The length of the longer leg of the triangle is sqrt(3) times the length of the shorter leg. This means that we have a 30^(∘)-60^(∘)-90^(∘) triangle. In this type of triangle, the measure of the angle opposite the shorter leg is 30^(∘). Next, to calculate the required angle θ, we will subtract the obtained value from 270^(∘), which is the measure of three quarters of a circle.
270^(∘) -30^(∘) = 240^(∘)
Finally, let's see the angle on the graph.
Alternative Solution
Using Trigonometry
The angle measure can also be found using right triangle trigonometry. Consider the right triangle we drew at the beginning of the solution.
In a right triangle, the tangent of an acute angle is equal to the ratio of the length of the opposite leg to the length of the adjacent leg.
tan θ = Opposite leg/Adjacent leg
In our diagram, we can see that the length of the opposite leg is 12, and the length of the adjacent leg is sqrt(3)2. Let's substitute these values into the above formula and solve for θ.
To find the value of θ we will use the inverse of the tangent function, tan^(- 1).
tan θ =1/sqrt(3) ⇔ θ =tan^(- 1) 1/sqrt(3)
Then, we will need to use a calculator.
θ =tan^(- 1) 1/sqrt(3) ⇒ θ=30^(∘)
As we can see, the angle θ is 30^(∘). To obtain the measure of the given angle, we need to subtract this value from 270^(∘), as we did in the solution.
270^(∘)-30^(∘) = 240^(∘)
