Question

emma is given the figure shown and asked to find the measurement of ∠a. which ratio could she use? a $sin a = \frac{24}{7}$ b $\tan a = \frac{7}{24}$ c $cos a = \frac{7}{25}$ d $sin a = \frac{7}{25}$

Explanation:

Step1: Recall Trigonometric Ratios

In a right - triangle, the trigonometric ratios are defined as:

• Sine of an angle: $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine of an angle: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent of an angle: $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$

For angle $A$ in the right - triangle $ABC$ (right - angled at $C$):

• The side opposite to $\angle A$ is $BC = a = 7$
• The side adjacent to $\angle A$ is $AC = b = 24$
• The hypotenuse is $AB = c = 25$

Step2: Analyze each option

• Option A: $\sin A=\frac{24}{7}$
• According to the sine formula, $\sin A$ should be $\frac{\text{opposite}}{\text{hypotenuse}}=\frac{7}{25}$, not $\frac{24}{7}$. So, option A is incorrect.
• Option B: $\tan A=\frac{7}{24}$
• According to the tangent formula, $\tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{7}{24}$. This is correct.
• Option C: $\cos A=\frac{7}{25}$
• According to the cosine formula, $\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}$, not $\frac{7}{25}$. So, option C is incorrect.
• Option D: $\sin A=\frac{24}{25}$
• According to the sine formula, $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{7}{25}$, not $\frac{24}{25}$. So, option D is incorrect.

Answer:

B. $\tan A=\frac{7}{24}$