Trigonometry and Sine Study Guide
Trigonometry and Sine
In this lesson, we introduce the sine function. We show how to evaluate the sine of an angle in a right triangle. Often, this is as easy as dividing the length of one side of the triangle by another. Sometimes we need to use the Pythagorean theorem to find one of the lengths.
The first trigonometric function is sine. The domain of this function consists of all angles x. There is no formula for sine that can be computed directly using x. Instead, we must follow a process. For now, we will use angles between 0° and 90°.
Step one: Start with an angle of measure x (Figure 5.1).
Step two: Make a right triangle with angle x (Figure 5.2).
Step three: Measure the length H of the triangle's hypotenuse and the length O of the side opposite the angle x (Figure 5.3). (The opposite side is one that isn't part of angle x.)
Step four: The sine of the angle x is the ratio of the two sides:
The length of the hypotenuse H and opposite side O will depend on the size of the triangle. In step two, we could have chosen a smaller or larger right triangle with angle x, as illustrated in Figure 5.4.
However, because the angles of a triangle add up to 180° (or π radians), the third angles of these triangles must all measure 90° – x degrees (or – x radians). Triangles with all the same angles are similar; thus, each one is a scale multiple of the original. Suppose the smaller triangle has scale factor k, and the larger triangle has scale factor K (see Figure 5.5).
If we use the small triangle, then
sin(x) = =
If we use the large triangle, then
sin(x) = =
In other words, the sine of the angle does not depend on the size of the triangle used.
It is tedious to actually draw a right triangle with the correct angle x, and then to measure the lengths of the sides. There are only a few "nice angles" for which this can be done precisely, which will be discussed in Lesson 9. Calculators can be used to estimate the sine of any angle; this will be covered in Lesson 12. For the time being, it is far easier to start with a triangle and then find the sine of each of its angles.
Find the sine of the angle x in Figure 5.6.
We might not know the exact measurement of angle x, in either degrees or radians, but we have everything we need to find its sine. The hypotenuse of this right triangle is H = 10 inches. The side that is opposite angle x is O = 6 inches. Thus,
sin(x) = =
Notice that the units used to measure the lengths will always cancel out like this. From now on, we will not worry about the feet, inches, or whatever is used to measure the lengths. The numbers that come out of the sine function have no particular units. They are merely ratios.
Find the sine of angle θ from Figure 5.7.
Here, the hypotenuse is H = 14 and the side opposite angle θ is O = 9, so
sin(θ) = =
Find the sine of the angle x illustrated in Figure 5.8.
Here, the hypotenuse is H = √34. The opposite side (the one that doesn't touch angle x) is O = 5. Thus,
We can find the sine of an angle even if only two sides of the triangle are known. If the two sides are the hypotenuse and opposite side, then we plug them into sin(x) = directly. Otherwise, we will need to use the Pythagorean theorem to find the length of the missing side.
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