Which right triangle contains an acute angle where the $\sin$ and $\cos$ of that angle are equal?

• A. A right triangle with legs of equal length (an isosceles right triangle)
• B. A right triangle with one leg twice as long as the other
• C. A right triangle with a hypotenuse twice as long as one leg
• D. A right triangle with angles $30$°, $60$°, and $90$°

Answer: (A)

Verified step by step guidance

1. Recall that in a right triangle, the sine of an acute angle \( \theta \) is defined as the ratio of the length of the side opposite \( \theta \) to the hypotenuse: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). Similarly, the cosine of \( \theta \) is the ratio of the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
2. To find when \( \sin(\theta) = \cos(\theta) \), set the two ratios equal: \( \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{adjacent}}{\text{hypotenuse}} \). Since the hypotenuse is common and nonzero, this simplifies to \( \text{opposite} = \text{adjacent} \).
3. This equality means the two legs of the right triangle adjacent to the right angle must be of equal length, which characterizes an isosceles right triangle.
4. In an isosceles right triangle, the two acute angles are equal, each measuring \( 45^\circ \), because the angles must sum to \( 90^\circ \) (since the right angle is \( 90^\circ \)).
5. Therefore, the right triangle where \( \sin(\theta) = \cos(\theta) \) for an acute angle \( \theta \) is the one with legs of equal length, also known as an isosceles right triangle.