https://file.aiquickdraw.com/mathbot/file/1725977965281zynr7b5u.jpg

The expression provided in the image is:

\[
(\cos 0^\circ + \cos 45^\circ + \sin 30^\circ) \times (\sin 90^\circ - \sin 45^\circ + \cos 60^\circ)
\]

Let's simplify this step by step.

### Step 1: Evaluate the trigonometric functions
- \(\cos 0^\circ = 1\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\sin 90^\circ = 1\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
- \(\cos 60^\circ = \frac{1}{2}\)

### Step 2: Substitute these values into the expression
Substitute the values into the expression:

\[
(1 + \frac{\sqrt{2}}{2} + \frac{1}{2}) \times (1 - \frac{\sqrt{2}}{2} + \frac{1}{2})
\]

### Step 3: Simplify each part
Let's simplify each part:

1. First bracket:
   \[
   1 + \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{2}{2} + \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{2 + \sqrt{2} + 1}{2} = \frac{3 + \sqrt{2}}{2}
   \]

2. Second bracket:
   \[
   1 - \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{2 - \sqrt{2} + 1}{2} = \frac{3 - \sqrt{2}}{2}
   \]

### Step 4: Multiply the two brackets
Now, multiply the two simplified expressions:

\[
\frac{3 + \sqrt{2}}{2} \times \frac{3 - \sqrt{2}}{2}
\]

This is in the form of \((a + b)(a - b) = a^2 - b^2\):

\[
\frac{(3 + \sqrt{2})(3 - \sqrt{2})}{4} = \frac{3^2 - (\sqrt{2})^2}{4} = \frac{9 - 2}{4} = \frac{7}{4}
\]

### Final Answer:
The simplified result is:

\[
\frac{7}{4}
\]

Would you like to know more details or have any questions?

---

Here are some related questions you might find interesting:

1. What are the steps to evaluate trigonometric identities?
2. How do you derive the value of trigonometric functions at standard angles?
3. Can you show examples of using the difference of squares in algebra?
4. How do you expand and simplify expressions using trigonometric identities?
5. How would this expression change if the angles were different?

**Tip:** When simplifying expressions involving trigonometric functions, always try to recognize patterns such as the difference of squares or sum/difference identities. This will make your calculations much faster!

(cos 0° + cos 45° + sin 30°)(sin 90° - sin 45° + cos 60°)

Learn how to simplify the trigonometric expression (cos 0° + cos 45° + sin 30°)(sin 90° - sin 45° + cos 60°) using basic trigonometric values and the difference of squares. This example is suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation