A tower $150$ meter high is situated at the top of a hill. At a point $650$ meter down the hill the angle between the surface of the hill and the line of sight to the top of the tower is $12^\circ$ $30$ minutes. Find the inclination of the hill to a horizontal plane.
How would I solve this problem?
I made a right triangle but so I know one of the angle is $90^\circ$ the other one must $77^\circ$ degrees $30$ minutes and faces $650$ meter and the other is $12^\circ$ $30$ minutes and faces $150$ meters.
But what would I do?
Best Answer
For the diagram, please refer to Ross Millikan's answer. I am stealing his picture and labels. The main difference between our answers is that in the one below, we can proceed directly by calculator.
Let $\phi =\angle CDB$. By the Sine Law for $\triangle BDC$, $$\frac{\sin\phi}{650}=\frac{\sin 12.5^\circ}{150}.$$
Now we can find $\phi$ to excellent accuracy with the calculator. I assume you know how to do that (calculate $\sin\phi$, press the $\sin^{-1}$ button). To $1$ decimal place, $\phi \approx 69.7^\circ$.
It follows that $\angle ABD$ is, to $1$ decimal place, $90^\circ-69.7\circ=20.3^\circ$. To find the angle the hill makes with the horizontal, subtract $12.5^\circ$. We get $7.8^\circ$. Convert the $0.8$ part to minutes if you wish.
Alternately, let $\theta=\angle ABD$. Then $\sin\phi=\cos\theta$, so the Sine Law gives directly $\frac{\cos\theta}{650}=\frac{\sin 12.5^\circ}{150}$.