In a right triangle, angle $A$ is acute. The side opposite $A$ has length $a$, the side adjacent to $A$ has length $b$, and the hypotenuse has length $c$. Which expressions give $sinA$ and $cosA$?

• A. $sinA=\frac{c}{a},cosA=\frac{c}{b}$
• B. $sinA=\frac{a}{b},cosA=\frac{c}{b}$
• C. $sinA=\frac{b}{c},cosA=\frac{a}{c}$
• D. $sinA=\frac{a}{c},cosA=\frac{b}{c}$

Answer: (D)

Verified step by step guidance

1. Recall the definitions of sine and cosine in a right triangle relative to an acute angle \(A\): \(\sin A\) is the ratio of the length of the side opposite angle \(A\) to the hypotenuse, and \(\cos A\) is the ratio of the length of the side adjacent to angle \(A\) to the hypotenuse.
2. Identify the sides in the triangle: the side opposite angle \(A\) is \(a\), the side adjacent to angle \(A\) is \(b\), and the hypotenuse is \(c\).
3. Write the expression for \(\sin A\) using the definition: \(\sin A = \frac{a}{c}\), since \(a\) is opposite \(A\) and \(c\) is the hypotenuse.
4. Write the expression for \(\cos A\) using the definition: \(\cos A = \frac{b}{c}\), since \(b\) is adjacent to \(A\) and \(c\) is the hypotenuse.
5. Verify that these expressions match the correct answer choice, confirming that \(\sin A = \frac{a}{c}\) and \(\cos A = \frac{b}{c}\).