sin(85)*cos(90) cos(85)*sin(90)*cos(-30-180)

Let's evaluate the given expression step by step.

The expression is:

\[
\sin(85^\circ) \cdot \cos(90^\circ) + \cos(85^\circ) \cdot \sin(90^\circ) \cdot \cos(-30^\circ - 180^\circ)
\]

### Step 1: Evaluate Trigonometric Values
1. **\(\cos(90^\circ)\):**
   \[
   \cos(90^\circ) = 0
   \]
   Therefore, the term \(\sin(85^\circ) \cdot \cos(90^\circ)\) becomes:
   \[
   \sin(85^\circ) \cdot 0 = 0
   \]

2. **\(\sin(90^\circ)\):**
   \[
   \sin(90^\circ) = 1
   \]

3. **\(\cos(-30^\circ - 180^\circ)\):**
   \[
   \cos(-30^\circ - 180^\circ) = \cos(-210^\circ)
   \]
   \[
   \cos(-210^\circ) = \cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2}
   \]

### Step 2: Substitute Values into the Expression
Now substitute the values back into the expression:

\[
0 + \cos(85^\circ) \cdot 1 \cdot -\frac{\sqrt{3}}{2}
\]

\[
= -\frac{\sqrt{3}}{2} \cdot \cos(85^\circ)
\]

### Final Answer:
The simplified expression is:

\[
-\frac{\sqrt{3}}{2} \cdot \cos(85^\circ)
\]

This is the final result of the given expression.

Would you like more details or have any questions?

---

Here are some related questions for further practice:

1. What is the value of \(\cos(85^\circ)\) in decimal form?
2. How do you calculate \(\cos(150^\circ)\) without using a calculator?
3. What are the properties of cosine and sine for angles greater than 90 degrees?
4. How does the sign of the cosine function change in different quadrants?
5. Can you prove that \(\cos(150^\circ) = -\frac{\sqrt{3}}{2}\) using the unit circle?

**Tip:** Remember that the cosine of an angle in the second quadrant is always negative, which can be useful when simplifying trigonometric expressions.

Learn how to evaluate the trigonometric expression sin(85)*cos(90) + cos(85)*sin(90)*cos(-30-180) step-by-step with detailed explanations. This problem involves trigonometric identities and calculations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation