Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length
using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle.
Rationalize the denominator if applicable.
-Find $\sin \mathrm { A }$ when $\mathrm { a } = 9$ and $\mathrm { b } = 5$.
A) $\frac { 9 \sqrt { 106 } } { 106 }$
B) $\frac { 5 \sqrt { 106 } } { 106 }$
C) $\frac { \sqrt { 106 } } { 5 }$
D) $\frac { \sqrt { 106 } } { 9 }$

Question

Quizplus · Accepted Answer

Using the Pythagorean theorem, we have $c^2 = a^2 + b^2$. Substituting the given values, we get $c^2 = 9^2 + 5^2 = 81 + 25 = 106$. Therefore, $c = \sqrt{106}$.Now, we can find $\sin A$ using the definition: $\sin A = \frac{opposite}{hypotenuse}$. In this case, the opposite side is $a = 9$ and the hypotenuse is $c = \sqrt{106}$. Substituting these values, we get $\sin A = \frac{9}{\sqrt{106}}$.Rationalizing the denominator, we get $\sin A = \frac{9}{\sqrt{106}} \cdot \frac{\sqrt{106}}{\sqrt{106}} = \frac{9\sqrt{106}}{106}$.Therefore, the correct answer is A.

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