Given a right triangle with hypotenuse of length $c$ and an acute angle $\theta$, express the lengths of the side adjacent to $\theta$ (labeled $a$) and the side opposite $\theta$ (labeled $b$) in terms of $c$ and $\theta$.

• A. $a=c\cdot \cos \left(\theta \right)$ and $b=c\cdot \sin \left(\theta \right)$
• B. $a=\frac{1}{c\cdot \sin \left(\theta \right)}$ and $b=\frac{1}{c\cdot \cos \left(\theta \right)}$
• C. $a=c\cdot \tan \left(\theta \right)$ and $b=c\cdot \cot \left(\theta \right)$
• D. $a=c\cdot \sin \left(\theta \right)$ and $b=c\cdot \cos \left(\theta \right)$

Answer: (A)

Verified step by step guidance

1. Recall that in a right triangle, the hypotenuse is the side opposite the right angle and is the longest side, labeled as \(c\) here.
2. Identify the sides relative to the acute angle \(\theta\): the side adjacent to \(\theta\) is labeled \(a\), and the side opposite \(\theta\) is labeled \(b\).
3. Use the definitions of the trigonometric functions cosine and sine for angle \(\theta\): \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}\) and \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}\).
4. Solve each equation for the side lengths \(a\) and \(b\) by multiplying both sides by \(c\): \(a = c \cdot \cos(\theta)\) and \(b = c \cdot \sin(\theta)\).
5. Thus, the lengths of the sides adjacent and opposite to angle \(\theta\) are expressed in terms of \(c\) and \(\theta\) as \(a = c \cdot \cos(\theta)\) and \(b = c \cdot \sin(\theta)\).