Applications of the Tangent Ratio

Tangent

In a right triangle, the tangent of angle $\theta$ is the ratio between the length of the opposite side and the adjacent side. This is very different from sine, which compares the opposite side to the hypotenuse, or cosine, which compares the adjacent side to the hypotenuse.

For example, if we have a right triangle with angle $\theta$, then:

The value of tangent changes with the angle. For example, $\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$.

Heights Distances Shadows and Slopes

Use the tangent ratio when a right triangle relates an angle to its opposite and adjacent sides. Typical quantities include:

1. Height of objects that are difficult to measure directly
2. Distance between two points that cannot be accessed
3. Length of an object's shadow
4. Slope of a surface

Measuring Height through Shadows

Suppose we want to measure the height of a vertical tree or building on level ground. When sunlight is treated as parallel, the object, its shadow, and a sun ray form a right triangle. We can then:

1. Measure the horizontal shadow length, the adjacent side.
2. Measure or obtain the sun's elevation angle $\theta$.
3. Use $\tan \theta$ to calculate the vertical height, the opposite side.

Methods for Calculating with Tangent

There are two ways we can use to solve problems using tangent. The first works from a known side and a known angle and finds the other side, the second works from two known sides and finds the angle.

Similar Triangle Comparison

We can use the principle of similar triangles to solve problems. If we have two triangles with the same shape (similar), then the ratio of their sides will be the same.

For example, if we have shadows from three objects of different heights (child, teenager, and adult), we can create the equation:

Let $h_1$, $h_2$, and $h_3$ be the heights of the child, teenager, and adult, respectively. Write $s_1$, $s_2$, and $s_3$ for their corresponding shadow lengths.

If a child and a teenager stand on the same level ground at the same time, the sunlight gives their height-shadow triangles the same acute angle. Suppose the child's height is $114 \text{ cm}$, the child's shadow is $200 \text{ cm}$, and the teenager's height is $148 \text{ cm}$. Then:

The measured reference triangle has $\tan \theta=114/200=0.57$, so its implied sun elevation is $\theta=\arctan(0.57)\approx29.68^\circ$.

Shadow Length from the Tangent Ratio

The tangent ratio relates the height to the shadow length:

When the sun elevation is $30^\circ$ and the teenager is $148 \text{ cm}$ tall, the tangent ratio gives the shadow length:

This result is slightly shorter than $259.65 \text{ cm}$ because $30^\circ$ is slightly larger than the measured reference angle $29.68^\circ$. A larger elevation angle produces a shorter shadow for the same height.