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

We want to find the value of \( \cos 120^\circ + \cos 45^\circ \).

Let's break this down step by step:

### Step 1: Find \( \cos 120^\circ \)

- \( 120^\circ \) lies in the second quadrant.
- The reference angle for \( 120^\circ \) is \( 180^\circ - 120^\circ = 60^\circ \).
- In the second quadrant, cosine is negative, so:
  \[
  \cos 120^\circ = -\cos 60^\circ
  \]
- We know that \( \cos 60^\circ = \frac{1}{2} \), so:
  \[
  \cos 120^\circ = -\frac{1}{2}
  \]

### Step 2: Find \( \cos 45^\circ \)

- We know that \( \cos 45^\circ = \frac{1}{\sqrt{2}} \).

### Step 3: Add the two results

Now, add the values of \( \cos 120^\circ \) and \( \cos 45^\circ \):
\[
\cos 120^\circ + \cos 45^\circ = -\frac{1}{2} + \frac{1}{\sqrt{2}}
\]

### Step 4: Combine terms

To combine these, we need a common denominator. The least common denominator is \( 2\sqrt{2} \). Let's rewrite both fractions:

- Rewrite \( -\frac{1}{2} \) as \( -\frac{\sqrt{2}}{2\sqrt{2}} \)
- Rewrite \( \frac{1}{\sqrt{2}} \) as \( \frac{2}{2\sqrt{2}} \)

Now, add them together:
\[
-\frac{\sqrt{2}}{2\sqrt{2}} + \frac{2}{2\sqrt{2}} = \frac{-\sqrt{2} + 2}{2\sqrt{2}}
\]

Thus, the value of \( \cos 120^\circ + \cos 45^\circ \) in surd form is:
\[
\frac{2 - \sqrt{2}}{2\sqrt{2}}
\]

Would you like more details or explanations on this solution?

Here are 5 follow-up questions:
1. How do you find the reference angle for any angle in trigonometry?
2. What are the signs of cosine in each of the four quadrants?
3. Why is \( \cos 45^\circ \) expressed as \( \frac{1}{\sqrt{2}} \)?
4. How do you rationalize a denominator in a fraction?
5. Can you explain the significance of special angles like \( 45^\circ \) and \( 60^\circ \) in trigonometry?

**Tip**: In the unit circle, angles in different quadrants share the same reference angle, but their sine and cosine values differ based on the quadrant.

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

This guide explains how to find the value of cos 120° + cos 45° without using a calculator or tables, leaving the answer in surd form. The solution breaks down the cosine function for angles in different quadrants and simplifies the result with a step-by-step method. Suitable for students in Grades 9-12.

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