Question

Math

Posted 5 months ago

```
Malia tried to prove that $\cos (\theta)=\sin \left(90^{\circ}-\theta\right)$ using the following diagram. Her proof is not correct.
\begin{tabular}{lll}
Statement & Reason \\
\hline 1 & $m \angle C=90^{\circ}-\theta$ & The acute angles in a right triangle are complementary.
\end{tabular}
$2 \sin \left(90^{\circ}-\theta\right)=\frac{A C}{B C} \quad$ Definition of sine.
$3 \cos (\theta)=\frac{A C}{B C} \quad$ Definition of cosine.
$4 \cos (\theta)=\sin \left(90^{\circ}-\theta\right) \quad$ Substitution
What is the first mistake in Malia's proof?
Choose 1 answer:
(A) The measures of $\angle B$ and $\angle C$ add up to $180^{\circ}$, not $90^{\circ}$.
(B) Malia used the wrong sides in her ratio for $\sin \left(90^{\circ}-\theta\right)$.
(C) Malia used the wrong sides in her ratio for $\cos (\theta)$.
(D) Malia substituted a value that wasn't equivalent to the one she replaced.
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

step 2

Confirm that the trigonometric identities $cos(\theta) = sin(90° - \theta)$ and $sin(90° - \theta) = cos(\theta)$ are correct

step 3

Examine Statement 1: $m\angle C = 90° - \theta$. This is correct as the angles in a right triangle are complementary

step 4

Examine Statement 2: $sin(90° - \theta) = \frac{AC}{BC}$. This is the correct definition of sine for angle C

step 5

Examine Statement 3: $cos(\theta) = \frac{AC}{BC}$. This is incorrect as the definition of cosine for angle B should be $\frac{BC}{AC}$ or $\frac{AB}{AC}$ depending on the triangle orientation

step 6

Identify the first mistake in Malia's proof, which is in Statement 3, where the sides used for the cosine ratio are incorrect

C

Key Concept

Trigonometric Ratios

Explanation

The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse, not the opposite side to the adjacent side as used by Malia in her proof.

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