A board rests against a wall. The angle that the board makes with the ground is 60...
Check the final answer first, then review the worked steps.
Problem
A board rests against a wall. The angle that the board makes with the ground is 60 degrees. How far is the base of the board away from the wall? Select the correct trig ratio and distance from wall.
Answer
D. $\cos 60^{\circ} = \frac{x}{12} ; x = 6$ feet
Step-by-step solution
- Identify the problem type: This is a trigonometry problem involving a right-angled triangle.
- Visualize the scenario: Imagine a right-angled triangle where the board is the hypotenuse, the wall is the vertical side (opposite to the angle with the ground), and the ground is the horizontal side (adjacent to the angle with the ground). The angle between the board and the ground is given as $60^\circ$. The length of the board (hypotenuse) is 12 feet. We need to find the distance from the base of the board to the wall, which is the adjacent side.
- Determine the relevant trigonometric ratio: We have the hypotenuse and the angle, and we need to find the adjacent side. The trigonometric ratio that relates the adjacent side, the hypotenuse, and the angle is the cosine function: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.
- Set up the equation: In this case, $\theta = 60^\circ$, the hypotenuse is 12 feet, and the adjacent side is the distance we want to find, let's call it $x$. So, the equation is $\cos(60^\circ) = \frac{x}{12}$.
- Solve for x: We know that $\cos(60^\circ) = \frac{1}{2}$. Therefore, we have $\frac{1}{2} = \frac{x}{12}$. To solve for $x$, multiply both sides of the equation by 12: $x = 12 \times \frac{1}{2} = 6$ feet.
- Select the correct option: Based on the equation and the calculated value of $x$, option D is the correct choice, as it states $\cos 60^\circ = \frac{x}{12}$ and $x = 6$ feet.