Sure! Let's go through the steps to solve the problem.
1. Understanding the problem:
- We are given an angle of elevation from point C to point B which is [tex]\( 42.9^\circ \)[/tex].
- We are also given the angle of depression from point B to point A which is [tex]\( 27.7^\circ \)[/tex].
- The distance between points D and A (denoted as [tex]\( DA \)[/tex]) is 7.23 meters.
- We need to calculate the lengths [tex]\( BC \)[/tex] and [tex]\( AC \)[/tex].
2. Convert the angles from degrees to radians:
The angles need to be converted to radians to use in trigonometric formulae.
3. Calculate length [tex]\( BC \)[/tex]:
- Angle of elevation from C to B is [tex]\( 42.9^\circ \)[/tex].
- We can use the tangent function: [tex]\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)[/tex] where [tex]\( \theta \)[/tex] is the angle.
- Here, [tex]\( BC \)[/tex] is the opposite side (vertical length from B to C) and [tex]\( DA \)[/tex] is the adjacent side.
- Thus, [tex]\( BC = DA \cdot \tan(42.9^\circ) \)[/tex].
4. Calculate length [tex]\( AC \)[/tex]:
- Since [tex]\( AC \)[/tex] forms the hypotenuse of the right triangle [tex]\( \triangle DAC \)[/tex].
- We can use the angle of depression [tex]\( 27.7^\circ \)[/tex] and find the relation using the cosine function: [tex]\( \cos(\phi) = \frac{\text{adjacent}}{\text{hypotenuse}} \)[/tex].
- Here, [tex]\( DA \)[/tex] is considered as the base (adjacent side) and [tex]\( AC \)[/tex] is the hypotenuse.
- Thus, [tex]\( AC = \frac{BC}{\cos(27.7^\circ)} \)[/tex], but more importantly, using trigonometric relationships.
5. Plugging in the calculations:
- From our calculations,
- [tex]\( BC = 6.72 \)[/tex] meters.
- [tex]\( AC = 7.59 \)[/tex] meters.
6. Rounding off the answers:
- We are to round off the lengths to two decimal places.
The lengths are:
- [tex]\( BC = 6.72 \)[/tex] meters.
- [tex]\( AC = 7.59 \)[/tex] meters.
So, lengths [tex]\( BC \)[/tex] and [tex]\( AC \)[/tex] given the angle of elevation from C to B and the angle of depression from B to A with distance [tex]\( DA = 7.23 \)[/tex] meters are approximately 6.72 meters and 7.59 meters respectively, rounded to two decimal places.
