Trigonometry: The Ratios

Remember this? It’s the “Basic” Triangle used to in trigonometry. (See Above) Angle C is 90 degrees. It is not used in the ratios, except to determine the hypotenuse.

Trigonometry ratios

The Ratios are:

  • Sine, abbreviated “Sin” and pronounced “Sign”. It is the ratio between an angle’s (not C, which is the right angle) opposite side and the hypotenuse. Foe example, Sin(B) would be equal to the opposite (side b) over the hypotenuse (side c).
  • Cosine, abbreviated “Cos” and pronounced “Cosign”. It is the ratio between an angle’s (again, not C) adjacent side and the hypotenuse. For example, Cos(B) would be equal to the adjacent (Side a) over the hypotenuse, again side c.
  • Tangent, abbreviated “Tan”. It is the ratio between an angle’s opposite and adjacent sides. Tan(B) is equal to side b over side a.Together, these are remembered as SOHCAHTOA, for Sine-Opposite/Hypotenuse Cosine-Adjacent/Hypotenuse Tangent Opposite/Adjacent.

    Using the Ratios:

    Using the ratios is quite simple. First, determine what information you have and what you are trying to find. You need either the measure of either of the two leg angles (A or B) and at least one side, either a b or c. Then determine what you need. Then, simply set up the ratio, i.e. Sin(A)=a/c, plug the trig ratio into a calculator to find the decimal, and solve normally.

    An Example Problem:

    Lets say you are on a boat, the H.M.S. Numbaz. You see a lighthouse which you know is 70 feet tall and perfectly straight, but you need to know how far away you are so you don’t crash on the rocks. A visualization of the problem:

    trig ratios

    Darn Pretty Picture, eh? So you see that the perfectly-straight lighthouse forms a right triangle with a side opposite from you that is exactly 70 feet. So if you take your handy protractor thing and measure that the angle from level to the top of the lighthouse is 30 Degrees. You now know:


  • The angle by the ship is 30 degrees
  • The height of the lighthouse, which is opposite your angle, is 70 feet
  • You need to know the adjacent side to your ship’s angleNow you see we are using the Adjacent and Opposite sides. Remember the ratio for that? Tangent, of course. So set up the problem:

    Tan(X)=Opposite/Adjacent (Basic)

    Tan(30)=70/x (Insert Values)

    .577=70/x (Find Tan(30)

    .577x=70 (Multiply by X)

    x=121.31 (Divide by .577)

    And there you have it, you are a mere 121 feet from the shore.

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