Without Geometry There Would Be No Trigonometry
Trigonometry is a subject that has to be studied after some background in geometry. Why? Because trigonometry has its whole basis in triangles and angle measures and circles. Geometric studies acquaint you with properties of triangles that are necessary to understand trig concepts — everything makes more sense. On the other hand, when you study trig, you learn even more geometry — things your teacher or book never told you. What are these overlapping topics?
In a circle, if you draw a segment from the center, perpendicular to a chord of the circle, that segment bisects the chord. What’s the big deal? In geometry, you then can prove that you have two congruent right triangles when you draw the radii to the endpoints of the chord. In trigonometry, you can determine the values of the trig functions of the angles in the triangles.
Given a circle, you can determine the length of an arc and the area of a sector. In geometry, when an arc is determined by a central angle, you find the arc length by multiplying the radius times the angle measure (in radians). The area of the sector is found by taking half the product of the square of the radius times the measure of the angle (in radians). Central angles play a huge role in trigonometry. A central angle has its vertex at the center of the circle (surprise, surprise). The unit circle and all the central angles formed on that circle are critical to defining the trig functions for all angles.
An inscribed angle with rays that go through the endpoints of the diameter is a 90-degree angle. What you find in geometry is that you can prove that any triangle constructed with two vertices on the ends of a diameter and the third anywhere else on the circle is always a right triangle. In trigonometry, you go even further to talk about the angle measure of any inscribed angle (an angle with its vertex on the circle). And it doesn’t end there!
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Who doesn’t recognize the Pythagorean theorem? Where would trigonometry be without Pythagoras? Nowhere. In geometry, you find the basic theorem and many other properties, such as what happens in a right triangle when you draw an altitude from the hypotenuse to the right angle (means and extremes). In trigonometry, the basic functions are first determined from the lengths of the sides of a right triangle. And most applications revolve around using Pythagoras’s gift.
In an isosceles triangle, dropping an altitude to the non-congruent side forms two congruent triangles. The proof of this statement in geometry can go in many directions. Some mathematicians use hypotenuse-side of a right triangle, and others use side-side-side (SSS), after invoking another property of that altitude. In trigonometry, this property is very helpful when determining lengths and angles in regular polygons.
Segments drawn from the circumcenter of a triangle to its vertices form three isosceles triangles. The circumcenter is the point either inside or outside a triangle that is at the intersection of the three altitudes of the triangle. The circumcenter is the center of the circle that you can draw through all three vertices of the triangle. In geometry, you may also learn about a triangle’s centroid (intersection of the medians), orthocenter (intersection of the perpendicular bisectors of the sides), and incenter (intersection of the angle bisectors). In trigonometry, all those congruent triangle sides make for interesting relationships between functions of angles.