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tags:
Cosine Fundamental Trigonometric Identity Sine Trigonometry Trigonometry Series

Using Identities to Find Exact Trig Values Exercise 3

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem: Use a Double Angle Identity to find the exact value of
\[10\sin{75^\circ}\cos{75^\circ}\]

Solution: \(\dfrac{5}{2}\)
 

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 \[\begin{aligned} 5\sin{150^\circ} = 5 \left( \frac{1}{2} \right) = \frac{5}{2} \end{aligned}\]

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