Questions: Evaluate the six trigonometric functions of the acute angle θ. sin θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cos θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) tan θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) csc θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) sec θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) cot θ= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

The given right triangle has:

• Opposite side to angle θ: \( \sqrt{10} \)
• Adjacent side to angle θ: \( 3\sqrt{5} \)
• Hypotenuse: We need to calculate it using the Pythagorean theorem.

Using the Pythagorean theorem: \[ \text{Hypotenuse} = \sqrt{(\sqrt{10})^2 + (3\sqrt{5})^2} \] \[ = \sqrt{10 + 45} \] \[ = \sqrt{55} \]

\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \] \[ = \frac{\sqrt{10}}{\sqrt{55}} \] \[ = \frac{\sqrt{10}}{\sqrt{55}} \times \frac{\sqrt{55}}{\sqrt{55}} \] \[ = \frac{\sqrt{550}}{55} \] \[ = \frac{\sqrt{25 \times 22}}{55} \] \[ = \frac{5\sqrt{22}}{55} \] \[ = \frac{\sqrt{22}}{11} \]

\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \] \[ = \frac{3\sqrt{5}}{\sqrt{55}} \] \[ = \frac{3\sqrt{5}}{\sqrt{55}} \times \frac{\sqrt{55}}{\sqrt{55}} \] \[ = \frac{3\sqrt{275}}{55} \] \[ = \frac{3\sqrt{25 \times 11}}{55} \] \[ = \frac{15\sqrt{11}}{55} \] \[ = \frac{3\sqrt{11}}{11} \]

\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \] \[ = \frac{\sqrt{10}}{3\sqrt{5}} \] \[ = \frac{\sqrt{10}}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} \] \[ = \frac{\sqrt{50}}{15} \] \[ = \frac{5\sqrt{2}}{15} \] \[ = \frac{\sqrt{2}}{3} \]

Final Answer