697 7.3 Sum and Difference Identities (Modeling) Radio Tuners See Example 6. Let the energy stored in the inductor be given by L1t2 = 3 cos2 6,000,000t and let the energy stored in the capacitor be given by C1t2 = 3 sin2 6,000,000t, where t is time in seconds. The total energy E in the circuit is given by E1t2 = L1t2 + C1t2. 103. Graph L, C, and E in the window 30, 10-64 by 3-1, 44, with Xscl = 10-7 and Yscl = 1. Interpret the graph. 104. Make a table of values for L, C, and E starting at t = 0, incrementing by 10-7. Interpret the results. 105. Use a fundamental identity to derive a simplified expression for E1t2. 7.3 Sum and Difference Identities ■ Cosine Sum and Difference Identities ■ Cofunction Identities ■ Sine andTangent Sum and Difference Identities ■ Applications of the Sum and Difference Identities ■ Verifying an Identity Cosine Sum and Difference Identities Several examples presented earlier have shown that cos1A −B2 does not equal cos A −cos B. For example, if A = p 2 and B = 0, then cos1A - B2 = cos a p 2 - 0b = cos p 2 = 0, while cos A - cos B = cos p 2 - cos 0 = 0 - 1 = -1. To derive a formula for cos1A - B2, we start by locating angles A and B in standard position on a unit circle, with B6A. Let S and Q be the points where the terminal sides of angles A and B, respectively, intersect the circle. Let P be the point 11, 02, and locate point R on the unit circle so that angle POR equals the difference A - B. See Figure 4. O x y (cos(A – B), sin(A – B)) R B A A – B (cos A, sin A) S P (1, 0) Q (cos B, sin B) Figure 4 Because point Q is on the unit circle, the x-coordinate of Q is the cosine of angle B, while the y-coordinate of Q is the sine of angle B. Q has coordinates 1cos B, sin B2.
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