Find the signs of the six trigonometric function values for the given angle, \( -242^{\circ} \) The trigonometric function value of \( \sin \left(-242^{\circ}\right) \) is positive, \( \cos \left(-242^{\circ}\right) \) is negative, and \( \tan \left(-242^{\circ}\right) \) is negative. The trigonometric function value of \( \csc \left(-242^{\circ}\right) \) is \( \square \sec \left(-242^{\circ}\right) \) is Part 2 of 2

To determine the signs of all six trigonometric functions for the angle \( -242^{\circ} \), let's analyze the angle step by step. ### Step 1: Determine the Equivalent Positive Angle First, find the positive coterminal angle for \( -242^{\circ} \): \[ -242^{\circ} + 360^{\circ} = 118^{\circ} \] So, \( -242^{\circ} \) is coterminal with \( 118^{\circ} \). ### Step 2: Identify the Quadrant The angle \( 118^{\circ} \) lies in the **Second Quadrant** (between \( 90^{\circ} \) and \( 180^{\circ} \)). ### Step 3: Determine the Signs of the Trigonometric Functions in the Second Quadrant | Function | Sign in Second Quadrant | |----------|-------------------------| | \( \sin \) | Positive | | \( \cos \) | Negative | | \( \tan \) | Negative | | \( \csc \) | Positive | | \( \sec \) | Negative | | \( \cot \) | Negative | ### Step 4: Apply to \( -242^{\circ} \) 1. **Sine and Cosecant:** - \( \sin(-242^{\circ}) = \sin(118^{\circ}) \) is **Positive**. - \( \csc(-242^{\circ}) = \csc(118^{\circ}) \) is **Positive**. 2. **Cosine and Secant:** - \( \cos(-242^{\circ}) = \cos(118^{\circ}) \) is **Negative**. - \( \sec(-242^{\circ}) = \sec(118^{\circ}) \) is **Negative**. 3. **Tangent and Cotangent:** - \( \tan(-242^{\circ}) = \tan(118^{\circ}) \) is **Negative**. - \( \cot(-242^{\circ}) = \cot(118^{\circ}) \) is **Negative**. ### Summary of Signs - \( \sin(-242^{\circ}) \) : **Positive** - \( \cos(-242^{\circ}) \) : **Negative** - \( \tan(-242^{\circ}) \) : **Negative** - \( \csc(-242^{\circ}) \) : **Positive** - \( \sec(-242^{\circ}) \) : **Negative** - \( \cot(-242^{\circ}) \) : **Negative** ### Final Answer - **Cosecant:** \( \csc(-242^{\circ}) \) is **positive**. - **Secant:** \( \sec(-242^{\circ}) \) is **negative**.