Answer :
To determine which expression shows the height of the sail in a right triangle, let's break down the problem.
In this scenario, the sail forms a right triangle where we need to find the height (opposite side) using the given angle (35°) and a known side (which would typically be the base or the hypotenuse).
Let's denote:
- \( \theta \) = 35° (the given angle)
- \( \text{height} \) = the opposite side to the angle \(\theta\)
- \( \text{base} \) = the side adjacent to the angle \(\theta\) (given to be 8 meters here)
The trigonometric function that relates the opposite side and the adjacent side with the angle is the tangent function:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, \(\theta = 35^\circ\), the opposite side is the height we want to find, and the adjacent side is 8 meters.
We can set up the equation using the tangent function:
[tex]\[ \tan(35^\circ) = \frac{\text{height}}{8} \][/tex]
To find the height, solve for it by multiplying both sides of the equation by 8:
[tex]\[ \text{height} = 8 \cdot \tan(35^\circ) \][/tex]
Thus, the correct expression that shows the height, in meters, of the sail is:
[tex]\[ 8 \left( \tan(35^\circ) \right) \][/tex]
Among the given choices, this matches the second option:
[tex]\[ 8\left(\tan 35^\circ\right) \][/tex]
So, the correct answer is:
[tex]\[ 8\left(\tan 35^\circ\right) \][/tex]
In this scenario, the sail forms a right triangle where we need to find the height (opposite side) using the given angle (35°) and a known side (which would typically be the base or the hypotenuse).
Let's denote:
- \( \theta \) = 35° (the given angle)
- \( \text{height} \) = the opposite side to the angle \(\theta\)
- \( \text{base} \) = the side adjacent to the angle \(\theta\) (given to be 8 meters here)
The trigonometric function that relates the opposite side and the adjacent side with the angle is the tangent function:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, \(\theta = 35^\circ\), the opposite side is the height we want to find, and the adjacent side is 8 meters.
We can set up the equation using the tangent function:
[tex]\[ \tan(35^\circ) = \frac{\text{height}}{8} \][/tex]
To find the height, solve for it by multiplying both sides of the equation by 8:
[tex]\[ \text{height} = 8 \cdot \tan(35^\circ) \][/tex]
Thus, the correct expression that shows the height, in meters, of the sail is:
[tex]\[ 8 \left( \tan(35^\circ) \right) \][/tex]
Among the given choices, this matches the second option:
[tex]\[ 8\left(\tan 35^\circ\right) \][/tex]
So, the correct answer is:
[tex]\[ 8\left(\tan 35^\circ\right) \][/tex]
