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Sagot :
Certainly! Let's solve the problem step by step.
### 1. Given Data:
- Hypotenuse ([tex]\(c\)[/tex]) = 16.0 units
- One of the acute angles ([tex]\(\angle A\)[/tex]) = 35.0 degrees
### a) Finding the Other Acute Angle
In a right triangle, the sum of all the angles is always 180 degrees. Since one of the angles is 90 degrees (the right angle), and we are given one acute angle, the other acute angle ([tex]\(\angle B\)[/tex]) can be found as follows:
[tex]\[ \angle B = 90^\circ - \angle A \][/tex]
So we have:
[tex]\[ \angle B = 90^\circ - 35^\circ = 55.0^\circ \][/tex]
Thus, the other acute angle is [tex]\(55.0\)[/tex] degrees.
### b) Finding the Other Two Sides
To find the other two sides of the triangle, we will use trigonometric functions:
#### i) Finding the side opposite to [tex]\(\angle A\)[/tex] (let's call it [tex]\(a\)[/tex]):
Using the sine function:
[tex]\[ \sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
So,
[tex]\[ \sin(35.0^\circ) = \frac{a}{16.0} \][/tex]
Rearranging to solve for [tex]\(a\)[/tex]:
[tex]\[ a = 16.0 \times \sin(35.0^\circ) \][/tex]
Given that the result for the length of side [tex]\(a\)[/tex] rounded to one decimal place is [tex]\(9.2\)[/tex] units.
#### ii) Finding the side adjacent to [tex]\(\angle A\)[/tex] (let's call it [tex]\(b\)[/tex]):
Using the cosine function:
[tex]\[ \cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
So,
[tex]\[ \cos(35.0^\circ) = \frac{b}{16.0} \][/tex]
Rearranging to solve for [tex]\(b\)[/tex]:
[tex]\[ b = 16.0 \times \cos(35.0^\circ) \][/tex]
Given that the result for the length of side [tex]\(b\)[/tex] rounded to one decimal place is [tex]\(13.1\)[/tex] units.
### Summary:
1. The other acute angle [tex]\( \angle B \)[/tex] is [tex]\(55.0^\circ\)[/tex].
2. The side opposite [tex]\(\angle A\)[/tex] (side [tex]\(a\)[/tex]) is [tex]\(9.2\)[/tex] units.
3. The side adjacent to [tex]\(\angle A\)[/tex] (side [tex]\(b\)[/tex]) is [tex]\(13.1\)[/tex] units.
So, the final answers are:
- [tex]\(\angle B = 55.0^\circ\)[/tex]
- Side [tex]\(a = 9.2\)[/tex] units
- Side [tex]\(b = 13.1\)[/tex] units
### 1. Given Data:
- Hypotenuse ([tex]\(c\)[/tex]) = 16.0 units
- One of the acute angles ([tex]\(\angle A\)[/tex]) = 35.0 degrees
### a) Finding the Other Acute Angle
In a right triangle, the sum of all the angles is always 180 degrees. Since one of the angles is 90 degrees (the right angle), and we are given one acute angle, the other acute angle ([tex]\(\angle B\)[/tex]) can be found as follows:
[tex]\[ \angle B = 90^\circ - \angle A \][/tex]
So we have:
[tex]\[ \angle B = 90^\circ - 35^\circ = 55.0^\circ \][/tex]
Thus, the other acute angle is [tex]\(55.0\)[/tex] degrees.
### b) Finding the Other Two Sides
To find the other two sides of the triangle, we will use trigonometric functions:
#### i) Finding the side opposite to [tex]\(\angle A\)[/tex] (let's call it [tex]\(a\)[/tex]):
Using the sine function:
[tex]\[ \sin(\angle A) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
So,
[tex]\[ \sin(35.0^\circ) = \frac{a}{16.0} \][/tex]
Rearranging to solve for [tex]\(a\)[/tex]:
[tex]\[ a = 16.0 \times \sin(35.0^\circ) \][/tex]
Given that the result for the length of side [tex]\(a\)[/tex] rounded to one decimal place is [tex]\(9.2\)[/tex] units.
#### ii) Finding the side adjacent to [tex]\(\angle A\)[/tex] (let's call it [tex]\(b\)[/tex]):
Using the cosine function:
[tex]\[ \cos(\angle A) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
So,
[tex]\[ \cos(35.0^\circ) = \frac{b}{16.0} \][/tex]
Rearranging to solve for [tex]\(b\)[/tex]:
[tex]\[ b = 16.0 \times \cos(35.0^\circ) \][/tex]
Given that the result for the length of side [tex]\(b\)[/tex] rounded to one decimal place is [tex]\(13.1\)[/tex] units.
### Summary:
1. The other acute angle [tex]\( \angle B \)[/tex] is [tex]\(55.0^\circ\)[/tex].
2. The side opposite [tex]\(\angle A\)[/tex] (side [tex]\(a\)[/tex]) is [tex]\(9.2\)[/tex] units.
3. The side adjacent to [tex]\(\angle A\)[/tex] (side [tex]\(b\)[/tex]) is [tex]\(13.1\)[/tex] units.
So, the final answers are:
- [tex]\(\angle B = 55.0^\circ\)[/tex]
- Side [tex]\(a = 9.2\)[/tex] units
- Side [tex]\(b = 13.1\)[/tex] units
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