Questions: Clear my choice (triangle ABC) is a right triangle: If angle A=32 degrees, angle B=90 degrees, AC=9 and AB=5, choose the expression used to find BC. Select one: a. 5 tan 32 degrees b. 9/sin 32 degrees c. 9/tan 58 degrees d. 5 cos 58 degrees Use your calculator to find the measure of angle. (Round your answer to the nearest degree.).

Clear my choice
(triangle ABC) is a right triangle: If angle A=32 degrees, angle B=90 degrees, AC=9 and AB=5, choose the expression used to find BC.

Select one:
a. 5 tan 32 degrees
b. 9/sin 32 degrees
c. 9/tan 58 degrees
d. 5 cos 58 degrees

Use your calculator to find the measure of angle. (Round your answer to the nearest degree.).

Transcript text: Clear my choice ( (triangle $A B C($ ) is a right triangle: If $\angle A=32^{\circ}, \angle B=90^{\circ}, A C=9$ and $A B=5$, choose the expression used to find $B C$. Select one: a. $5 \tan 32^{\circ}$ b. $\frac{9}{\sin 32^{\circ}}$ c. $\frac{9}{\tan 58^{\circ}}$ d. $5 \cos 58^{\circ}$ Use your calculator to find the measure of angle. (Round your answer to the nearest degree.).

Solution

Solution Steps

To find the length of side $ BC $ in the right triangle $ \triangle ABC $, we can use trigonometric ratios. Given that $ \angle A = 32^\circ $ and $ \angle B = 90^\circ $, we can use the tangent function, which relates the opposite side to the adjacent side in a right triangle. The correct expression to find $ BC $ is $ AB \cdot \tan(\angle A) $.

Solution Approach

Identify the given values: $ \angle A = 32^\circ $, $ \angle B = 90^\circ $, $ AC = 9 $, and $ AB = 5 $.
Use the tangent function: $ \tan(\angle A) = \frac{BC}{AB} $.
Solve for $ BC $: $ BC = AB \cdot \tan(\angle A) $.

Step 1: Identify the Given Information

We are given a right triangle $ \triangle ABC $ with:

$ \angle A = 32^\circ $
$ \angle B = 90^\circ $
$ AC = 9 $
$ AB = 5 $

We need to find the expression used to determine $ BC $.

Step 2: Determine the Relationship

Since $ \angle B = 90^\circ $, $ \angle C = 58^\circ $ (because $ \angle A + \angle C = 90^\circ $).

Step 3: Use Trigonometric Ratios

To find $ BC $, we can use the trigonometric ratios involving the given sides and angles.

Option a: $ 5 \tan 32^\circ $

This option uses the tangent function: \[ \tan 32^\circ = \frac{BC}{AB} \] \[ BC = AB \cdot \tan 32^\circ \] \[ BC = 5 \cdot \tan 32^\circ \]

Option b: $ \frac{9}{\sin 32^\circ} $

This option uses the sine function: \[ \sin 32^\circ = \frac{BC}{AC} \] \[ BC = AC \cdot \sin 32^\circ \] \[ BC = 9 \cdot \sin 32^\circ \]

Option c: $ \frac{9}{\tan 58^\circ} $

This option uses the tangent function: \[ \tan 58^\circ = \frac{BC}{AC} \] \[ BC = AC \cdot \tan 58^\circ \] \[ BC = 9 \cdot \tan 58^\circ \]

Option d: $ 5 \cos 58^\circ $

This option uses the cosine function: \[ \cos 58^\circ = \frac{AB}{BC} \] \[ BC = \frac{AB}{\cos 58^\circ} \] \[ BC = \frac{5}{\cos 58^\circ} \]

Step 4: Evaluate the Correct Expression

From the above analysis, the correct expression to find $ BC $ is: \[ BC = 5 \cdot \tan 32^\circ \]

Final Answer

$\boxed{a. \, 5 \tan 32^\circ}$

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