The trigonometry of right triangles is of great help in calculating the measures of the elements that characterize a triangle and is, in general, a fundamental part of trigonometry. Usually, a student's first encounter with trigonometry occurs with the right triangle, and it is possible that, at first, it is confusing. These steps will shed some light on trigonometric functions and how they are employed.
Steps
Step 1. Know the 6 trigonometric functions
You must memorize the following:
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otherwise
Use Right Angled Trigonometry Step 1Bullet1 - abbreviated to "sin"
- opposite side / hypotenuse
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cosine
Use Right Angled Trigonometry Step 1Bullet2 - abbreviated to "cos"
- adjacent side / hypotenuse
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tangent
Use Right Angled Trigonometry Step 1Bullet3 - abbreviated to "tan"
- opposite side / adjacent side
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cosecant
Use Right Angled Trigonometry Step 1Bullet4 - abbreviated to "csc"
- hypotenuse / opposite side
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secant
Use Right Angled Trigonometry Step 1Bullet5 - abbreviated to "sec"
- hypotenuse / adjacent side
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cotangent
Use Right Angled Trigonometry Step 1Bullet6 - abbreviated to "cot"
- adjacent / opposite side
Step 2. Locate the patterns
If you are currently confused by the meaning of each word, don't worry, and don't fret trying to memorize everything. If you know the patterns, it's not too difficult:
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When writing trigonometric functions, abbreviations are always used. You will never write "cotangent" or "secant" in full. Seeing the abbreviation, you should hear the full name. Likewise, when you hear the full name, you should see the abbreviation. Note that, in all cases, with the exception of csc (cosecant), the abbreviation consists of the first three letters of the name. Csc is an exception because the first three letters, "cos", already serve to indicate the cosine; therefore, in this case, the first three consonants are used.
Use Right Angled Trigonometry Step 2Bullet1 -
You can remember the first three functions by memorizing the word "Soicaitoa". It's just a name you need to help you remember; if it helps, pretend it's that of an Aztec chieftain, but make sure you remember how to spell it. Basically, it's just an acronym for " sin orpost thepotenusa, cos todiacente thepotenusa, tan orpost todiacente. Note that if you insert the symbol of the division between two words that indicate the sides (for example, adjacent and hypotenuse, not so and adjacent), you get the ratio that determines the function.
Use Right Angled Trigonometry Step 2Bullet2 -
The last three functions are the reciprocal of the first three (not the inverse). Remember that any function without the prefix "co" is the reciprocal of the one with the prefix, and vice versa. Consequently, the functions csc, sec, and cot are the reciprocals of sin, cos, and tan, respectively. For example, the cot ratio is adjacent / opposite.
Use Right Angled Trigonometry Step 2Bullet3
Use Right Angled Trigonometry Step 3 Step 3. Know the elements of the triangle
At this point, you probably already know what the hypotenuse is, but you may be a little confused about the opposite and adjacent sides. Look at the diagram above: the names of these sides are right if you are using angle C. If you wanted to use angle A instead, the words "opposite" and "adjacent" in the diagram should be swapped.
Use Right Angled Trigonometry Step 4 Step 4. Understand what trigonometric functions are and when they are used
When the trigonometry of the right triangle was first discovered, it was understood that, given two similar right triangles (that is, whose angles are the same size), if you divide one side by another and do the same with the corresponding sides of the other triangle, you get the same values. Trigonometric functions were then developed so that the ratio for any given angle could be found. The sides were also given names, in order to more easily determine which angles to use. You can use trigonometric functions to determine the measurement of a side from one side and an angle, or you can use them to determine the measurement of an angle from the length of two sides.
Use Right Angled Trigonometry Step 5 Step 5. Understand what you need to solve
Identify the unknown value with an "x". This will help you set up the equation later on. Also make sure you have enough information to solve the triangle. You need the measurement of one corner and one side, or that of all three sides.
Use Right Angled Trigonometry Step 6 Step 6. Set up the report
Mark the opposite side, the adjacent side and the hypotenuse in relation to the marked angle (it does not matter if the sign is a number or an "x", as indicated in the previous step). Then take note of which sides you know or want to discover. Regardless of csc, sec, or cot, determine which relationship involves both sides you noted. You shouldn't use reciprocal functions, as calculators usually don't have a reciprocating button. But even if you could, there will almost never be a situation where you have to use them to solve a right triangle. After figuring out which function to use, write it down, followed by the value or variable of the triangle. Then write an "equal" sign followed by the sides included in the function (always in terms of opposite, adjacent and hypotenuse). Rewrite the equation, entering the length or variable of the sides contained in the function.
Use Right Angled Trigonometry Step 7 Step 7. Solve the equation
If the variable is outside the trig function (i.e. if you are solving a side), solve for the exact value of x, then enter the expression in the calculator to get a decimal approximation of the side length. If, on the other hand, the variable is inside the trig function (i.e. you are solving an angle), you should simplify the expression on the right, then enter the inverse of that trig function, followed by the expression. For example, if your equation is sin (x) = 2/4, simplify the term to the right to get 1/2, then type "sin-1"(this is just a single button, usually the second option of the trig function you want), followed by 1/2. Make sure you are in the correct mode when doing the calculations. If you want to get the angle in sexagesimal degrees, set the calculator in this mode; if you want to obtain it in radians, set it in radian mode; if you do not know how it is configured, set it in sexagesimal degrees. The value of x corresponds to the value of the side or angle you are interested in obtaining.
Advice
- The values of sin and cos are always between -1 and 1, but that of the tangent can be represented by any number. If you make a mistake using the inverse trig function, the value you get will likely be too large or too small. Check the report and try again. A common mistake is to swap sides in the relationship, such as using the hypotenuse / opposite side for the sin.
- sin-1 it is not the same as csc, cos-1 does not match sec, and tan-1 it is not the same as cot. The first is the inverse trig function (which means that if you enter the value of a ratio, you will get the corresponding angle), while the second is the reciprocal function (the ratio is inverted).
