Question: 4,9
39°
1.
Determine the lengths of the line segments AB
and BC. You will need to use your knowledge of
trigonometric ratios for these calculations.
INSTRUCTION: Round your answers to two
decimal places.
Answer:
AB = units
BC = units
Graph: a triangle CBA is shown with C at the left, A at the upper right, and B at the bottom; segment CA is labeled 4,9, the angle at A is 39°, and angle at B is a right angle.
1. **State the problem:** We have a right triangle CBA with right angle at B.
Given:
- Hypotenuse $CA = 4.9$ units
- Angle at $A = 39^\circ$
- Angle at $B = 90^\circ$
We need to find lengths $AB$ and $BC$.
2. **Recall trigonometric ratios:**
- $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
- $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
3. **Identify sides relative to angle $A$:**
- Opposite side to angle $A$ is $BC$
- Adjacent side to angle $A$ is $AB$
- Hypotenuse is $CA$
4. **Calculate $BC$ using sine:**
$$
BC = CA \times \sin(39^\circ) = 4.9 \times \sin(39^\circ)
$$
Calculate $\sin(39^\circ)$:
$$
\sin(39^\circ) \approx 0.6293
$$
So,
$$
BC = 4.9 \times 0.6293 = 3.08357
$$
Rounded to two decimals:
$$
BC \approx 3.08
$$
5. **Calculate $AB$ using cosine:**
$$
AB = CA \times \cos(39^\circ) = 4.9 \times \cos(39^\circ)
$$
Calculate $\cos(39^\circ)$:
$$
\cos(39^\circ) \approx 0.7771
$$
So,
$$
AB = 4.9 \times 0.7771 = 3.80779
$$
Rounded to two decimals:
$$
AB \approx 3.81
$$
**Final answers:**
- $AB = 3.81$ units
- $BC = 3.08$ units