Basic Trigonometric Ratios: Solving Triangles

• List the values of sin(α), cos(α), sin(β), and tan(β) for the triangle below, accurate to three decimal places:

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Once you've memorized the trig ratios, you can start using them to find other values. To do this, you'll likely need to use a calculator. If your calculator does not have keys or menu options with "SIN", "COS", and "TAN", then now is the time to upgrade. Make sure you know how to use the calculator, too; the owners manual should have clear instructions.

(Yes, I know you've got a full-featured calculator on your phone. You also have texting and messaging apps, which could be used for, shall we say, "unsanctioned assistance" on tests. So you'll probably need a stand-alone calculator, too. Don't assume that your instructor will be chill with your using your mobile during an exam.)

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With the six trig ratios for any right triangle, you can now begin to "solve" right triangles.

What does it mean to "solve" a right triangle?

"Solving a right triangle" refers to the process of taking partial information (such as the lengths of one of the triangle's sides, and the measure of one of the non-right angles), plugging this information into one or another of the six trigonometric ratios to create equations, and then solving these equations (often with the help of a calculator) to find the values of the missing side lengths or angle measures.

• In the triangle shown below, find the value of x, accurate to three decimal places.

Note: If your calculator displayed a value of 71.19813587... instead of 190.047286..., then check the "mode"; your calculator is set to "radians" rather than to "degrees". (You'll learn about radians later.)

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• For the triangle shown below, find the value of y, accurate to four decimal places.

• Find the values indicated by the letters in the diagram below. Give each answer correct to the nearest whole number.

If you use the pre-rounded value of 60 tan(30°), then your value for will be slightly different; it'll be more like 24.2559006.... This is why I discourage such severe rounding as "to the nearest whole number"; it can throw your values off by a lot.

Keep this in mind when you're doing your own work, especially for word problems. To get a usefully accurate answer, keeping as many decimal places as reasonably feasible can make a big difference by the end of the exercise.

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Note: Since the sine and cosine ratios involve dividing a leg (one of the shorter two sides) by the hypotenuse (which is always the longest side), the ratio values will never be more than 1, because (some number) / (a bigger number) is always going to be smaller than 1. But you can have right triangles that are really wide and short or really tall and skinny, so "opposite" and "adjacent" can have *very* different values. This tells you that the tangent ratio, being (opposite) / (adjacent), can have very large and very small values, depending on the triangle.

So if you get a value for sine or cosine that is larger than 1, you should go check your work. On the other hand, if you get a huge value for tangent for some wonky triangle, you shouldn't be surprised.