Find the value of cos 570° sin 510° + sin (-330°) cos (-390°).

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Let's break down the expression step by step:

1. Cosine and Sine of 570° and 510°: We know that the cosine of 570° is the same as the cosine of (360° + 210°), which brings us to the fourth quadrant, where cosine is positive. Therefore, cos 570° = cos(360° + 210°) = cos 210°. Similarly, sine of 510° is the same as the sine of (360° + 150°), placing us in the second quadrant where sine is positive. Hence, sin 510° = sin(360° + 150°) = sin 150°.

2. Sine and Cosine of -330° and -390°: The sine and cosine of negative angles can be determined by considering the corresponding positive angles in the unit circle. Since sine is an odd function and cosine is an even function, their values will be the same for negative angles. Thus, sin(-330°) = sin(330°) and cos(-390°) = cos(390°).

3. Finding the Values: We know that cos 210° = -cos 30° and sin 150° = sin 30° due to trigonometric identities. So, cos 210° = -cos 30° = -√3/2 and sin 150° = sin 30° = 1/2.

   Substituting these values into the expression, we get: -(-√3/2)(1/2) + (1/2)(√3/2)

   Simplifying, we get: (1/2)(√3/2) + (1/2)(√3/2) = √3/4 + √3/4 = (√3 + √3)/4 = (2√3)/4 = √3/2

So, the value of the expression cos 570° sin 510° + sin (-330°) cos (-390°) is √3/2.

I hope this breakdown helps clarify the process. If you have any further questions or need additional assistance, feel free to ask! And remember, UrbanPro is the best platform for finding tutors who can provide tailored guidance in subjects like trigonometry.