Is trigonometry part of synthetic approach to geometry? I used to think Trigonometry is an analytic way to solve math problems but now I'm reading a book with only synthetic methods to solve geometry problems and it contains trigonometry too.
I can't understand the difference between analytic  and synthetic approaches to geometry.
"I can't understand the difference between analytic and synthetic approaches to geometry."

I think a good way to understand the difference between the two is that in analytic geometry, mathematicians assign a a real number to each pair of points in whatever space they are working in (representing the distance between the given points).  Similarly for angle measure. When one assigns real numbers to represent things like angle and distance one has entered the realm of analytic geometry.

So, while precise definitions change from century to century (and even author to author in the same century), in synthetic geometry, we might claim that two angles are congruent, meaning one could be superimposed on the other with no mention of a numerical quantity.  In analytic geometry we might claim that two angles are equal meaning that the numbers used to measure those angle are the same.

Again, the precise meanings of particular words can vary depending on the setting, but the use of real numbers to describe geometric entities is the hallmark of the analytic approach.
"I can't understand the difference between analytic and synthetic approaches to geometry."

I think a good way to understand the difference between the two is that in analytic geometry, mathematicians assign a a real number to each pair of points in whatever space they are working in (representing the distance between the given points).  Similarly for angle measure. When one assigns real numbers to represent things like angle and distance one has entered the realm of analytic geometry.

So, while precise definitions change from century to century (and even author to author in the same century), in synthetic geometry, we might claim that two angles are congruent, meaning one could be superimposed on the other with no mention of a numerical quantity.  In analytic geometry we might claim that two angles are equal meaning that the numbers used to measure those angle are the same.

Again, the precise meanings of particular words can vary depending on the setting, but the use of real numbers to describe geometric entities is the hallmark of the analytic approach.