To solve $\cos(120^\circ + 45^\circ)$ without using a calculator or tables, we can use the angle addition formula for cosine:

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Here, $A = 120^\circ$ and $B = 45^\circ$. Substituting these values into the formula:

$\cos(120^\circ + 45^\circ) = \cos 120^\circ \cos 45^\circ - \sin 120^\circ \sin 45^\circ$

Step 1: Calculate $\cos 120^\circ$ and $\sin 120^\circ$

• $\cos 120^\circ = -\frac{1}{2}$ (since $120^\circ$ is in the second quadrant, where cosine is negative)
• $\sin 120^\circ = \frac{\sqrt{3}}{2}$ (since $120^\circ = 180^\circ - 60^\circ$)

Step 2: Calculate $\cos 45^\circ$ and $\sin 45^\circ$

• $\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Step 3: Substitute into the formula

Now substitute the values into the angle addition formula:

$\cos(120^\circ + 45^\circ) = \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right)$

Step 4: Simplify

$\cos(165^\circ) = -\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ $= -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$

Step 5: Combine terms

$\cos(165^\circ) = -\frac{\sqrt{2} + \sqrt{6}}{4}$

Thus, the value of $\cos(120^\circ + 45^\circ)$ in surd form is:

$\boxed{-\frac{\sqrt{2} + \sqrt{6}}{4}}$

Would you like further details or have any questions?

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Here are 5 related questions:

1. How would you apply the sine addition formula to solve $\sin(120^\circ + 45^\circ)$?
2. What are the exact values of trigonometric functions for angles in other quadrants?
3. Can you simplify $\cos(120^\circ + \theta)$ for any angle $\theta$?
4. How do trigonometric values change as angles move through different quadrants?
5. What is the geometric interpretation of cosine for angles greater than $90^\circ$?

Tip: For non-standard angles, always break them into familiar angles using addition or subtraction and apply known formulas.