Question

A ladder is leaning against a wall at an angle of 70° with the ground. The distance along the ground is 86cm. Find the length of the ladder

Answer 1

`\boxed{x = 251.4 cm}`

Explanation:

Part 1: Sketching the triangle

We are given the angle of elevation, 70°, and the distance along the ground, 86 centimeters. Our unknown is a ladder leaning against the building. Buildings are erected vertically, so the unknown side length is the hypotenuse of the triangle.

We can then sketch this triangle out to visualize it (attachment).

Part 2: Determining what trigonometric ratio can solve the problem

Now, we need to refer to our three trigonometric ratios:

`sin = (opposite)/(hypotenuse)`

`cos = (adjacent)/(hypotenuse)`

`tan = (opposite)/(adjacent)`

Visualizing the sketched triangle, we can assign the three sides their terms in correspondence to the known angle -- this angle cannot be the right angle because the hypotenuse is opposite of it.

Therefore, we know our unknown side length is the hypotenuse of the triangle and because the other side is bordering the 70° angle, it is the adjacent side.

By assigning the sides, we can see that we need to use the trigonometric function that utilizes both the hypotenuse and the adjacent side to find the angle. This is the cosine function.

Part 3: Solving for the unknown variable

Now that we have determined what side we need to solve for and what trigonometric function we are going to use to do so, we just need to plug it all into the equation.

The cosine function is provided:
`cos( \alpha) = (adjacent)/(hypotenuse)`, where
`\alpha` is the angle. We just need to plug in our values and solve for our unknown side; the hypotenuse.

`cos (70) = (86 cm)/(x)`, where x is the unknown side/the hypotenuse.

`x * cos (70) = (86 cm)/(x) * x` Multiply by x on both sides of the equation to eliminate the denominator and make the unknown easier to solve for.

`(xcos (70))/(cos(70)) = (86 cm)/(cos(70))`, Evaluate the second fraction because the first one cancels down to just the unknown, x.

`(86)/(cos(70)) = 251.4`, round to one decimal place.

Your final answer is
`\boxed{x=251.4cm}`.