Trigonometry Basics: Non-Acute Angles, Reference Angles, and Unit Circle

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is a fundamental subject that is widely used in various fields, including physics, engineering, and computer science. In this article, we will discuss nonacute angle trigonometry, reference angles, quadrantal angles, and the unit circle.

Nonacute Angle Trigonometry

Trigonometry traditionally deals with acute angles, which are angles less than 90 degrees. However, we can extend the definition of trigonometric functions to nonacute angles, which are angles greater than 90 degrees. To do this, we need to define how to determine the values for the sine and cosine of other angles.

Consider a right triangle drawn on the coordinate axes. The positive acute angle (\theta) will be the angle created between the (x) -axis and the hypotenuse of the triangle. The trigonometric functions of the angle of rotation and the reference angle will differ only in their sign.

For example, if (\sin\theta=\frac{3}{8},\theta) is in Quadrant II, then the reference angle is (\theta'=180^\circ-\theta=180^\circ-(-90^\circ)=270^\circ). The sine of the reference angle is (\sin\theta'=-\frac{1}{3}), and the cosine of the reference angle is (\cos\theta'=-\frac{3}{8}). The tangent of the reference angle is (\tan\theta'=\frac{1}{1}=1).

Reference Angles

When an angle greater than (90^\circ) is created on the coordinate axes, we can find the reference angle by dropping a perpendicular from the point of intersection to the (x) -axis. The angle created is the reference angle.

The process for finding reference angles depends on which quadrant the angle terminates in. In Quadrant II, the reference angle is (\theta'=180^\circ-\theta). In Quadrant III, the reference angle is (\theta'=\theta-180^\circ). In Quadrant IV, the reference angle is (\theta'=360^\circ-\theta).

For example, for the angle (-100^\circ), the reference angle is (-100^\circ+360^\circ=260^\circ). If (\sin\theta=\frac{3}{8}) and (\theta) is in Quadrant II, then the reference angle is (\theta'=180^\circ-\theta=180^\circ-(-90^\circ)=270^\circ).

Quadrantal Angles

Quadrantal angles are angles that lie in one of the four quadrants: Quadrant I ((0^\circ<\theta<90^\circ)), Quadrant II ((90^\circ<\theta<180^\circ)), Quadrant III ((180^\circ<\theta<270^\circ)), and Quadrant IV ((270^\circ<\theta<360^\circ)). The trigonometric functions of these angles can be found using the reference angles in each quadrant.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate axes. It is used in trigonometry to find the values of the trigonometric functions for angles in the range ([0^\circ,360^\circ]).

For example, the sine of an angle (\theta) is equal to the ratio of the opposite side to the hypotenuse of a right triangle inscribed in the unit circle. The cosine of an angle (\theta) is equal to the ratio of the adjacent side to the hypotenuse of a right triangle inscribed in the unit circle. The tangent of an angle (\theta) is equal to the ratio of the opposite side to the adjacent side of a right triangle inscribed in the unit circle.

In conclusion, trigonometry is a powerful tool for understanding the relationships between angles and sides of triangles. By extending the definition of trigonometric functions to nonacute angles and finding reference angles, we can apply this knowledge to a wide range of problems. The unit circle is a useful concept that provides a standardized way to find the values of the trigonometric functions for any angle in the range ([0^\circ,360^\circ]).