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

Question

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Review the given proof and the trigonometric identities provided by asksia-ll calculator. ‖ ‖ step 2 ⋮ Confirm that the trigonometric identities $cos( heta) = sin(90° - heta)$ and $sin(90° - heta) = cos( heta)$ are correct. ‖ ‖ step 3 ⋮ Examine Statement 1: $m\angle C = 90° - heta$. This is correct as the angles in a right triangle are complementary. ‖ ‖ step 4 ⋮ Examine Statement 2: $sin(90° - heta) = \frac{AC}{BC}$. This is the correct definition of sine for angle C. ‖ ‖ step 5 ⋮ Examine Statement 3: $cos( heta) = \frac{AC}{BC}$. 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.⚹◊What is the correct way to define sine and cosine in relation to the sides of the triangle in Malia's proof⍭ Generate me a similar question◊

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