Calculator Trigonometry
This lesson contains the essence of trigonometry but without the explanations. With a calculator, you can find any side of a right triangle if you know one side and an angle. The only disadvantage to using a calculator is that you get only approximate answers. However, several decimal places of accuracy will be close enough in any practical situation. If you don't have a calculator, it's a good idea to get one that has buttons that say SIN, COS, and TAN somewhere up near the top. A graphing calculator is unnecessary. A perfectly decent scientific calculator can be bought for just more than $10.
Suppose we have the right triangle in Figure 12.1. The hypotenuse has length H, one of the angles has measure θ, the side adjacent to θ has length A, and the opposite side has length O.
The Pythagorean theorem states that
A^{2} + O^{2} = H^{2}
For example, if A = 8 inches long and O = 5 inches long, then
H = √82 + 52 = √64 + 25 = √89 ≈ 9.43398 inches long
On an inexpensive calculator, you estimate √89 by hitting "8," then "9," and then the square root button "√" or "√x." On fancier calculators, like graphing calculators, you have to hit the square root button first, then type "89," and then close up with a right parenthesis ")" so that the screen reads "√(89) before hitting "Enter."
If you had any other right triangle with angle θ, then it would be a scaled version (either shrunk or enlarged) with each side some number K times larger, as shown in Figure 12.2.
For example, suppose we had a 30°60°90° triangle with a hypotenuse of length 20 inches. It would have to be a scale multiple K of the classic 30°60°90° triangle, shown in Figure 12.3.
Each side of the larger triangle is exactly 20 times bigger.
The essence of trigonometry is that the ratio of one side divided by another in Figure 12.1 depends only on the angle θ. These ratios are the trigonometric values of the angle θ, called sine, cosine, tangent, secant, cosecant, and cotangent.
To evaluate these ratios for a given angle θ, all you need is a calculator. The most important detail of all is to make sure that the calculator is set to "DEG" if the angle θ is given in degrees, like 34° or 14.2°. If θ is measured in radians, like or , then the calculator needs to be set to "RAD." Inexpensive calculators have a "DEG" button, which cycles between "DEG,""RAD," and "GRAD" (a type of angle measurement where a right angle measures 100 gradians). Usually one of these three is displayed in small letters somewhere on the screen. Fancier calculators sometimes require you to adjust between degrees and radians on an options or preferences menu. Most calculators start in degrees mode, but don't assume this.
Tip
If you don't like radians, just convert to degrees by multiplying by .
On an inexpensive calculator, type "30" and then hit the "SIN" button. If the screen reads 0.5, then the calculator is in degrees mode. If the screen reads –0.988031624, then the calculator is in radians mode. On a fancier calculator, you usually have to hit the "SIN" button first, type "30," then close the right parentheses ")" and hit "Enter."

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