How to Use Right Angled Trigonometry

Right angled trig is useful when dealing with triangles and is a fundamental part of trigonometry in general. Usually, right trig is a student's first encounter with trig, and it can be a bit confusing at first. These steps will clarify the trig ratios and how they are used.


  1. 1
    Know the 6 trig ratios. You need to memorize the following:
    • sine
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      • abbreviated as sin
      • opposite/hypotenuse
    • cosine
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      • abbreviated as cos
      • adjacent/hypotenuse
    • tangent
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      • abbreviated as tan
      • opposite/adjacent
    • cosecant
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      • abbreviated as csc
      • hypotenuse/opposite
    • secant
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      • abbreviated as sec
      • hypotenuse/adjacent
    • cotangent
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      • abbreviated as cot
      • adjacent/opposite
  2. 2
    Discover the patterns. Don't worry if you're confused about what everything means right now and don't freak out about memorizing everything. It's not too hard if you know the patterns:
    • The abbreviations are always used when writing out the trig functions. You will never write out cotangent or secant. When you see the abbreviation, you should hear the name. Likewise, when you hear the name, you should hear the abbreviation. Notice that in every case except csc (cosecant), the abbreviation is the first three letters of the name. Csc is an exception because the first three letters are "cos" which is already used. So instead it's the first three consonants.
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    • You can remember the first three ratios by the following: Sohcahtoa. Just think of it as a name to remember. Make him an Aztec Chieftain if it helps you remember it, just make sure you remember how to spell it. It's basically the first letter of "sin opposite hypotenuse, cos adjacent hypotenuse, tan opposite adjacent" Notice that if you insert the word over between any two words that aren't trig ratios (i.e. adjacent and hypotenuse, not cos and adjacent), after the name of each trig function is its ratio.
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    • The last three are just reciprocals of the first three (not inverses). Remember that anything without a prefix "co" has a reciprocal with the prefix, and anything with a prefix "co" has a reciprocal function without the prefix. Therefore, the csc, sec, and cot trig ratios are the reciprocals of the sin, cos, and tan ratios respectively. For example, cot's ratio is adjacent over opposite.
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    Know the parts of the triangle. You probably know what the hypotenuse is at this point, but you might be a bit confused about the opposite and adjacent sides. Look at the following diagram: These sides are correct when you are using angle C. If you wanted to use angle A, the words opposite and adjacent would be flipped in the diagram.
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    Understand what the trig ratios are and when they are used. When right angled trig was first discovered, it was realized that when you have two right triangles that are similar (meaning the angle measures are the same), if you divide one side by another and do the same with the corresponding sides of the other triangle, you would get the same values. The trig functions were then developed so that you could find the ratio for any given angle. The side names were also given to make it easier to determine which angles to use. You can use the trig ratios to determine a side measure given one of the sides and an angle, or you can use them to determine an angle measure given two side lengths.
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    Figure out what you want to solve. Mark the unknown value with an "x." This will help you set up the equation later. Also make sure you have enough information to solve the triangle. You need either an angle and a side or all three sides.
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    Set up the ratio. Label the opposite side, adjacent side, and hypotenuse with respect to the marked angle (it doesn't matter if the mark is a number or an "x" from the previous step). Then write down which sides you either know or want to find. Without considering csc, sec, or cot, determine which ratio involves both of the sides you wrote down. You shouldn't use the reciprocal ratios because there's usually no calculator button for them. Even if you could, there will almost never be a situation where you will need to use them to solve a right triangle. Once you know which ratio to use, write it down, followed by the value or variable of the triangle. Then write an equals sign followed by the sides the ratio encompasses (still in terms of opposite, adjacent, and hypotenuse). Rewrite the equation, filling in the side lengths/variable in the ratio.
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    Solve the equation. If the variable is outside the trig function (that means you were solving for a side), then just solve for the exact value of x then plug the expression into your calculator for a decimal approximation of the side length. If your variable is inside the argument of the trig function (that means you were solving for an angle), then you should simplify the expression on the right then plug the inverse of that trig function followed by the expression. For example, if your equation was sin(x)=2/4, then you would simplify the right hand side to get 1/2, then punch into your calculator "sin-1" (it's all one button, usually the second option for the trig function you want) followed by 1/2. Make sure when you do calculations, you are in the correct mode. If you want degrees, put your calculator in degree mode, if you want radians, put your calculator in radian mode, if you don't know what degrees or radians, put your calculator in degree mode. The value of x is the value of the side or angle you were trying to find.


  • The values of sin and cos are always between -1 and 1, but tangent can be any number. If you get an error on the inverse trig function, your value is probably too big or too small. Check your ratio and try again. A common error is to flip the sides in the ratio, such as using hypotenuse/opposite for sin.
  • sin-1 is not the same thing as csc, cos-1 is not the same thing as sec, and tan-1 is not the same thing as cot. The first is the inverse trig function, meaning that if you put in the value of a ratio, it will give you the corresponding angle, the second is the reciprocal function meaning the ratio is reversed.

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