4. The Tangent Ratio
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Repeat the process until you draw a triangle with a hypotenuse of sqrt(6). Then you can use the inverse tangent to approximate the measure of the angle.
Diagram:
The approximate measure: 22^(∘)
We are asked to continue the diagram according to the given instructions. Notice that the hypotenuse of the first right isosceles triangle is sqrt(2) by the 45^(∘)-45^(∘)-90^(∘) Theorem.
Next, the hypotenuse of the length sqrt(2) will become the leg of a second triangle with the remaining leg of the length 1. Before we draw the second triangle, let's evaluate the length of its hypotenuse c using the Pythagorean Theorem.
( sqrt(a) )^2 = a
1^a=1
Add terms
sqrt(LHS)=sqrt(RHS)
sqrt(a^2)=a
Now we will repeat the process for the third triangle. This time sqrt(3) will become the leg of this triangle with the remaining leg of the length 1. Again, let's evaluate the length of the hypotenuse of the third triangle d using the Pythagorean Theorem.
Since d represents the side length, we considered only the positive case when taking a square root of d^2. Therefore, the hypotenuse of the third triangle is 2. Let's draw this triangle.
Let's repeat the process until we draw a triangle with a leg of sqrt(6).
Finally let's recall that in a right triangle the tangent of an angle is the ratio between the leg opposite this angle to the leg adjacent this angle. Having this in mind, we will highlight the angle A that has a tangent of 1sqrt(6).
To approximate the measure of the highlighted angle A we will use the inverse tangent of an angle. To do this we will use a calculator. tan∠A=1/sqrt(6) ⇓ A=tan ^(-1)1/sqrt(6) ≈ 22^(∘) The measure of this angle is approximately 22^(∘).