Sure, let's solve this step by step.
1. Calculating the Third Angle:
- In any triangle, the sum of the angles is [tex]\( 180^{\circ} \)[/tex].
- In a right triangle, one of these angles is always [tex]\( 90^{\circ} \)[/tex].
- Given one angle is [tex]\( 35^{\circ} \)[/tex], let's denote this angle as [tex]\( \angle A \)[/tex].
- Let's denote the right angle as [tex]\( \angle B = 90^{\circ} \)[/tex].
To find the third angle ([tex]\(\angle C\)[/tex]):
[tex]\[
\angle C = 180^{\circ} - 90^{\circ} - 35^{\circ} = 55^{\circ}
\][/tex]
2. Calculate the Length of the Hypotenuse:
- Let's denote the adjacent side to the [tex]\( 35^{\circ} \)[/tex] angle (usually given) as [tex]\( \text{adjacent} = 7 \)[/tex] units.
- We use the cosine function, which relates the adjacent side and the hypotenuse in a right triangle:
[tex]\[
\cos(35^{\circ}) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
Solving for the hypotenuse:
[tex]\[
\text{hypotenuse} = \frac{\text{adjacent}}{\cos(35^{\circ})}
\][/tex]
The hypotenuse is approximately:
[tex]\[
\text{hypotenuse} \approx 8.5454 \text{ units}
\][/tex]
3. Calculate the Length of the Missing Side [tex]\( x \)[/tex] (Opposite side):
- We use the tangent function, which relates the opposite side and the adjacent side in a right triangle:
[tex]\[
\tan(35^{\circ}) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Solving for the opposite side:
[tex]\[
\text{opposite} = \text{adjacent} \times \tan(35^{\circ})
\][/tex]
The opposite side [tex]\( x \)[/tex] is approximately:
[tex]\[
x \approx 4.9015 \text{ units}
\][/tex]
In summary:
1. The third angle is [tex]\( 55^{\circ} \)[/tex].
2. The length of the hypotenuse is approximately [tex]\( 8.5454 \)[/tex] units.
3. The length of the missing side [tex]\( x \)[/tex] (opposite the [tex]\( 35^{\circ} \)[/tex] angle) is approximately [tex]\( 4.9015 \)[/tex] units.
