The blades of a wind turbine are turning at a steady rate - Scottish Highers Maths - Question 9 - 2015

To rework the expression into the required form, we can use the compound angle formula for sine:

$k ext{sin}(x - a) = k( ext{sin}(x) ext{cos}(a) - ext{cos}(x) ext{sin}(a)).$

We want to match:

• $k ext{cos}(a) = 36$
• $k ext{sin}(a) = -15$.

Squaring both equations and adding them gives:

$k^2 ext{cos}^2(a) + k^2 ext{sin}^2(a) = 36^2 + (-15)^2$

Simplifying yields:

$k^2 = 1296 + 225 = 1521,$

thus:

$k = ext{sqrt}(1521) = 39.$

Now we can find $a$:

From $k ext{cos}(a) = 36$, we have:

$ext{cos}(a) = \frac{36}{39} = \frac{12}{13},$

From $k ext{sin}(a) = -15$, we find:

$ext{sin}(a) = \frac{-15}{39} = -\frac{5}{13}.$

By using the identity $\text{tan}(a) = \frac{\text{sin}(a)}{\text{cos}(a)}$, we can now calculate:

$\text{tan}(a) = \frac{-5/13}{12/13} = -\frac{5}{12}.$

This implies $a$ is in the fourth quadrant, hence:

$a = \tan^{-1}\left(-\frac{5}{12}\right).$