Sine squared alpha plus cosine (60 degrees plus alpha) cosine (60 degrees minus alpha).

| January 6, 2021 | education

A trigonometric expression sin²α + cos (60 ° + α) * cos (60 ° – α) is given, which we denote by T. However, there is no accompanying requirement for it. Apparently, the authors of the assignment wanted to simplify this trigonometric expression (if possible, and calculate its value), which we denote by T.
Let’s use the formulas cos (α + β) = cosα * cosβ – sinα * sinβ (cosine of the sum) and cos (α – β) = cosα * cosβ + sinα * sinβ (cosine of the difference). Then, we have: T = sin²α + cos (60 ° + α) * cos (60 ° – α) = sin²α + (cos60 ° * cosα – sin60 ° * sinα) * (cos60 ° * cosα + sin60 ° * sinα).
Let’s turn to the table of the main values ​​of sines, cosines, tangents and cotangents and find the corresponding values: cos60 ° = ½ and sin60 ° = √ (3) / 2. Therefore, T = sin²α + (½ * cosα – √ (3) / 2 * sinα) * (½ * cosα + √ (3) / 2 * sinα).
Using the abbreviated multiplication formula (a – b) * (a + b) = a² – b² (difference of squares), we get: Т = sin²α + (½ * cosα) ² – (√ (3) / 2 * sinα) ² = sin²α + ¼ * cos²α – ¾ * sin²α = (1 – ¾) * sin²α + ¼ * cos²α = ¼ * (sin²α + cos²α).
The formula sin2α + cos2α = 1 (basic trigonometric identity) will allow us to finally assert that T = ¼ * 1 = ¼.
Answer: ¼.



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