The diagram shows an eight-foot solar panel mounted on the roof of a house. The roof is inclined at 20 degrees to the horizontal. And for maximum yield, the solar panel is placed at 38 degrees to the horizontal. The solar panel is held in position by a vertical support. How long should the support be to hold the solar panel at an inclination of 38 degrees? Give your answer to one decimal place.
We want to calculate the length of the vertical support of the solar panel. Let’s enlarge the triangle formed by the solar panel and the roof. We can see that we know the length of one of its sides to be eight foot. Let’s call this side lowercase 𝑎. The angle directly opposite we can denote with the letter capital 𝐴. We can also easily calculate the measure of the angle at 𝐵.
The two angles given to us are measured from the horizontal. We can use the fact then that these lines are parallel. And since corresponding angles are equal, we can calculate the measure of the angle at 𝐵 by subtracting 20 from 38. 38 minus 20 is 18. So the measure of the angle at 𝐵 is 18 degrees.
We can also calculate the measure of the angle at 𝐴. By constructing a third line representing the horizontal, we know that the angle between the horizontal and the vertical support must be 90 degrees. Alternate angles are equal. So we can calculate the measure of the angle at 𝐴 by adding 90 and 20. 90 plus 20 is 110. So the measure of the angle at 𝐴 is 110 degrees.
So we have a non-right-angled triangle with two known angles and one known side, for which we want to calculate the length of a second side. That’s the side marked with the lowercase 𝑏 on our diagram. We can use the law of sines to do so. Remember, we usually only need to use two parts of this equation.
Since we know the measure of the angles at 𝐴 and 𝐵, we’ll use the first two parts: 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵. Substituting these values into the formula gives us eight divided by sin 110 is equal to 𝑏 divided by sin 18.
To solve this equation, we’ll multiply both sides by sin 18. 𝑏 is, therefore, equal to eight over sin 110 multiplied by sin of 18. That’s 2.6307. Rounding the number to one decimal place and including the relevant units, we get that the length of the support is 2.6 metres.