Question

for $\triangle abc$, which two relationships are true?
$\cos(\theta) = \frac{ab}{ac} = \sin(90 - \theta)$
$\sin(\theta) = \frac{bc}{ac} = \cos(90 - \theta)$
$\sin(\theta) = \frac{ab}{ac} = \cos(90 - \theta)$
$\cos(\theta) = \frac{ac}{bc} = \sin(\theta)$
$\sin(\theta) = \frac{ab}{ac} = \cos(\theta)$
$\cos(\theta) = \frac{ab}{bc} = \sin(90 - \theta)$

Explanation:

Step1: Recall Trigonometric Ratios

In a right - triangle \( \triangle ABC \) with right angle at \( B \), for angle \( \theta \) at \( A \):

• \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AC} \)
• \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AB}{AC} \)
• Also, we know the co - function identity \( \sin(90 - \alpha)=\cos\alpha \) and \( \cos(90 - \alpha)=\sin\alpha \). For \( \alpha=\theta \), \( \sin(90 - \theta)=\cos\theta \) and \( \cos(90 - \theta)=\sin\theta \)

Step2: Analyze Each Option

• Option 1: \( \cos(\theta)=\frac{AB}{AC} \) and \( \sin(90 - \theta)=\cos\theta \), so this is true.
• Option 2: \( \sin(\theta)=\frac{BC}{AC} \) and \( \cos(90 - \theta)=\sin\theta \), so this is true.
• Option 3: \( \sin(\theta)=\frac{AB}{AC} \) is wrong because \( \sin\theta=\frac{BC}{AC} \), and \( \cos(90 - \theta)=\sin\theta \), but the first part is wrong.
• Option 4: \( \cos(\theta)=\frac{AC}{BC} \) is wrong because \( \cos\theta=\frac{AB}{AC} \), and \( \sin(\theta)=\sin\theta \) (this part is not relevant to the wrong first part), so this is wrong.
• Option 5: \( \sin(\theta)=\frac{AB}{AC} \) is wrong, and \( \cos(\theta)=\cos\theta \) (this part is not relevant to the wrong first part), so this is wrong.
• Option 6: \( \cos(\theta)=\frac{AB}{BC} \) is wrong because \( \cos\theta=\frac{AB}{AC} \), and \( \sin(90 - \theta)=\cos\theta \), but the first part is wrong.

Answer:

The two true relationships are \( \cos(\theta)=\frac{AB}{AC}=\sin(90 - \theta) \) (first option) and \( \sin(\theta)=\frac{BC}{AC}=\cos(90 - \theta) \) (second option)