To find the six trigonometric (circular) functions of $240^\circ$, let's start by noting that $240^\circ$ is in the third quadrant, where both sine and cosine are negative.

Step-by-Step Solution:

1. Determine the reference angle:

   • The reference angle for $240^\circ$ is $240^\circ - 180^\circ = 60^\circ$.

2. Evaluate the trigonometric functions:

   • Sine and cosine are negative in the third quadrant.
   • Tangent and cotangent are positive in the third quadrant.

Using these values for $60^\circ$: $\sin(60^\circ) = \frac{\sqrt{3}}{2}, \quad \cos(60^\circ) = \frac{1}{2}, \quad \tan(60^\circ) = \sqrt{3}$

Calculating the Six Functions:

1. Sine: $\sin(240^\circ) = -\sin(60^\circ) = -\frac{\sqrt{3}}{2}$

2. Cosine: $\cos(240^\circ) = -\cos(60^\circ) = -\frac{1}{2}$

3. Tangent: $\tan(240^\circ) = \frac{\sin(240^\circ)}{\cos(240^\circ)} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}$

4. Cosecant (reciprocal of sine): $\csc(240^\circ) = \frac{1}{\sin(240^\circ)} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$

5. Secant (reciprocal of cosine): $\sec(240^\circ) = \frac{1}{\cos(240^\circ)} = \frac{1}{-\frac{1}{2}} = -2$

6. Cotangent (reciprocal of tangent): $\cot(240^\circ) = \frac{1}{\tan(240^\circ)} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Final Answer

The six trigonometric functions of $240^\circ$ are:

• $\sin(240^\circ) = -\frac{\sqrt{3}}{2}$
• $\cos(240^\circ) = -\frac{1}{2}$
• $\tan(240^\circ) = \sqrt{3}$
• $\csc(240^\circ) = -\frac{2\sqrt{3}}{3}$
• $\sec(240^\circ) = -2$
• $\cot(240^\circ) = \frac{\sqrt{3}}{3}$

Would you like further explanation on any of these steps?

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Related Questions:

1. What are the six trigonometric functions of $300^\circ$?
2. How do the signs of trigonometric functions vary in each quadrant?
3. How can you find the reference angle for any given angle?
4. What are the values of the trigonometric functions at $120^\circ$?
5. Why are tangent and cotangent positive in the third quadrant?

Tip:

Always remember that the sign of trigonometric functions depends on the quadrant of the angle, while the reference angle helps find the values.