Use a sum or difference formula to find the exact value of the trigonometric function.

(Type an exact answer, using radicals as needed. Simplify your answer.)

Ask by Bob Lindsey. in the United States

Jan 23,2025

Question

Use a sum or difference formula to find the exact value of the trigonometric function.

(Type an exact answer, using radicals as needed. Simplify your answer.)

Ask by Bob Lindsey. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the exact value of $$ \tan 105^{\circ} $$ using the sum formula, follow these steps:

1. **Express $$ 105^{\circ} $$ as a sum of known angles:**
   $$
   105^{\circ} = 60^{\circ} + 45^{\circ}
   $$

2. **Apply the tangent addition formula:**
   $$
   \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}
   $$
   Let $$ a = 60^{\circ} $$ and $$ b = 45^{\circ} $$, so:
   $$
   \tan 105^{\circ} = \frac{\tan 60^{\circ} + \tan 45^{\circ}}{1 - \tan 60^{\circ} \tan 45^{\circ}}
   $$

3. **Substitute the known values:**
   $$
   \tan 60^{\circ} = \sqrt{3}, \quad \tan 45^{\circ} = 1
   $$
   $$
   \tan 105^{\circ} = \frac{\sqrt{3} + 1}{1 - \sqrt{3} \cdot 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}
   $$

4. **Rationalize the denominator:**
   Multiply the numerator and the denominator by the conjugate of the denominator $$ (1 + \sqrt{3}) $$:
   $$
   \tan 105^{\circ} = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}
   $$

So, the exact value is:

$$
\tan 105^{\circ} = \boxed{-2 - \sqrt{3}}
$$
$$ \tan 105^{\circ} = -2 - \sqrt{3} $$

Use a sum or difference formula to find the exact value of the trigonometric function.

(Type an exact answer, using radicals as needed. Simplify your answer.)

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Use a sum or difference formula to find the exact value of the trigonometric function. (Type an exact answer, using radicals as needed. Simplify your answer.)

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Use a sum or difference formula to find the exact value of the trigonometric function.

(Type an exact answer, using radicals as needed. Simplify your answer.)