Sure, let's go step-by-step.
### 1. Determining the Third Angle:
In any triangle, the sum of the interior angles is [tex]\(180^\circ\)[/tex]. Since we have a right triangle, one angle is [tex]\(90^\circ\)[/tex]. Given one of the other angles is [tex]\(35^\circ\)[/tex], we can find the third angle as follows:
[tex]\[ \text{Third angle} = 180^\circ - 90^\circ - 35^\circ \][/tex]
[tex]\[ \text{Third angle} = 55^\circ \][/tex]
So, the third angle is [tex]\(55^\circ\)[/tex].
### 2. Calculating the Length of the Hypotenuse:
Now, let's assume you are given the length of the side opposite the [tex]\(35^\circ\)[/tex] angle is [tex]\(5\)[/tex] units, and we need to find the hypotenuse.
We use the sine function, which is given by:
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
For [tex]\( \theta = 35^\circ \)[/tex],
[tex]\[ \sin(35^\circ) = \frac{5}{\text{hypotenuse}} \][/tex]
We can solve for the hypotenuse:
[tex]\[ \text{hypotenuse} = \frac{5}{\sin(35^\circ)} \][/tex]
Since we know the result from the calculations:
[tex]\[ \text{hypotenuse} \approx 8.717 \][/tex]
Hence, the length of the hypotenuse is approximately [tex]\(8.717\)[/tex] units.
### 3. Calculating the Length of the Missing Side ([tex]\(x\)[/tex]):
Next, we need to find the adjacent side to the [tex]\(35^\circ\)[/tex] angle, which is the missing side [tex]\(x\)[/tex].
We use the cosine function, which is given by:
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
For [tex]\( \theta = 35^\circ \)[/tex],
[tex]\[ \cos(35^\circ) = \frac{x}{8.717} \][/tex]
We can solve for [tex]\(x\)[/tex]:
[tex]\[ x = 8.717 \times \cos(35^\circ) \][/tex]
From the result of the calculations:
[tex]\[ x \approx 7.141 \][/tex]
Hence, the length of the missing side [tex]\(x\)[/tex] is approximately [tex]\(7.141\)[/tex] units.
To summarize:
1. The third angle is [tex]\(55^\circ\)[/tex].
2. The hypotenuse is approximately [tex]\(8.717\)[/tex] units.
3. The missing side ([tex]\(x\)[/tex]) is approximately [tex]\(7.141\)[/tex] units.
