Question

malia tried to prove that \\(\cos(\theta) = \sin(90^\circ - \theta)\\) using the following diagram. her proof is not correct.
diagram: right triangle with right angle at a, angle \\(\theta\\) at b, vertices a, b, c

statement	reason
2. \\(\sin(90^\circ - \theta) = \frac{ac}{bc}\\)	definition of sine.
3. \\(\cos(\theta) = \frac{ac}{bc}\\)	definition of cosine.
4. \\(\cos(\theta) = \sin(90^\circ - \theta)\\)	substitution

what is the first mistake in malias proof?

Explanation:

To determine the first mistake, we analyze each step:

1. Step 1: In a right triangle, acute angles are complementary, so \(m\angle C = 90^\circ - \theta\) is correct.
2. Step 2: The definition of sine for an angle in a right triangle is \(\sin(\alpha)=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle C = 90^\circ - \theta\), the opposite side to \(\angle C\) is \(AB\), and the hypotenuse is \(BC\). So \(\sin(90^\circ - \theta)=\sin(\angle C)=\frac{AB}{BC}\), not \(\frac{AC}{BC}\). Thus, the reason "Definition of sine" is misapplied here as the ratio is incorrect.
3. Step 3: For \(\angle B=\theta\), \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AB}{BC}\) (wait, no, adjacent to \(\theta\) is \(AB\)? Wait, no, in triangle \(ABC\) with right angle at \(A\), sides: \(AB\) and \(AC\) are legs, \(BC\) is hypotenuse. For \(\angle B = \theta\), adjacent side is \(AB\), opposite is \(AC\), hypotenuse \(BC\). So \(\cos(\theta)=\frac{AB}{BC}\), but in step 3, it's written as \(\frac{AC}{BC}\), which is also a mistake, but step 2 is the first mistake. Wait, no, let's re - check step 2: \(\sin(90^\circ - \theta)\) is \(\sin(\angle C)\). For \(\angle C\), opposite side is \(AB\), hypotenuse \(BC\), so \(\sin(\angle C)=\frac{AB}{BC}\), but Malia wrote \(\frac{AC}{BC}\), so step 2 has the wrong ratio for sine.

Answer:

The first mistake is in step 2. The correct definition of sine for \(\angle C = 90^\circ-\theta\) is \(\sin(90^\circ - \theta)=\sin(\angle C)=\frac{AB}{BC}\) (opposite over hypotenuse), but Malia used \(\frac{AC}{BC}\) which is incorrect. So the first mistake is in the second step where the ratio for \(\sin(90^\circ - \theta)\) is wrong (using \(AC\) as the opposite side instead of \(AB\) when applying the definition of sine).