Evaluate the following expression by drawing the unit circle and the appropriate right triangle. 

secant left parenthesis StartFraction negative 5 pi Over 6 EndFraction right parenthesissec−5π6

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Part 1

secant left parenthesis StartFraction negative 5 pi Over 6 EndFraction right parenthesissec−5π6equals=enter your response here

​(Simplify your​ answer, including any radicals. Use integers or fractions for any numbers in the​ expression.)

To evaluate \(\sec\left(-\frac{5\pi}{6}\right)\), let's follow the steps:

### 1. Draw the Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. The angle \(-\frac{5\pi}{6}\) is measured clockwise from the positive x-axis because it's a negative angle.

### 2. Locate the Angle \(-\frac{5\pi}{6}\) on the Unit Circle
- The angle \(\frac{5\pi}{6}\) corresponds to 150° in degrees. Moving clockwise, \(-\frac{5\pi}{6}\) is located in the third quadrant of the unit circle.
- The reference angle for \(-\frac{5\pi}{6}\) is \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\), or 30°.

### 3. Determine the Coordinates of the Corresponding Point on the Unit Circle
- In the third quadrant, the cosine value is negative.
- For the reference angle of \(\frac{\pi}{6}\) (or 30°), the coordinates on the unit circle are:
  \[
  \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)
  \]

### 4. Calculate the Secant
- The secant function is the reciprocal of the cosine function:
  \[
  \sec\left(-\frac{5\pi}{6}\right) = \frac{1}{\cos\left(-\frac{5\pi}{6}\right)}
  \]
- The cosine of \(-\frac{5\pi}{6}\) is the x-coordinate, \(\cos\left(-\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}\).
  \[
  \sec\left(-\frac{5\pi}{6}\right) = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}}
  \]

### 5. Simplify the Answer
- Simplify the expression by rationalizing the denominator:
  \[
  \sec\left(-\frac{5\pi}{6}\right) = -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}
  \]

### Final Answer:
\[
\sec\left(-\frac{5\pi}{6}\right) = -\frac{2\sqrt{3}}{3}
\]

Would you like further details or have any questions?

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Here are five related questions to explore this topic further:
1. What is the value of \(\sec\left(\frac{5\pi}{6}\right)\)?
2. How do you determine the quadrant in which an angle lies on the unit circle?
3. What is the significance of reference angles in trigonometry?
4. Can you explain why secant is the reciprocal of cosine?
5. How does rationalizing the denominator affect the value of an expression?

**Tip:** When working with trigonometric functions, always consider the sign of the function based on the quadrant in which the angle lies.

Evaluate the following expression by drawing the unit circle and the appropriate right triangle: sec(−5π/6). Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.

Learn how to evaluate the secant of −5π/6 by drawing the unit circle and using the appropriate right triangle. This detailed solution simplifies the expression and includes key trigonometric concepts, suitable for high school students in Grades 10-12.

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