without using tables or a calculator find the value of cos(120+45) leave your answer in surd form

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.

Without using tables or a calculator find the value of cos(120+45) leave your answer in surd form.

Learn how to solve cos(120° + 45°) using the angle addition formula. This step-by-step guide covers trigonometric concepts, including exact values for angles and simplifying the result in surd form. Suitable for students in Grades 10-12.

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