Given a right triangle with an angle of $40°$ and a hypotenuse of length $10$, which equation can be used to find the length of the side adjacent to the $40°$ angle?

• A. $10\times \cos (40°)=\mathrm{ac}$
• B. $10\times \sin (40°)=\mathrm{ac}$
• C. $\mathrm{ac}=\frac{10}{\sin (40°)}$
• D. $\mathrm{ac}=\frac{10}{\tan (40°)}$

Answer: (A)

Verified step by step guidance

1. Identify the sides of the right triangle relative to the 40° angle: the hypotenuse is given as 10, the side adjacent to the 40° angle is what we want to find (let's call it \( ac \)), and the opposite side is not needed for this problem.
2. Recall the definition of cosine in a right triangle: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \). Here, \( \theta = 40^\circ \), the adjacent side is \( ac \), and the hypotenuse is 10.
3. Set up the equation using the cosine definition: \( \cos(40^\circ) = \frac{ac}{10} \).
4. To solve for \( ac \), multiply both sides of the equation by 10: \( 10 \times \cos(40^\circ) = ac \).
5. This equation \( 10 \times \cos(40^\circ) = ac \) can be used to find the length of the side adjacent to the 40° angle.