If $\sec \alpha = -\frac{9}{7}$ And $\tan \alpha = -\frac{4 \sqrt{2}}{7}$, Find $\csc \alpha$.

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If sec⁡α=−97\sec \alpha = -\frac{9}{7} and tan⁡α=−427\tan \alpha = -\frac{4 \sqrt{2}}{7}, find csc⁡α\csc \alpha

In trigonometry, the secant, tangent, and cosecant functions are reciprocal functions of the sine, cosine, and tangent functions, respectively. Given the values of sec⁡α\sec \alpha and tan⁡α\tan \alpha, we can find the value of csc⁡α\csc \alpha using the relationships between these trigonometric functions.

Reciprocal Trigonometric Functions

The reciprocal trigonometric functions are defined as follows:

  • sec⁡α=1cos⁡α\sec \alpha = \frac{1}{\cos \alpha}
  • csc⁡α=1sin⁡α\csc \alpha = \frac{1}{\sin \alpha}
  • tan⁡α=sin⁡αcos⁡α\tan \alpha = \frac{\sin \alpha}{\cos \alpha}

We can use these relationships to find the value of csc⁡α\csc \alpha.

Finding csc⁡α\csc \alpha

We are given that sec⁡α=−97\sec \alpha = -\frac{9}{7} and tan⁡α=−427\tan \alpha = -\frac{4 \sqrt{2}}{7}. We can use these values to find the value of csc⁡α\csc \alpha.

First, we can find the value of cos⁡α\cos \alpha using the definition of the secant function:

cos⁡α=1sec⁡α=1−97=−79\cos \alpha = \frac{1}{\sec \alpha} = \frac{1}{-\frac{9}{7}} = -\frac{7}{9}

Next, we can find the value of sin⁡α\sin \alpha using the definition of the tangent function:

sin⁡α=tan⁡αcos⁡α=(−427)(−79)=28263\sin \alpha = \tan \alpha \cos \alpha = \left(-\frac{4 \sqrt{2}}{7}\right)\left(-\frac{7}{9}\right) = \frac{28 \sqrt{2}}{63}

Now, we can find the value of csc⁡α\csc \alpha using the definition of the cosecant function:

csc⁡α=1sin⁡α=128263=63282\csc \alpha = \frac{1}{\sin \alpha} = \frac{1}{\frac{28 \sqrt{2}}{63}} = \frac{63}{28 \sqrt{2}}

To simplify this expression, we can rationalize the denominator by multiplying the numerator and denominator by 2\sqrt{2}:

csc⁡α=63282⋅22=63256\csc \alpha = \frac{63}{28 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{63 \sqrt{2}}{56}

Therefore, the value of csc⁡α\csc \alpha is 63256\frac{63 \sqrt{2}}{56}.

In this article, we used the reciprocal trigonometric functions to find the value of csc⁡α\csc \alpha given the values of sec⁡α\sec \alpha and tan⁡α\tan \alpha. We first found the value of cos⁡α\cos \alpha using the definition of the secant function, then found the value of sin⁡α\sin \alpha using the definition of the tangent function. Finally, we found the value of csc⁡α\csc \alpha using the definition of the cosecant function. The value of csc⁡α\csc \alpha is 63256\frac{63 \sqrt{2}}{56}.

  • [1] "Trigonometry" by Michael Corral, 2018.
  • [2] "Precalculus" by James Stewart, 2019.
  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: 18.01 Single Variable Calculus
  • Wolfram Alpha: Trigonometry

In our previous article, we discussed how to find the value of csc⁡α\csc \alpha given the values of sec⁡α\sec \alpha and tan⁡α\tan \alpha. In this article, we will answer some common questions related to trigonometry and reciprocal functions.

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

A: The secant, tangent, and cosecant functions are reciprocal functions of the sine, cosine, and tangent functions, respectively. This means that:

  • sec⁡α=1cos⁡α\sec \alpha = \frac{1}{\cos \alpha}
  • csc⁡α=1sin⁡α\csc \alpha = \frac{1}{\sin \alpha}
  • tan⁡α=sin⁡αcos⁡α\tan \alpha = \frac{\sin \alpha}{\cos \alpha}

Q: How do I find the value of csc⁡α\csc \alpha given the values of sec⁡α\sec \alpha and tan⁡α\tan \alpha?

A: To find the value of csc⁡α\csc \alpha, you can follow these steps:

  1. Find the value of cos⁡α\cos \alpha using the definition of the secant function: cos⁡α=1sec⁡α\cos \alpha = \frac{1}{\sec \alpha}.
  2. Find the value of sin⁡α\sin \alpha using the definition of the tangent function: sin⁡α=tan⁡αcos⁡α\sin \alpha = \tan \alpha \cos \alpha.
  3. Find the value of csc⁡α\csc \alpha using the definition of the cosecant function: csc⁡α=1sin⁡α\csc \alpha = \frac{1}{\sin \alpha}.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that describe the relationship between the angles and side lengths of a right triangle. However, the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, while the cosine function is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Q: How do I use the reciprocal trigonometric functions to solve problems?

A: The reciprocal trigonometric functions can be used to solve problems in a variety of ways. For example, you can use the reciprocal functions to find the value of a trigonometric function given the values of other trigonometric functions. You can also use the reciprocal functions to solve equations involving trigonometric functions.

Q: What are some common applications of trigonometry and reciprocal functions?

A: Trigonometry and reciprocal functions have a wide range of applications in fields such as physics, engineering, and computer science. Some common applications include:

  • Calculating distances and angles in right triangles
  • Modeling periodic phenomena such as sound waves and light waves
  • Solving equations involving trigonometric functions
  • Analyzing and visualizing data using trigonometric functions

In this article, we answered some common questions related to trigonometry and reciprocal functions. We discussed the relationship between the secant, tangent, and cosecant functions, and provided step-by-step instructions for finding the value of csc⁡α\csc \alpha given the values of sec⁡α\sec \alpha and tan⁡α\tan \alpha. We also discussed the difference between the sine and cosine functions, and provided examples of how to use the reciprocal trigonometric functions to solve problems.

  • [1] "Trigonometry" by Michael Corral, 2018.
  • [2] "Precalculus" by James Stewart, 2019.
  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: 18.01 Single Variable Calculus
  • Wolfram Alpha: Trigonometry