Questions: Find the exact value of csc[2 tan^(-1)(-3/4)].

Solution Steps

To find the exact value of \(\csc \left[2 \tan ^{-1}\left(-\frac{3}{4}\right)\right]\), we can use trigonometric identities. First, let \(\theta = \tan^{-1}\left(-\frac{3}{4}\right)\). Then, we need to find \(\csc(2\theta)\). We can use the double angle identity for sine: \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\). We can find \(\sin(\theta)\) and \(\cos(\theta)\) using the right triangle definition based on \(\tan(\theta) = -\frac{3}{4}\). Finally, \(\csc(2\theta) = \frac{1}{\sin(2\theta)}\).

Let \(\theta = \tan^{-1}\left(-\frac{3}{4}\right)\). This means that \(\tan(\theta) = -\frac{3}{4}\).

Using the definition of tangent, we can represent \(\theta\) in a right triangle where the opposite side is \(-3\) and the adjacent side is \(4\). The hypotenuse can be calculated as: \[ \text{hypotenuse} = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 \] Thus, we have: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-3}{5} = -0.6 \] \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} = 0.8 \]

Using the double angle identity for sine: \[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2 \left(-\frac{3}{5}\right) \left(\frac{4}{5}\right) = -\frac{24}{25} = -0.96 \]

The cosecant function is the reciprocal of the sine function: \[ \csc(2\theta) = \frac{1}{\sin(2\theta)} = \frac{1}{-\frac{24}{25}} = -\frac{25}{24} \approx -1.0417 \]

Final Answer