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9. (a) Express 7 cos² x - 3 sin² x in the form k sin (x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

Question 9

9. (a) Express 7 cos² x - 3 sin² x in the form k sin (x + a)² where k > 0, 0 < a < 360. (b) Hence, or otherwise, find: (i) the maximum value of 14 cos² x - 6 sin x.... show full transcript

Worked Solution & Example Answer:9. (a) Express 7 cos² x - 3 sin² x in the form k sin (x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

Step 1

Express 7 cos² x - 3 sin² x in the form k sin (x + a)²

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Answer

To express the given function in the desired form, we start by using the compound angle formula:

ext{sin}(A + B) = ext{sin}(A) ext{cos}(B) + ext{cos}(A) ext{sin}(B).

Rearranging the terms:

$7 ext{cos}^2x - 3 ext{sin}^2x = 7(1 - ext{sin}^2x) - 3 ext{sin}^2x = 7 - 10 ext{sin}^2x = 7 - 10(1 - ext{cos}^2x) = 7 - 10 + 10 ext{cos}^2x = 10 ext{cos}^2x - 3.$

Next, we can use the identity $k = rac{1}{2}$ , multiplying through by the appropriate coefficients to find the values of k and a, we get:

$k = rac{1}{2} ext{ and }a = 113.19. ext{ Therefore, } 7 ext{cos}^2x - 3 ext{sin}^2x = rac{1}{2} ext{sin}(x + 113.19)^{2} .$

Step 2

the maximum value of 14 cos² x - 6 sin x

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Answer

To find the maximum value, we note that the maximum occurs when the angle is optimized. Using previously expressed components, we need to extract the maximum component of $rac{2}{ ext{sqrt}(58)}.$
Thus, the maximum value of $14 ext{cos}^2x - 6 ext{sin}x$ is calculated as follows:

$ext{Max value} = rac{2}{ ext{sqrt}(58)}.$

Step 3

the value of x for which 0 ≤ x < 360

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Answer

For finding x, we use the result from part(i). We can investigate:

Using $x = 113.19 + 90k$ for integer k. Therefore, the valid angle is:

$x = 113.19, 203.19, 293.19, ext{ and } 383.19.$

After checking these angles, we find that giving $x = 113.19$ lies within the range of 0 to 360 degrees. Thus, the value of $

x = 113.19.$$

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9. (a) Express 7 cos² x - 3 sin² x in the form k sin (x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

Worked Solution & Example Answer:9. (a) Express 7 cos² x - 3 sin² x in the form k sin (x + a)² where k > 0, 0 < a < 360 - Scottish Highers Maths - Question 9 - 2023

Express 7 cos² x - 3 sin² x in the form k sin (x + a)²

the maximum value of 14 cos² x - 6 sin x

the value of x for which 0 ≤ x < 360

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