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Chapter 5: Problem 8

Use the given values to find the values (if possible) of all six trigonometric functions. $$\csc \theta=\frac{25}{7}, \tan \theta=\frac{7}{24}$$

Short Answer

Expert verified
\(\sin \theta = \frac{7}{25}\), \(\cos \theta = \frac{24}{25}\), \(\tan \theta = \frac{7}{24}\), \(\csc \theta = \frac{25}{7}\), \(\sec \theta = \frac{25}{24}\), \(\cot \theta = \frac{24}{7}\)

Step by step solution

01

Calculating Sin 胃

We start the problem by finding the value of \(\sin \theta\). Since \(\sin \theta\) is the reciprocal of \(\csc \theta\), we find \(\sin \theta = \frac{1}{\csc \theta} =\frac{1}{\frac{25}{7}} = \frac{7}{25}\).
02

Calculating Cos 胃

Next, we find the value of \(\cos \theta\). We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Substituting the value of \(\sin \theta\) we obtained, we get \(\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{7}{25}\right)^2\). Taking the square root of each side to solve for \(\cos \theta\) gives us two possible solutions: \(\cos \theta = \frac{24}{25}\) or \(\cos \theta = -\frac{24}{25}\). Since we know that \(\tan \theta = \frac{7}{24} > 0\) and that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we must have \(\cos \theta = \frac{24}{25}\) since the signs of sine and cosine must be the same.
03

Calculating other trigonometric functions

Now that we have \(\sin \theta = \frac{7}{25}\) and \(\cos \theta = \frac{24}{25}\), we can easily find the other trigonometric functions. The cosecant \( \csc \theta = \frac{1}{\sin \theta} = \frac{25}{7}\) is the given one. The secant \(\sec \theta = \frac{1}{\cos \theta} = \frac{25}{24}\), and the cotangent \(\cot \theta = \frac{1}{\tan \theta} = \frac{24}{7}\).

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