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Right Triangle Trig

Solving Triangles with SOH CAH TOA

Right Triangle Trig is the study of the relationships between the sides and angles of right-angled triangles. A right triangle is any triangle that contains a 90-degree angle. Because the angles and sides are linked in predictable ways, we can use specific ratios to find missing information.

This branch of math is essential for construction, engineering, and navigation. If you know one side and one angle, finding the rest is easy using three magic words: SOH CAH TOA.

1. Labeling the Triangle

Before doing any math, you must label the sides of your triangle relative to the angle you are interested in (often called Theta, θ).

  • Hypotenuse: The longest side. It is always opposite the right angle (90°).
  • Opposite: The side directly across from the angle θ. It does not touch the angle.
  • Adjacent: The side next to the angle θ (that is not the hypotenuse).

2. The SOH CAH TOA Ratios

These three acronyms help you remember the definitions of the sine, cosine, and tangent functions.

SOH (Sine)

Sin(θ) = Opposite / Hypotenuse

CAH (Cosine)

Cos(θ) = Adjacent / Hypotenuse

TOA (Tangent)

Tan(θ) = Opposite / Adjacent

3. Finding a Missing Side

If you know one angle and one side length, you can calculate the length of any other side.

Example: A ladder leans against a wall making an angle of 60° with the ground. The ladder is 10 meters long (Hypotenuse). How high up the wall does it reach (Opposite)?

  1. Identify the sides: We have Hypotenuse (10) and want Opposite (x).
  2. Choose the ratio: "Opposite" and "Hypotenuse" means we use SOH (Sine).
  3. Set up the equation:
    sin(60°) = x / 10
  4. Solve:
    0.866 = x / 10
    x = 10 × 0.866 = 8.66 meters

4. Finding a Missing Angle (Inverse Trig)

If you know two sides but do not know the angle, you use the Inverse Trigonometric Functions. These are usually written as sin-1, cos-1, and tan-1.

Example: A triangle has an Opposite side of 5 and an Adjacent side of 5. What is the angle θ?

  1. Identify the sides: Opposite (5) and Adjacent (5).
  2. Choose the ratio: "Opposite" and "Adjacent" means we use TOA (Tangent).
  3. Set up the equation:
    tan(θ) = 5 / 5 = 1
  4. Use the inverse function:
    θ = tan-1(1)
    θ = 45°

5. Angles of Elevation and Depression

Right Triangle Trig is frequently used in word problems involving line of sight.

[Image of angle of elevation and depression diagram]
  • Angle of Elevation: The angle measured upwards from the horizontal (looking up at a bird).
  • Angle of Depression: The angle measured downwards from the horizontal (looking down from a cliff).

Conclusion

Right Triangle Trig gives you the power to measure the unreachable. By correctly labeling your triangle and choosing the right ratio from SOH CAH TOA, you can solve for any unknown side or angle with precision.