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Lesson 146: Intro to Trigonometry
Trigonometry is a vast field of study in math. In this and the next lesson, we will learn the basics, which should be all that is covered by your state's math exit exam.
We use trigonometry to help us find the measurements of angles and sides of right triangles where we are only given incomplete information. This lesson has some definitions that need to be memorized.
In a right triangle, we define the sine of an angle as the ratio of the length of the leg opposite that angle, to the length of the hypotenuse. Sine is often abbreviated as sin, and opposite and hypotenuse are often abbreviated as opp and hyp, respectively. We can say that sin = opp/hyp.
In a right triangle, we define the cosine of an angle as the ratio of the length of the leg adjacent that angle, to the length of the hypotenuse. Cosine of often abbreviated as cos, and adjacent is often abbreviated as adj. We can say that cos = adj/hyp.
In
a right triangle, we define the tangent of an angle
as the ratio of the length of the leg opposite that angle, to the
length of the leg adjacent that angle. Tangent is often
abbreviated as tan. We can say that
tan = opp/adj.
There is a special acronym that helps us remember these ratios: SOH-CAH-TOA. Practice pronouncing it the way it is written. From this, we are reminded that sin = opp/hyp, cos = adj/hyp, and tan = opp/adj. Be sure to memorize that special acronym. If it helps, write it down on your test paper the moment you begin your test.
In the above example, we can see that sin A = 3/5. cos A = 4/5. tan A = 3/4. Of course we could do the division to convert each of those ratios to a decimal. With this knowledge, we can use any of these three ratios to help us figure out the measure of angle A. Let's use the sine, but we could have used either of the other two ratios. sin A = 3/5 = 0.6. One method is to use a table of trigonometric values which will probably be provided on an exam, or in your textbook. In the table, you would see columns for the sin, cos, and tan of every angle measurement. It would be easy to see that in this case, angle A equals 37°, rounded to the nearest degree.
Another method would be to use your scientific calculator.
Each calculator requires that you enter this problem in a different
way. On some calculators, you'll need to enter
arcsin(0.6), which means, "give me the measure of the angle
whose sine equals 0.6". You'll need to know how to enter the arcsin command on your particular model of calculator. On
other calculators, you'll have to enter sin-1(0.6),
which means the same thing. Again, you'll have to know how to
enter that on your particular model of calculator. It might
involve pressing a SHIFT, INV, or 2ND key, or similar.
You can use these trigonometric ratios along with algebra to solve problems involving right triangles when you have incomplete information. For example, one of the sides or angles might be labeled x, and you'll need to find its value. As long as you have some known values to work with, you can usually set up an equation using the appropriate trigonometric ratio and the values that you do have. Then you can determine the unknown value.
In the next lesson you'll have some practice with this in the context of word problems. For now, though, it is essential that you memorize the trig ratios presented in this lesson.
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