Under our general definitions of trigonometry functions, we can expand these formulas to hold for all angles. For example, we assume that α and β are two angles with 0◦ ≤ α, β < 90◦ and α +β > 90◦. We set α = 90◦ − α and β = 90◦ − β. Then α and β are angles between 0◦ and 90◦ with a sum of less than 90◦. By the addition formulas we developed earlier, we have cos(α + β) = cos 180◦ − (α + β ) = − cos(α + β ) = − cos α cos β + sin α sin β = − cos(90◦ − α ) cos(90◦ − β ) + sin(90◦ − α )sin(90◦ − β ) = − sin α sin β + cos α cos β = cos α cos β − sin α sin β. Thus, the addition formula for the cosine function holds for angles α and β with 0◦ ≤ α, β < 90◦ and α + β > 90◦. Similarly, we can show that all the addition formulas developed earlier hold for all angles α and β. Furthermore, we can prove the subtraction formulas sin(α − β) = sin α cos β − cos α sin β, cos(α − β) = cos α cos β + sin α sin β, tan(α − β) = tan α − tan β 1 + tan α tan β . We call these, collectively, the addition and subtraction formulas. Various forms of the double-angle and triple-angle formulas are special cases of the addition and subtraction formulas. Double-angle formulas lead to various forms of the halfangle formulas. It is also not difficult to check the product-to-sum formulas by the addition and subtraction formulas. We leave this to the reader. For angles α and 1. Trigonometric Fundamentals 13 β, by the addition and subtraction formulas, we also have sin α + sin β = sin α + β 2 + α − β 2 + sin α + β 2 − α − β 2 = sin α + β 2 cos α − β 2 + cos α + β 2 sin α − β 2 + sin α + β 2 cos α − β 2 − cos α + β 2 sin α − β 2 = 2 sin α + β 2 cos α − β 2 , which is one of the sum-to-product formulas. Similarly, we obtain various forms of the sum-to-product formulas and difference-to-product formulas. Example 1.4. Let a and b be nonnegative real numbers. (a) Prove that there is a real number x such that sin x + a cos x = b if and only if a2 − b2 + 1 ≥ 0. (b) If sin x + a cos x = b, express |a sin x − cos x| in terms of a and b. We set the units of the x axis to be degrees. The graph of y = sin x looks like a wave, as shown in Figure 1.12. (This is only part of the graph. The graph extends infinitely in both directions along the x axis.) For example, the point A = (1, x◦) corresponds to the point A1 = (x,sin x) on the curve y = sin x. If two points B1 and C1 are 360 from each other in the x direction, then they have the same y value, and they correspond to the same point B = C on the unit circle. (This is the correspondence of the identity sin(x◦+360◦) = sin x◦.)Also, the graph is symmetric about line x = 90. (This

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2 (x - 1) The triangles are equal in area. Calculate the value of x. Expand fully and simplify ac (x - 1) 2. Solve the equation X-5

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