Given:$ \cos \theta = -\frac{4}{5}, \quad \theta \text{ Is In Quadrant II} }$1. Find { \sec \theta$}$ ${ \sec \theta = -\frac{5 {4} }$2. Find { \csc \theta$}$: $[ \csc \theta = \square

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on finding the secant and cosecant of an angle in Quadrant II, given the cosine of the angle.

Understanding Quadrant II

Quadrant II is one of the four quadrants in the Cartesian coordinate system. It is located in the upper-left region of the coordinate plane, where both the x-coordinate and the y-coordinate are positive. In trigonometry, Quadrant II is associated with angles that lie between 90° and 180°.

Recalling Trigonometric Identities

Before we proceed, let's recall some essential trigonometric identities:

  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

These identities will be useful in finding the secant and cosecant of an angle in Quadrant II.

Finding Secant

Given that cos⁡θ=−45\cos \theta = -\frac{4}{5} and θ\theta is in Quadrant II, we can use the identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} to find the secant of the angle.

$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$

Therefore, the secant of the angle is −54-\frac{5}{4}.

Finding Cosecant

To find the cosecant of the angle, we need to find the sine of the angle first. Since θ\theta is in Quadrant II, we can use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find the sine of the angle.

$\sin^2 \theta + \cos^2 \theta = 1$
$\sin^2 \theta + \left(-\frac{4}{5}\right)^2 = 1$
$\sin^2 \theta + \frac{16}{25} = 1$
$\sin^2 \theta = 1 - \frac{16}{25}$
$\sin^2 \theta = \frac{9}{25}$
$\sin \theta = \pm \sqrt{\frac{9}{25}}$
$\sin \theta = \pm \frac{3}{5}$

Since θ\theta is in Quadrant II, the sine of the angle is positive. Therefore, sin⁡θ=35\sin \theta = \frac{3}{5}.

Now that we have the sine of the angle, we can use the identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to find the cosecant of the angle.

$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$

Therefore, the cosecant of the angle is 53\frac{5}{3}.

Conclusion

In this article, we have found the secant and cosecant of an angle in Quadrant II, given the cosine of the angle. We have used trigonometric identities and the Pythagorean identity to find the secant and cosecant of the angle. The secant of the angle is −54-\frac{5}{4}, and the cosecant of the angle is 53\frac{5}{3}.

Discussion

The secant and cosecant functions are reciprocal functions of the cosine and sine functions, respectively. They are used to find the length of the hypotenuse of a right triangle and the length of the opposite side of an angle, respectively. In this article, we have seen how to find the secant and cosecant of an angle in Quadrant II, given the cosine of the angle.

References

  • "Trigonometry" by Michael Corral
  • "Precalculus" by James Stewart

Further Reading

  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Precalculus: Mathematics for Calculus" by James Stewart

Glossary

  • Secant: The reciprocal of the cosine function.
  • Cosecant: The reciprocal of the sine function.
  • Quadrant II: The upper-left region of the Cartesian coordinate system.
  • Pythagorean identity: The identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.

FAQs

  • Q: What is the secant of an angle in Quadrant II? A: The secant of an angle in Quadrant II is the reciprocal of the cosine of the angle.
  • Q: What is the cosecant of an angle in Quadrant II? A: The cosecant of an angle in Quadrant II is the reciprocal of the sine of the angle.
  • Q: How do I find the secant and cosecant of an angle in Quadrant II? A: To find the secant and cosecant of an angle in Quadrant II, use the identities sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, respectively.
    Trigonometric Functions: Q&A ==============================

Q: What is the difference between the secant and cosecant functions?

A: The secant function is the reciprocal of the cosine function, while the cosecant function is the reciprocal of the sine function.

Q: How do I find the secant of an angle in Quadrant II?

A: To find the secant of an angle in Quadrant II, use the identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}, where cos⁡θ\cos \theta is the cosine of the angle.

Q: How do I find the cosecant of an angle in Quadrant II?

A: To find the cosecant of an angle in Quadrant II, use the identity csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}, where sin⁡θ\sin \theta is the sine of the angle.

Q: What is the relationship between the secant and cosecant functions?

A: The secant and cosecant functions are reciprocal functions, meaning that they are related by the identity sec⁡θcsc⁡θ=1\sec \theta \csc \theta = 1.

Q: Can I use the Pythagorean identity to find the secant and cosecant of an angle?

A: Yes, you can use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find the secant and cosecant of an angle. However, you will need to use the identity sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to find the secant and cosecant of the angle.

Q: How do I find the secant and cosecant of an angle in Quadrant III?

A: To find the secant and cosecant of an angle in Quadrant III, use the identities sec⁡θ=−1cos⁡θ\sec \theta = -\frac{1}{\cos \theta} and csc⁡θ=−1sin⁡θ\csc \theta = -\frac{1}{\sin \theta}, respectively.

Q: Can I use a calculator to find the secant and cosecant of an angle?

A: Yes, you can use a calculator to find the secant and cosecant of an angle. However, you will need to enter the angle in radians or degrees, depending on the calculator you are using.

Q: What is the range of the secant and cosecant functions?

A: The range of the secant function is all real numbers, while the range of the cosecant function is all real numbers except for zero.

Q: Can I use the secant and cosecant functions to solve trigonometric equations?

A: Yes, you can use the secant and cosecant functions to solve trigonometric equations. However, you will need to use the identities sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to rewrite the equation in terms of the cosine and sine functions.

Q: How do I graph the secant and cosecant functions?

A: To graph the secant and cosecant functions, use a graphing calculator or a graphing software. You can also use the identities sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to rewrite the functions in terms of the cosine and sine functions.

Q: Can I use the secant and cosecant functions to model real-world problems?

A: Yes, you can use the secant and cosecant functions to model real-world problems. For example, you can use the secant function to model the height of a projectile, while you can use the cosecant function to model the length of a shadow.

Q: What are some common applications of the secant and cosecant functions?

A: Some common applications of the secant and cosecant functions include:

  • Modeling the height of a projectile
  • Modeling the length of a shadow
  • Modeling the distance between two points on a circle
  • Modeling the angle between two lines

Q: Can I use the secant and cosecant functions to solve optimization problems?

A: Yes, you can use the secant and cosecant functions to solve optimization problems. For example, you can use the secant function to maximize the height of a projectile, while you can use the cosecant function to minimize the length of a shadow.

Q: How do I use the secant and cosecant functions to solve optimization problems?

A: To use the secant and cosecant functions to solve optimization problems, you will need to use the identities sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to rewrite the problem in terms of the cosine and sine functions. You can then use calculus to find the maximum or minimum value of the function.

Q: Can I use the secant and cosecant functions to solve differential equations?

A: Yes, you can use the secant and cosecant functions to solve differential equations. For example, you can use the secant function to solve the differential equation dydx=sec⁡x\frac{dy}{dx} = \sec x, while you can use the cosecant function to solve the differential equation dydx=csc⁡x\frac{dy}{dx} = \csc x.

Q: How do I use the secant and cosecant functions to solve differential equations?

A: To use the secant and cosecant functions to solve differential equations, you will need to use the identities sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} and csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} to rewrite the differential equation in terms of the cosine and sine functions. You can then use calculus to find the solution to the differential equation.