Solve 10.752 = 9.13 x 9.8 x 0.01 x cos(a) + 9.13 x 9.8 x sin(a) for the angle a.

Step-by-step solution

1. 1

   Factor out the repeated constant. Both terms share $9.13\cdot9.8 = 89.474$, so dividing the whole equation by it gives $0.01\cos\alpha_m + \sin\alpha_m = \frac{10.752}{89.474} = 0.12016899.$ This is the standard $a\cos\alpha+b\sin\alpha=c$ form with $a=0.01$, $b=1$.

2. 2

   Introduce the auxiliary angle. Write $\sin\alpha_m+0.01\cos\alpha_m = R\sin(\alpha_m+\varphi)$, where $R = \sqrt{1^2+0.01^2} = \sqrt{1.0001} = 1.00005000,\qquad \tan\varphi = \frac{0.01}{1},\ \varphi = \arctan 0.01 = 0.5729387^\circ.$ Combining the two waves into one is what makes the equation solvable in closed form.

3. 3

   Reduce to a single sine equation. $\sin(\alpha_m+\varphi) = \frac{0.12016899}{1.00005000} = 0.12016298.$

4. 4

   Take both branches of the arcsine. In $[0^\circ,360^\circ)$ a sine equation has two solutions: $\alpha_m+\varphi = 6.9015087^\circ \quad\text{or}\quad 180^\circ-6.9015087^\circ = 173.0984913^\circ.$ Keep enough digits here: the auxiliary angle and this arcsine are both subtracted, so an error in either propagates straight into $\alpha_m$.

5. 5

   Subtract the auxiliary angle. $\alpha_m = 6.9015087^\circ - 0.5729387^\circ = 6.32857^\circ \quad\text{or}\quad \alpha_m = 173.0984913^\circ - 0.5729387^\circ = 172.52555^\circ.$ In radians these are $0.110454$ and $3.011139$.

6. 6

   Verify both roots in the original equation. At $\alpha_m = 6.32857^\circ$: $0.89474\cos + 89.474\sin = 0.88929+9.86271 = 10.75200$. At $\alpha_m = 172.52555^\circ$: $0.89474(-0.9915030)+89.474(0.1300841) = -0.88714+11.63914 = 10.75200$. Both check exactly; the first is the physically small angle usually wanted.