5. Given that \( \sin \theta>0 \) and that \( \cot \theta=\frac{7}{5} \), find the other five trig functions. Sin theta \( =5 / \operatorname{sqrt} 74, \cos \) theta \( =7 / \) sqrt 74 , tan theta \( =5 / 7 \), csc theta \( = \) frac/sqrt 74 \( a^{\wedge} 2+b^{\wedge} 2=c^{\wedge} 2 \)

To find the other five trigonometric functions given that \( \sin \theta > 0 \) and \( \cot \theta = \frac{7}{5} \), we can follow these steps: 1. **Understanding Cotangent**: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{7}{5} \] This means that for every 7 units of the adjacent side (cosine), there are 5 units of the opposite side (sine) in a right triangle. 2. **Finding Sine and Cosine**: We can use the Pythagorean theorem to find the hypotenuse. The sides of the triangle are: - Opposite side (for sine): \( 5 \) - Adjacent side (for cosine): \( 7 \) The hypotenuse \( h \) can be calculated as: \[ h = \sqrt{(5^2 + 7^2)} = \sqrt{25 + 49} = \sqrt{74} \] Now we can find sine and cosine: \[ \sin \theta = \frac{5}{\sqrt{74}}, \quad \cos \theta = \frac{7}{\sqrt{74}} \] 3. **Finding Tangent**: Tangent is the ratio of sine to cosine: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/\sqrt{74}}{7/\sqrt{74}} = \frac{5}{7} \] 4. **Finding Cosecant**: Cosecant is the reciprocal of sine: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{74}}{5} \] 5. **Finding Secant**: Secant is the reciprocal of cosine: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{74}}{7} \] 6. **Finding Cotangent**: Cotangent is already given as: \[ \cot \theta = \frac{7}{5} \] Now we have all six trigonometric functions: - \( \sin \theta = \frac{5}{\sqrt{74}} \) - \( \cos \theta = \frac{7}{\sqrt{74}} \) - \( \tan \theta = \frac{5}{7} \) - \( \csc \theta = \frac{\sqrt{74}}{5} \) - \( \sec \theta = \frac{\sqrt{74}}{7} \) - \( \cot \theta = \frac{7}{5} \) Thus, the final results are: - \( \sin \theta = \frac{5}{\sqrt{74}} \) - \( \cos \theta = \frac{7}{\sqrt{74}} \) - \( \tan \theta = \frac{5}{7} \) - \( \csc \theta = \frac{\sqrt{74}}{5} \) - \( \sec \theta = \frac{\sqrt{74}}{7} \) - \( \cot \theta = \frac{7}{5} \)