To find the six circular (or trigonometric) functions of $-405^\circ$, let's start by simplifying the angle to an equivalent one within the standard range $0^\circ \leq \theta < 360^\circ$.

Step 1: Simplify $-405^\circ$

1. Since $-405^\circ$ is a negative angle, add $360^\circ$ repeatedly until the angle falls within the range of $0^\circ$ to $360^\circ$.

$-405^\circ + 360^\circ = -45^\circ$ $-45^\circ + 360^\circ = 315^\circ$

Thus, $-405^\circ$ is equivalent to $315^\circ$.

Step 2: Determine the Six Trigonometric Functions of $315^\circ$

Since $315^\circ$ is in the fourth quadrant, we know:

• Sine is negative
• Cosine is positive
• Tangent is negative

Now, using $315^\circ = 360^\circ - 45^\circ$, we find:

1. Sine ($\sin(315^\circ)$): $\sin(315^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$

2. Cosine ($\cos(315^\circ)$): $\cos(315^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$

3. Tangent ($\tan(315^\circ)$): $\tan(315^\circ) = -\tan(45^\circ) = -1$

4. Cosecant ($\csc(315^\circ)$): $\csc(315^\circ) = \frac{1}{\sin(315^\circ)} = -\frac{1}{\frac{\sqrt{2}}{2}} = -\sqrt{2}$

5. Secant ($\sec(315^\circ)$): $\sec(315^\circ) = \frac{1}{\cos(315^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

6. Cotangent ($\cot(315^\circ)$): $\cot(315^\circ) = \frac{1}{\tan(315^\circ)} = -1$

Summary of the Six Trigonometric Functions of $-405^\circ$ (or $315^\circ$):

• $\sin(-405^\circ) = -\frac{\sqrt{2}}{2}$
• $\cos(-405^\circ) = \frac{\sqrt{2}}{2}$
• $\tan(-405^\circ) = -1$
• $\csc(-405^\circ) = -\sqrt{2}$
• $\sec(-405^\circ) = \sqrt{2}$
• $\cot(-405^\circ) = -1$

Would you like a deeper breakdown or clarification on any of these functions?

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1. What are the trigonometric functions of $-45^\circ$ and why are they equivalent to those of $315^\circ$?
2. How can trigonometric identities be applied to angles larger than $360^\circ$ or smaller than $0^\circ$?
3. What is the significance of the quadrant in determining the sign of trigonometric functions?
4. How can we verify these values on the unit circle?
5. Why is it useful to reduce angles to within the $0^\circ$ to $360^\circ$ range?

Tip: For angles greater than $360^\circ$ or less than $0^\circ$, use angle reduction by adding or subtracting $360^\circ$ until within the standard range for quicker function evaluation.