Questions: Sketch a right triangle corresponding to the trigonometric function of the acute csc(θ) = 10 sin(θ) = □ cos(θ) = □ tan(θ) = □ sec(θ) = □ cot(θ) = □

Solution Steps

To solve the problem, we need to find the values of the trigonometric functions for the given angle \(\theta\) using the provided information. We know that \(\csc(\theta) = 10\), which means \(\sin(\theta) = \frac{1}{\csc(\theta)}\). Using the Pythagorean identity, we can find \(\cos(\theta)\) and then use the definitions of the other trigonometric functions to find their values.

Given that \(\csc(\theta) = 10\), we can find \(\sin(\theta)\) using the relationship: \[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{10} = 0.1 \]

Using the Pythagorean identity: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] we can substitute \(\sin(\theta)\): \[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - (0.1)^2 = 1 - 0.01 = 0.99 \] Thus, we find: \[ \cos(\theta) = \sqrt{0.99} \approx 0.99499 \]

Using the definitions of the trigonometric functions: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{0.1}{0.99499} \approx 0.1005 \] \[ \sec(\theta) = \frac{1}{\cos(\theta)} \approx \frac{1}{0.99499} \approx 1.0050 \] \[ \cot(\theta) = \frac{1}{\tan(\theta)} \approx \frac{1}{0.1005} \approx 9.9499 \]

Final Answer