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Tangent (SOH CAH TOA)

The Relationship of Rise over Run in Trigonometry

In trigonometry, the Tangent function (tan) is one of the three primary ratios used to analyze right-angled triangles. It is unique because it relates the two straight sides of the triangle (Opposite and Adjacent) without involving the diagonal Hypotenuse.

[Image of tangent ratio in right triangle]

Tangent is particularly famous because it directly relates to the concept of Slope or Gradient in algebra. It tells you how steep a line is—the "Rise" over the "Run."

1. The TOA Formula

The acronym TOA stands for:

  • Tangent
  • Opposite
  • Adjacent

This gives us the fundamental formula:

tan(θ) = Opposite / Adjacent

2. When to Use Tangent

You use the Tangent ratio when:

  • You know the Opposite and Adjacent sides and need to find the angle.
  • You know one leg of the triangle and an angle, and you need to find the other leg.
  • You do not know (and do not need to find) the Hypotenuse.

3. Example: Finding a Missing Side

Imagine you are standing 20 meters away from a tall tree. You look up at the top of the tree at an angle of elevation of 45°. How tall is the tree?

  1. Identify Sides:
    • Distance from tree (Adjacent) = 20m.
    • Height of tree (Opposite) = x.
  2. Choose Ratio: We have Opposite and Adjacent, so we use TOA.
  3. Equation:
    tan(45°) = x / 20
  4. Solve: Since tan(45°) is exactly 1:
    1 = x / 20
    x = 20 meters. The tree is 20 meters tall.

4. Example: Finding a Missing Angle

Suppose you have a ramp that rises 3 meters high over a horizontal distance of 4 meters. What is the angle of the ramp?

  1. Identify Sides: Opposite (Rise) = 3, Adjacent (Run) = 4.
  2. Equation:
    tan(θ) = 3 / 4 = 0.75
  3. Inverse Tangent: Use the tan-1 button on your calculator.
    θ = tan-1(0.75)
    θ ≈ 36.87°

5. Connection to Slope

In coordinate geometry, the slope of a line is defined as Change in Y / Change in X. Notice that in a right triangle drawn on a graph:

  • The Opposite side is the Change in Y.
  • The Adjacent side is the Change in X.

Therefore, Slope = tan(θ), where θ is the angle the line makes with the x-axis.

Conclusion

The Tangent (SOH CAH TOA) ratio is a versatile tool bridging geometry and algebra. Whether calculating the steepness of a hill or the height of a building from a distance, TOA provides the solution using simple division.