Using Identities to Find Exact Trig Values Exercise 3
Exercises
Solution Method: Use the Double Angle Identities to find the exact value of \(10\sin{75^\circ}\cos{75}^\circ\)
The Double Angle Identities are obtained from the sum identities in the case where the values being summed are the same. As you can see, there are multiple equivalencies for \(\cos{2x}\) due to the Pythagorean Identity.
Double Angle Identities \[\begin{aligned} \sin{2x}&=2\sin{x}\cos{x}\\ \cos{2x}&=\cos^2{x}-\sin^2{x} \\ \cos{2x}&=1-2\sin^2{x} \\ \cos{2x}&= 2\cos^2{x}-1 \\ \tan{2x}&=\frac{2\tan{x}}{1-\tan^2{x}} \end{aligned}\]
Glancing over the Double Angle Identities, I see the closest one to the form of \(10\sin{75^\circ}\cos{75}^\circ\) is \(\sin{2x}=2\sin{x}\cos{x}\). I just need to factor out a 5 to make the coefficient a 2. \[\begin{aligned} 10\sin{75^\circ}\cos{75}^\circ &= 5(2\sin{75^\circ}\cos{75}^\circ) \\ &= 5(\sin(2(75^\circ)) \\ &= 5\sin{150^\circ} \end{aligned}\] Then 150is a special angle value so I recall from the unit circle that \[\begin{aligned} 5\sin{150^\circ} = 5 \left( \frac{1}{2} \right) = \frac{5}{2} \end{aligned}\]
