Photo AI

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

96%

114 rated

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

99%

104 rated

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

96%

101 rated

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.$$

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

Join the Scottish Highers students using SimpleStudy...