Find the exaet value of each of the remaining tigenometrio functions of 0 . Rationalize denominafors when applicable. \( \tan 0=\frac{4}{3} \), given that 0 is in quadrant I (Bimplify your answer, ineluding any radicals, Use integers of fractions for any numbers in the expression.) 13. The function is undeffned. Belect the correct ehoice below and, if necessary, fill in the answer box to complete your choice. A. \( \cos 0=\frac{3}{5} \) (Bimplify your answer, including any radicals. Use integers of fractions for any numbers in the expression.) B. The function is undefined. Seleet the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \operatorname{ese} 0=\frac{5}{4} \) (Simplify your answer, including any radicals. Use integers of fractions for any numbers in the expression.) B. The function is undeffned. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sec 0= \) \( \square \) (Simplify your answer, including any radicals, Use integers or fractions for any numbers in the expression.) B. The function is undefined. Viow an example Get more help :

To find the exact values of the remaining trigonometric functions given that \(\tan \theta = \frac{4}{3}\) and \(\theta\) is in quadrant I, we can use the definitions of the trigonometric functions and the Pythagorean identity. ### Step 1: Identify the sides of the triangle From the tangent function, we know: \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \] This means we can consider a right triangle where the opposite side is 4 and the adjacent side is 3. ### Step 2: Calculate the hypotenuse Using the Pythagorean theorem: \[ \text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] ### Step 3: Calculate the remaining trigonometric functions Now we can find the values of \(\sin \theta\), \(\cos \theta\), \(\sec \theta\), and \(\csc \theta\). 1. **Sine**: \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \] 2. **Cosine**: \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5} \] 3. **Cosecant**: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{4} \] 4. **Secant**: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{5}{3} \] 5. **Cotangent**: \[ \cot \theta = \frac{1}{\tan \theta} = \frac{3}{4} \] ### Summary of Results - \(\sin \theta = \frac{4}{5}\) - \(\cos \theta = \frac{3}{5}\) - \(\tan \theta = \frac{4}{3}\) - \(\csc \theta = \frac{5}{4}\) - \(\sec \theta = \frac{5}{3}\) - \(\cot \theta = \frac{3}{4}\) ### Final Answers 1. **Cosine**: \[ \cos \theta = \frac{3}{5} \] (Choice A) 2. **Cosecant**: \[ \csc \theta = \frac{5}{4} \] (Choice A) 3. **Secant**: \[ \sec \theta = \frac{5}{3} \] (Choice A) All functions are defined in quadrant I, and we have rationalized the denominators where applicable.