Could someone tell me how my book got this answer?

Question

anonymous · Answer

attachment

alyssa_xo · Answer

using a calculator http://www.wolframalpha.com/input/?i=tan+inverse+0.75

mathstudent55 · Answer

In a right triangle, the sine, cosine, and tangent of one of the acute angles are different ratios of the lengths of sides of the triangle.

mathstudent55 · Answer

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mathstudent55 · Answer

In the figure above, you see a right triangle.
One of the acute angles is called angle A.
For angle A, one leg is called the opposite leg, the other leg is called the adjacent leg. The longest side of the triangle is the hypotenuse.

mathstudent55 · Answer

The tangent ratio is defined as follows:

$\tan A = \dfrac{opp}{adj}$

The tangent of angle A is the ratio of the lengths of the opposite leg to the adjacent leg.

mathstudent55 · Answer

Now let's look at your example.

mathstudent55 · Answer

You are looking for the measure of angle A.
There are two side lengths that you are given.
Side BC is the opposite leg to angle A.
Side AB is the adjacent leg to angle A.
Since the two sides that are given are the adjacent and opposite legs, that means you use the tangent.
According to the definition above,

$\tan A = \dfrac{opp}{adj} $

We are given 
opp = 15, and 
adj = 20,

so we substitute these values into the definition of tangent:

$\tan A = \dfrac{15}{20}$

$\tan A = 0.75$

mathstudent55 · Answer

Now we know that angle A is the angle whose tangent is 0.75.
We need to find the measure of angle A.
For that we use the inverse tangent function.

The tangent function gives us the ratio of the lengths of the sides if we know what the lengths of the sides are.

The inverse tangent function woks in reverse. If we know what the ratio of the lengths of the sides is, it tells us what the angle is.

mathstudent55 · Answer

If

$\tan A = 0.75$, then the inverse function is used as

$A = \tan^{-1} 0.75$

Now we use a calculator, and we enter 0.75 and we press the inverse tangent key to get

$A = 36.8698976...$

Then we round it off tot he nearest tenth of a degree, so the answer is

$A = 36.9^o$

anonymous · Answer

@mathstudent55 I don't understand how you get, "A=36.8698976..." When I enter:
$$\tan^{-1} 0.75$$ I get .6435011088

mathstudent55 · Answer

There are different units for measuring angles.
The more famous one is degrees.
Another unit that is used is radians.
The correct answer in degrees is 36.9 deg, as I wrote.
Your answer is correct; it's just that it is in radians.
0.6435 rad = 36.9 deg
Your calculator is in radian mode.
If you set your calculator to degree mode, then you will get the same answer I got.