Trigonometric Values of Angles Study Guide
Trigonometric Values of Angles
Some very interesting and important functions are formed by dividing the length of one side of a right triangle by the length of another side. These functions are called trigonometric because they come from the geometry of a triangle. The domain consists of the measures x of angles. Let H represent the length of the hypotenuse, A represent the length of the side adjacent to the angle x, and the letter O represent the length of the side opposite (away) from the angle x. A right triangle with angle x is depicted in Figure 4.1.
Some people remember the first three trigonometric functions by saying "Oliver Had A Heap Of Apples" to remember the , , and of sin(x), cos(x), and tan(x). Others say SOA CAH TOA to remember sin(x) = , cos(x) = , and tan(x) = .
The six trigonometric functions, sine (abbreviated sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot), are defined for each angle x by dividing the following sides:
The first thing to notice is that all of the functions can be obtained from just sin(x) and cos(x) using the following trigonometric identities.
Thus, all of the trigonometric functions can be evaluated for an angle x if the sin (x) and cos(x) are known.
The next thing to notice is that the Pythagorean theorem, which, stated in terms of the sides O, A, and H, is O2 + A2 = H2. And, if we divide everything by H2, we get the following:
Thus, (sin(x))2 + (cos(x))2 = 1. To save on parentheses, we often write this as sin2(x) + cos2(x) = 1. Because no particular value of x was used in the calculations, this is true for every value of x.
Drawing triangles and measuring their sides is an impractical and inaccurate method to calculate the values of trigonometric functions. Most people use calculators instead. Although, when using a calculator, it is very important to make sure that it is set to the same format for measuring angles that you are already using: that is, degrees or radians.
There are 360 degrees in a circle, possibly because ancient peoples thought that there were 360 days in a year. As the earth went around the sun, the position of the sun against the background stars moved one degree every day. The 2π radians in a circle correspond to the distance around a circle of radius 1. Because radians already correspond to a distance, there is no need for conversions when calculating with radians. Mathematicians thus use radians almost exclusively.
- To convert from degrees to radians, multiply by
- To convert from radians to degrees, multiply by
To convert from degrees to radians, multiply by To convert from radians to degrees, multiply by
Convert 45° into radians.
Convert radians into degrees.
Trigonometric Values of Nice Angles
There are a few nice angles for which the trigonometric functions can be easily calculated. If °, then the two legs of the triangle are equal. If the hypotenuse is H = 1, then we have what you can see in Figure 4.2.
By the Pythagorean theorem, A2 + A2 = 1, so 2A2 = 1 and . This means that O = A = . If we rationalize the denominator, we get . Thus:
Another nice angle is x = 60° = , because it is found in equilateral triangles such as those seen in Figure 4.3. This triangle can be cut in half to form the triangle shown in Figure 4.4.
By the Pythagorean theorem, , so . Thus, . This means that:
We can flip that last triangle around to calculate the trigonometric functions for the other angle x = 30° = (see Figure 4.5).
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