I am catching up over summer on math units that I have to rewrite. In my textbook it asks me to find the base of two triangles and add them together. the smallest angle of the larger triangle is 30 degrees. The opposite side to the angle is 40cm. it gives me Tan 30 = 40/x

1/square root of 3 = 40/x
40 *(squareroot of 3)

My question is, how did my textbook get 1/sq.rt. of 3 as Tan 30?

Because that is the value of tan 30 . The short side length is half the hypotenuse for a 30-60-90 triangle.

If the hypotenuse length is 1, the Second longest side length must be sqrt[3/4]
That makes the tangent of 30 degrees 1/(sqrt3)

Thank you!

Next question.

What if instead they gave me the hypotenuse? 2/(sqrt2? Still tan 30.

To understand how your textbook obtained the value of 1/sqrt(3) as the tangent of 30 degrees, we need to look at the definition and properties of the tangent function and the special right triangle related to the angle 30 degrees.

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the given relationship is tan(30) = 40/x, where x represents the length of the adjacent side.

Now, let's consider a special right triangle known as a 30-60-90 triangle. In such a triangle, one of the angles is 30 degrees, and the other two angles are 60 degrees and 90 degrees. It has specific side ratios that are always true.

In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows:

- The side opposite the 30-degree angle is always half the length of the hypotenuse.
- The side opposite the 60-degree angle is always sqrt(3) times the length of the side opposite the 30-degree angle.
- The hypotenuse is always twice the length of the side opposite the 30-degree angle.

Using these ratios, we can deduce that the side opposite the 30-degree angle in this triangle is (1/2) times the hypotenuse. Since the textbook equation relates the side opposite the angle to x, we can equate it to (1/2) times the hypotenuse, which is 40 cm in this case.

(1/2) * 40 = 40/x

Now, simplify the equation:

20 = 40/x

To find x, divide both sides of the equation by 20:

20/20 = (40/x)/20

1 = 2/x

Next, cross-multiply:

1 * x = 2 * 1

x = 2

Therefore, x = 2 cm.

To summarize, the textbook used the properties of a 30-60-90 triangle to determine that the side opposite the 30-degree angle is (1/2) times the hypotenuse. By substituting the given value of 40 cm for the side opposite the angle, the equation was set up to find the length of the adjacent side (x). Simplifying the equation led to the value of x being 2 cm.