How to calculate the height of a triangle using trigonometry

Answer:

As for the trig functions (sine, cosine, and tangent), you can use the angle you are interested in, θ, and the lengths of the sides of the triangle based off of angle θ to find an unknown length in the triangle.

sinθ =

cosθ =

tanθ =

Essentially, if you know the measure of one of the angles in the triangle (not including 90°, and the measure of at least one side length, you can select the proper trig function that will allow you to find the unknown length, or height, of the flagpole.

Ex:

Let's say you are standing 18 feet away from the base of the flagpole, and you know (somehow) that an imaginary line is formed from the top of the flagpole to your feet (this imaginary, slanted line is our hypotenuse) and that the angle between that imaginary line and ground is 30°.  

We will use 30° as our θ. In this situation, the unknown side length (the height of the flagpole) is opposite from this angle we have chosen. So it is our "opposite". We know that the distance between you and the base of the flagpole is 18 feet. Because this is not that slanted, imaginary line we formed before, this is not the hypotenuse. This is a side length that is adjacent to where we are standing, where our angle θ is. So 18 feet is our "adjacent".

The trig function that includes both "opposite" and "adjacent" is the tangent function, or tanθ = .

We have selected our trig function:

tanθ =

Substitute in the known values. θ = 30° adjacent = 18

tan(30°) =

Multiply both sides of the equation by 18, to remove 18 from the right side of the equation:

(18)·tan(30°) = ·18

The 18's cancel because that side becomes and a number divided by itself is equal to 1. And multiplying the opposite by 1 still equals the opposite.

So we are left with:

(18) · tan(30°) = opposite

At this point, simply input tan(30°) into your calculator (make sure your calculator is in degrees, not radians, by checking MODE), then multiply the result by 18.

This should equal about 10.3 = opposite. This means that 10.3 feet is the height of the flagpole.