693 7.2 Verifying Trigonometric Identities There are usually several ways to verify a given identity. Another way to begin verifying the identity in Example 5 is to work on the left as follows. sec a + tan a sec a - tan a Left side of given equation in Example 5 = 1 cos a + sin a cos a 1 cos a - sin a cos a Fundamental identities = 1 + sin a cos a 1 - sin a cos a Add and subtract fractions. = 1 + sin a cos a , 1 - sin a cos a Simplify the complex fraction. Use the definition of division. = 1 + sin a cos a # cos a 1 - sin a Multiply by the reciprocal. = 1 + sin a 1 - sin a Multiply and write in lowest terms. Compare this with the result shown in Example 5 for the right side to see that the two sides indeed agree. EXAMPLE 6 Applying a Pythagorean Identity to Electronics Tuners in radios select a radio station by adjusting the frequency. A tuner may contain an inductor L and a capacitor C, as illustrated in Figure 3. The energy stored in the inductor at time t is given by L1t2 = k sin2 2pFt and the energy stored in the capacitor is given by C1t2 = k cos2 2pFt, where F is the frequency of the radio station and k is a constant. The total energy E in the circuit is given by E1t2 = L1t2 + C1t2. Show that E is a constant function. (Data from Weidner, R., and R. Sells, Elementary Classical Physics, Vol. 2, Allyn & Bacon.) SOLUTION E1t2 = L1t2 + C1t2 Given equation = k sin2 2pFt + k cos2 2pFt Substitute. = k3sin2 2pFt + cos2 2pFt4 Factor out k. = k112 sin2 u + cos2 u = 1 1Here u = 2pFt.2 = k Identity property Because k is a constant, E1t2 is a constant function. S Now Try Exercise 105. C L An Inductor and a Capacitor Figure 3
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