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Trigonometry

Applications of Tan θ Trigonometric Comparison

Nabil Akbarazzima Fatih

Mathematics

What is Tangent?

In a right triangle, the tangent of angle θ 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.

Visualization of Tangent (tan θ)
Slide the slider to see how tangent changes as the angle changes.
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°

For example, if we have a right triangle with angle θ, then:

tan⁡θ=opposite sideadjacent side\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}

The value of tangent changes according to the angle. For example, tan⁡30°=0.57 or 13\tan 30° = 0.57 \text{ or } \frac{1}{\sqrt{3}}.

Applications of Tangent in Daily Life

The tangent trigonometric ratio is very useful for solving various real-life problems, especially when we want to calculate:

  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

Imagine we want to measure the height of a tree, building, or other tall object. We can use the tangent ratio with the following steps:

  1. Measure the length of the object's shadow (adjacent side)
  2. Measure or know the sun's angle of elevation (θ)
  3. Use the tan θ formula to calculate the object's height (opposite side)
object height=shadow length×tan⁡θ\text{object height} = \text{shadow length} \times \tan \theta
Application of Tangent in Measurement
Example of a triangle formed when measuring height with shadows.
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°

Methods for Calculating with Tangent

There are two ways we can use to solve problems using tangent:

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:

child’s heightchild’s shadow length=teenager’s heightteenager’s shadow length=adult’s heightadult’s shadow length\frac{\text{child's height}}{\text{child's shadow length}} = \frac{\text{teenager's height}}{\text{teenager's shadow length}} = \frac{\text{adult's height}}{\text{adult's shadow length}}

With this equation, if we know the child's height (e.g., 114 cm), their shadow length (200 cm), and the teenager's height (148 cm), we can calculate the teenager's shadow length:

114 cm200 cm=148 cmx cm\frac{114 \text{ cm}}{200 \text{ cm}} = \frac{148 \text{ cm}}{x \text{ cm}}
x cm=148 cm×200 cm114 cm=29,600 cm114 cm=259.65 cmx \text{ cm} = \frac{148 \text{ cm} \times 200 \text{ cm}}{114 \text{ cm}} = \frac{29,600 \text{ cm}}{114 \text{ cm}} = 259.65 \text{ cm}

Using the Tangent Formula

Another more direct way is to use the tangent trigonometric ratio. We know that:

tan⁡θ=opposite sideadjacent side\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}

For example, if we have a sun elevation angle of 30° (tan 30° = 0.57) and want to calculate the shadow length of a teenager with height 148 cm:

tan⁡θ=teenager’s heightteenager’s shadow length\tan \theta = \frac{\text{teenager's height}}{\text{teenager's shadow length}}
tan⁡30°=148 cmx cm\tan 30° = \frac{148 \text{ cm}}{x \text{ cm}}
0.57=148 cmx cm0.57 = \frac{148 \text{ cm}}{x \text{ cm}}
x cm=148 cm0.57=259.65 cmx \text{ cm} = \frac{148 \text{ cm}}{0.57} = 259.65 \text{ cm}
30 Degree Angle with Tangent 0.57
Visualization of a triangle with a 30° angle as in the calculation example.
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°
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