If Tan ⁡ Θ = − 4 3 \tan \theta = -\frac{4}{3} Tan Θ = − 3 4 ​ And Sin ⁡ Θ \textless 0 \sin \theta \ \textless \ 0 Sin Θ \textless 0 , Find Csc ⁡ Θ \csc \theta Csc Θ .

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If tan⁡θ=−43\tan \theta = -\frac{4}{3} and sin⁡θ \textless 0\sin \theta \ \textless \ 0, find csc⁡θ\csc \theta

In trigonometry, the tangent function is defined as the ratio of the sine and cosine functions. Given the value of the tangent function, we can use trigonometric identities to find the values of other trigonometric functions. In this article, we will use the given value of the tangent function to find the value of the cosecant function.

Understanding the Given Information

We are given that tan⁡θ=−43\tan \theta = -\frac{4}{3} and sin⁡θ \textless 0\sin \theta \ \textless \ 0. The negative value of the sine function indicates that the angle θ\theta is in the third or fourth quadrant. Since the tangent function is negative, the angle θ\theta must be in the second or third quadrant.

Recalling Trigonometric Identities

To find the value of the cosecant function, we need to recall the trigonometric identity that relates the tangent and cosecant functions. The identity is:

tan⁡θ=sin⁡θcos⁡θ=1csc⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{1}{\csc \theta}

Finding the Value of the Cosecant Function

We can rearrange the identity to solve for the cosecant function:

csc⁡θ=1tan⁡θ\csc \theta = \frac{1}{\tan \theta}

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

csc⁡θ=1−43=−34\csc \theta = \frac{1}{-\frac{4}{3}} = -\frac{3}{4}

In this article, we used the given value of the tangent function to find the value of the cosecant function. We recalled the trigonometric identity that relates the tangent and cosecant functions and rearranged it to solve for the cosecant function. The final answer is −34\boxed{-\frac{3}{4}}.

  • The value of the cosecant function is negative because the angle θ\theta is in the third or fourth quadrant.
  • The value of the cosecant function is reciprocal of the value of the tangent function.
  • The trigonometric identity that relates the tangent and cosecant functions is a fundamental concept in trigonometry.

The value of the cosecant function has real-world applications in various fields such as:

  • Physics: The cosecant function is used to describe the motion of objects in circular motion.
  • Engineering: The cosecant function is used to design and analyze electrical circuits.
  • Navigation: The cosecant function is used to determine the position and orientation of objects in navigation systems.
  • Incorrectly assuming the quadrant of the angle: The quadrant of the angle must be determined based on the given information.
  • Not recalling the trigonometric identity: The trigonometric identity that relates the tangent and cosecant functions must be recalled to solve the problem.
  • Not checking the units: The units of the answer must be checked to ensure that they are correct.
  • Use the trigonometric identity to solve the problem: The trigonometric identity that relates the tangent and cosecant functions can be used to solve the problem.
  • Check the units: The units of the answer must be checked to ensure that they are correct.
  • Use a calculator to check the answer: A calculator can be used to check the answer and ensure that it is correct.
    Q&A: If tan⁡θ=−43\tan \theta = -\frac{4}{3} and sin⁡θ \textless 0\sin \theta \ \textless \ 0, find csc⁡θ\csc \theta

Q: What is the value of the cosecant function if tan⁡θ=−43\tan \theta = -\frac{4}{3} and sin⁡θ \textless 0\sin \theta \ \textless \ 0?

A: The value of the cosecant function is −34\boxed{-\frac{3}{4}}.

Q: Why is the value of the cosecant function negative?

A: The value of the cosecant function is negative because the angle θ\theta is in the third or fourth quadrant.

Q: How do I recall the trigonometric identity that relates the tangent and cosecant functions?

A: The trigonometric identity that relates the tangent and cosecant functions is:

tan⁡θ=sin⁡θcos⁡θ=1csc⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{1}{\csc \theta}

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

A: The tangent and cosecant functions are reciprocal functions. The value of the cosecant function is the reciprocal of the value of the tangent function.

Q: How do I determine the quadrant of the angle?

A: The quadrant of the angle must be determined based on the given information. In this case, the angle θ\theta is in the third or fourth quadrant because the sine function is negative.

Q: What are some real-world applications of the cosecant function?

A: The cosecant function has real-world applications in various fields such as:

  • Physics: The cosecant function is used to describe the motion of objects in circular motion.
  • Engineering: The cosecant function is used to design and analyze electrical circuits.
  • Navigation: The cosecant function is used to determine the position and orientation of objects in navigation systems.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

  • Incorrectly assuming the quadrant of the angle: The quadrant of the angle must be determined based on the given information.
  • Not recalling the trigonometric identity: The trigonometric identity that relates the tangent and cosecant functions must be recalled to solve the problem.
  • Not checking the units: The units of the answer must be checked to ensure that they are correct.

Q: How can I check my answer?

A: A calculator can be used to check the answer and ensure that it is correct.

Q: What are some tips and tricks for solving this problem?

A: Some tips and tricks for solving this problem include:

  • Use the trigonometric identity to solve the problem: The trigonometric identity that relates the tangent and cosecant functions can be used to solve the problem.
  • Check the units: The units of the answer must be checked to ensure that they are correct.
  • Use a calculator to check the answer: A calculator can be used to check the answer and ensure that it is correct.

In this article, we have answered some frequently asked questions about the problem of finding the value of the cosecant function if tan⁡θ=−43\tan \theta = -\frac{4}{3} and sin⁡θ \textless 0\sin \theta \ \textless \ 0. We have provided explanations and examples to help clarify the concepts and provide a better understanding of the problem.