Math Snap

STEP 1

1. We need to find the six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent.
2. The angle given is $585^\circ$.
3. Trigonometric functions are periodic, and we can use this property to simplify calculations.

STEP 2

1. Reduce the angle $585^\circ$ to an equivalent angle between $0^\circ$ and $360^\circ$.
2. Determine the reference angle and quadrant for the equivalent angle.
3. Calculate the sine and cosine of the equivalent angle.
4. Use sine and cosine to find the remaining trigonometric ratios.

STEP 3

Reduce $585^\circ$ by subtracting multiples of $360^\circ$ to find an equivalent angle:
$$585^\circ - 360^\circ = 225^\circ$$ The equivalent angle is $225^\circ$.

STEP 4

Determine the reference angle and quadrant for $225^\circ$:
- $225^\circ$ is in the third quadrant.
- The reference angle is $225^\circ - 180^\circ = 45^\circ$.

STEP 5

Calculate the sine and cosine of $225^\circ$ using the reference angle $45^\circ$:
- In the third quadrant, both sine and cosine are negative.
- $\sin(225^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$
- $\cos(225^\circ) = -\cos(45^\circ) = -\frac{\sqrt{2}}{2}$

Calculate the remaining trigonometric ratios:
- $\tan(225^\circ) = \frac{\sin(225^\circ)}{\cos(225^\circ)} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$
- $\csc(225^\circ) = \frac{1}{\sin(225^\circ)} = -\sqrt{2}$
- $\sec(225^\circ) = \frac{1}{\cos(225^\circ)} = -\sqrt{2}$
- $\cot(225^\circ) = \frac{1}{\tan(225^\circ)} = 1$
The exact values of the six trigonometric ratios for $585^\circ$ are:
$$\sin(585^\circ) = -\frac{\sqrt{2}}{2}, \quad \cos(585^\circ) = -\frac{\sqrt{2}}{2}, \quad \tan(585^\circ) = 1$$ $$\csc(585^\circ) = -\sqrt{2}, \quad \sec(585^\circ) = -\sqrt{2}, \quad \cot(585^\circ) = 1$$