https://file.aiquickdraw.com/mathbot/file/17263645425561811rmz1.jpg

Let's evaluate the expression step by step:

\[
1 + \cos(90^\circ) + \tan(45^\circ) + \frac{\sin(315^\circ)}{\csc(150^\circ)}
\]

1. **Cosine of 90 degrees**:
   \[
   \cos(90^\circ) = 0
   \]
   
2. **Tangent of 45 degrees**:
   \[
   \tan(45^\circ) = 1
   \]

3. **Sine of 315 degrees**:
   \[
   \sin(315^\circ) = \sin(360^\circ - 45^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}
   \]

4. **Cosecant of 150 degrees**:
   \[
   \csc(150^\circ) = \frac{1}{\sin(150^\circ)} = \frac{1}{\sin(180^\circ - 30^\circ)} = \frac{1}{\sin(30^\circ)} = \frac{1}{\frac{1}{2}} = 2
   \]

5. **Simplifying the fraction**:
   \[
   \frac{\sin(315^\circ)}{\csc(150^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{2} = -\frac{\sqrt{2}}{4}
   \]

Now, let's put everything together:

\[
1 + 0 + 1 + \left(-\frac{\sqrt{2}}{4}\right) = 2 - \frac{\sqrt{2}}{4}
\]

So, the final result is:

\[
2 - \frac{\sqrt{2}}{4}
\]

Would you like further details or have any questions?

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Here are 5 questions to expand your understanding:
1. What are the exact values of sine, cosine, and tangent for common angles like 0°, 30°, 45°, and 90°?
2. How do we calculate the values of trigonometric functions for angles greater than 90°?
3. Why is the sine of 315° negative, and how do reference angles work in trigonometry?
4. How is the cosecant function related to the sine function, and how can we use that relationship in different quadrants?
5. What are the common identities involving tangent and cosecant?

**Tip:** Always check the quadrant of the angle to determine the sign of trigonometric functions (positive or negative).

1 + cos(90°) + tan(45°) + sin(315°) ÷ csc(150°)

Solve the trigonometric expression: 1 + cos(90°) + tan(45°) + sin(315°) ÷ csc(150°) using key trigonometric identities. This problem involves cosine, tangent, sine, and cosecant functions with angles in degrees. Suitable for students in Grades 10-12.

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