From our investigation of the signs for various quadrants, negative angles from the 1 st quadrant will be in the 4th quadrant. The second method of finding the formula for difference angles uses the sum formula already obtained, but makes B negative. In reasoning similar to that which was used for the sum angles, presented here somewhat abbreviated, are the sine and cosine formulas: First, use a geometric construction, such as the one that was used for sum angles, reversing it so that (A - B) is the angle B subtracted from the angle A. Now, you have two ways to obtain formulas for difference angles. You will begin to see a pattern to the way these trigonometric ratios for angles vary. The hypotenuse is always the rotating vector (r). What was called the adjacent is always the horizontal (x). As you move into bigger angles in the remaining quadrants, the opposite side is always the vertical (y). In the first quadrant, the sides were defined in the ratios for sine, cosine, and tangent. Also how the equivalent angle in the first quadrant "switches" as the vector passes from one quadrant to the next. Here, the signs of the three ratios have been tabulated for the four quadrants. So, the sign of the ratios can be figures for the various quadrants. So, the sine of an angle is y/r, the cosine x/r, and the tangent y/x. Horizontal elements are x: positive to the right, negative to the left.
Progressively larger angles are defined by a rotating vector, starting from zero and rotating counterclockwise. Now, use this method for plotting graphs. 0-90 degrees is the 1st quadrant, 90-180 the 2nd, 180-270 the 3rd, and 270-360 the 4th.ĭrawing in lines to represent the quadrant boundaries, with 0 or 360 horizontal to the right, 90 vertical up, 180 horizontal to the left, and 270 vertical down. Since the circle is commonly divided into 360 degrees, the quadrants are named by 90-degree segments. To simplify classification of angles according to size, they are divided into quadrants.Ī quadrant is a quarter of a circle. Other triangles with obtuse angles (over 90 degrees) might go over 180 degrees in later problems. So far, ratios of acute angles (between 0 and 90 degrees) have been considered. If you use your pocket calculator for evaluation, it will probably make no difference whether you simplify the expressions first or just plow through it! Everything depends on the calculator: some do make a difference, some don't! Sin(A + B) is the two parts of the opposite - all divided by the hypotenuse (9). Since the side marked "opposite" (7) is in both the numerator and denominator when cos A and sin B are multiplied together, cos A sin B is the top part of the original opposite - for (A + B) - divided by the main hypotenuse (8). The opposite over the main hypotenuse (7) is sin B. The top part of the opposite (6), over the longest of that shaded triangle, is cos A. Similar right triangles with an angle A show that the top angle, marked A, also equals the original A. The shaded angle is A, because the line on its top side is parallel to the base line. The middle line is in both the numerator and denominator, so each cancels and leaves the lower part of the opposite over the hypotenuse (4). The line between the two angles divided by the hypotenuse (3) is cos B. The lower part, divided by the line between the angles (2), is sin A. The big angle, (A + B), consists of two smaller ones, A and B, The construction (1) shows that the opposite side is made of two parts. The easiest way to find sin(A + B), uses the geometrical construction shown here. Wanted sine, cosine, or tangent, of whole angle (A + B) A sine or cosine can never be greater than 1, so a value of 1.2071 must be wrong. The most that the numerator can be is equal to the denominator.
Why? the ratio has the hypotenuse as its denominator. You know that no sine (or cosine) can be more than 1. The formula for what sin(A + B) does equal.įirst to show that removing parentheses doesn't "work." Here: make A 30 degrees and B 45 degrees. It doesn't work like removing the parentheses in algebra.Ģ. Sin(A + B) is not equal to sin A + sin B. In this chapter you need to learn two things:ġ. As the examples showed, sometimes we need angles other than 0, 30, 45, 60, and 90 degrees.