Given That $\operatorname{Csc} \theta = \frac{3}{2}$ And $\frac{\pi}{2} \leqslant \theta \leqslant \pi$, Find All Five Other Trigonometric Functions.

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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 values of the five other trigonometric functions given the value of the cosecant function.

Recalling the Definitions

Before we dive into the problem, let's recall the definitions of the six trigonometric functions:

  • Sine (sin): the ratio of the length of the opposite side to the length of the hypotenuse
  • Cosine (cos): the ratio of the length of the adjacent side to the length of the hypotenuse
  • Tangent (tan): the ratio of the length of the opposite side to the length of the adjacent side
  • Cosecant (csc): the ratio of the length of the hypotenuse to the length of the opposite side
  • Secant (sec): the ratio of the length of the hypotenuse to the length of the adjacent side
  • Cotangent (cot): the ratio of the length of the adjacent side to the length of the opposite side

The Problem

We are given that Csc⁡θ=32\operatorname{Csc} \theta = \frac{3}{2} and π2⩽θ⩽π\frac{\pi}{2} \leqslant \theta \leqslant \pi. Our goal is to find the values of the other five trigonometric functions.

Using the Pythagorean Identity

We can use the Pythagorean identity to find the value of the sine function. The Pythagorean identity states that:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Since we are given the value of the cosecant function, we can rewrite the Pythagorean identity as:

(1Csc⁡θ)2+cos⁡2θ=1\left(\frac{1}{\operatorname{Csc} \theta}\right)^2 + \cos^2 \theta = 1

Substituting the given value of the cosecant function, we get:

(132)2+cos⁡2θ=1\left(\frac{1}{\frac{3}{2}}\right)^2 + \cos^2 \theta = 1

Simplifying the expression, we get:

49+cos⁡2θ=1\frac{4}{9} + \cos^2 \theta = 1

Subtracting 49\frac{4}{9} from both sides, we get:

cos⁡2θ=59\cos^2 \theta = \frac{5}{9}

Taking the square root of both sides, we get:

cos⁡θ=±53\cos \theta = \pm \frac{\sqrt{5}}{3}

Since π2⩽θ⩽π\frac{\pi}{2} \leqslant \theta \leqslant \pi, we know that cos⁡θ\cos \theta is negative. Therefore, we can write:

cos⁡θ=−53\cos \theta = -\frac{\sqrt{5}}{3}

Finding the Sine Function

Now that we have the value of the cosine function, we can use the Pythagorean identity to find the value of the sine function. We can rewrite the Pythagorean identity as:

sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta

Substituting the value of the cosine function, we get:

sin⁡2θ=1−(−53)2\sin^2 \theta = 1 - \left(-\frac{\sqrt{5}}{3}\right)^2

Simplifying the expression, we get:

sin⁡2θ=1−59\sin^2 \theta = 1 - \frac{5}{9}

Subtracting 59\frac{5}{9} from 11, we get:

sin⁡2θ=49\sin^2 \theta = \frac{4}{9}

Taking the square root of both sides, we get:

sin⁡θ=±23\sin \theta = \pm \frac{2}{3}

Since π2⩽θ⩽π\frac{\pi}{2} \leqslant \theta \leqslant \pi, we know that sin⁡θ\sin \theta is negative. Therefore, we can write:

sin⁡θ=−23\sin \theta = -\frac{2}{3}

Finding the Tangent Function

Now that we have the values of the sine and cosine functions, we can use the definition of the tangent function to find its value. The tangent function is defined as:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Substituting the values of the sine and cosine functions, we get:

tan⁡θ=−23−53\tan \theta = \frac{-\frac{2}{3}}{-\frac{\sqrt{5}}{3}}

Simplifying the expression, we get:

tan⁡θ=25\tan \theta = \frac{2}{\sqrt{5}}

Rationalizing the denominator, we get:

tan⁡θ=255\tan \theta = \frac{2\sqrt{5}}{5}

Finding the Secant Function

Now that we have the value of the cosine function, we can use the definition of the secant function to find its value. The secant function is defined as:

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

Substituting the value of the cosine function, we get:

sec⁡θ=1−53\sec \theta = \frac{1}{-\frac{\sqrt{5}}{3}}

Simplifying the expression, we get:

sec⁡θ=−35\sec \theta = -\frac{3}{\sqrt{5}}

Rationalizing the denominator, we get:

sec⁡θ=−355\sec \theta = -\frac{3\sqrt{5}}{5}

Finding the Cotangent Function

Now that we have the values of the sine and cosine functions, we can use the definition of the cotangent function to find its value. The cotangent function is defined as:

cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}

Substituting the values of the sine and cosine functions, we get:

cot⁡θ=−53−23\cot \theta = \frac{-\frac{\sqrt{5}}{3}}{-\frac{2}{3}}

Simplifying the expression, we get:

cot⁡θ=52\cot \theta = \frac{\sqrt{5}}{2}

Conclusion

In this article, we have found the values of the five other trigonometric functions given the value of the cosecant function. We used the Pythagorean identity to find the value of the sine function, and then used the definitions of the tangent, secant, and cotangent functions to find their values. We have shown that:

  • sin⁡θ=−23\sin \theta = -\frac{2}{3}
  • cos⁡θ=−53\cos \theta = -\frac{\sqrt{5}}{3}
  • tan⁡θ=255\tan \theta = \frac{2\sqrt{5}}{5}
  • sec⁡θ=−355\sec \theta = -\frac{3\sqrt{5}}{5}
  • cot⁡θ=52\cot \theta = \frac{\sqrt{5}}{2}

Introduction

In our previous article, we explored the world of trigonometric functions and found the values of the five other trigonometric functions given the value of the cosecant function. In this article, we will answer some of the most frequently asked questions about trigonometric functions.

Q: What is the difference between sine and cosine?

A: The sine and cosine functions are two of the six trigonometric functions. The sine function is the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function is the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I remember the order of the trigonometric functions?

A: One way to remember the order of the trigonometric functions is to use the acronym "SOH-CAH-TOA". This stands for:

  • Sine = Opposite over Hypotenuse
  • Cosine = Adjacent over Hypotenuse
  • Tangent = Opposite over Adjacent
  • Cosecant = Hypotenuse over Opposite
  • Secant = Hypotenuse over Adjacent
  • Cotangent = Adjacent over Opposite

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

This identity can be used to find the value of one trigonometric function given the value of another.

Q: How do I use the Pythagorean identity to find the value of a trigonometric function?

A: To use the Pythagorean identity to find the value of a trigonometric function, follow these steps:

  1. Write down the Pythagorean identity.
  2. Substitute the value of the known trigonometric function into the identity.
  3. Simplify the expression to find the value of the unknown trigonometric function.

Q: What is the difference between the tangent and cotangent functions?

A: The tangent and cotangent functions are two of the six trigonometric functions. The tangent function is the ratio of the length of the opposite side to the length of the adjacent side, while the cotangent function is the ratio of the length of the adjacent side to the length of the opposite side.

Q: How do I use the definitions of the trigonometric functions to find their values?

A: To use the definitions of the trigonometric functions to find their values, follow these steps:

  1. Write down the definition of the trigonometric function.
  2. Substitute the values of the known trigonometric functions into the definition.
  3. Simplify the expression to find the value of the unknown trigonometric function.

Q: What are some common applications of trigonometric functions?

A: Trigonometric functions have numerous applications in various fields, including:

  • Physics: Trigonometric functions are used to describe the motion of objects in terms of their position, velocity, and acceleration.
  • Engineering: Trigonometric functions are used to design and analyze the performance of mechanical systems, such as gears and pulleys.
  • Navigation: Trigonometric functions are used to determine the position and orientation of objects in space, such as the location of a ship or aircraft.
  • Computer Science: Trigonometric functions are used in computer graphics and game development to create realistic 3D models and animations.

Conclusion

In this article, we have answered some of the most frequently asked questions about trigonometric functions. We have covered topics such as the difference between sine and cosine, the Pythagorean identity, and the definitions of the trigonometric functions. We have also discussed some common applications of trigonometric functions in various fields.