frac{4+2\sqrt3}{-2} \] \[ = -(2+\sqrt3) \] \[ \boxed{-(2+\sqrt3)} \]