Trigonometric functions: FAQ

Trigonometric functions are used as models in a wide variety of fields, including engineering, physics, astronomy, and navigation. For example, engineers might use trigonometric functions to describe vibrations or waves, while astronomers might use them to model the orbits of planets.

Practice with our Interpreting trigonometric graphs in context exercise.

Practice with our Modeling with sinusoidal functions exercise.

The unit circle is a circle with a radius of $1$‍ centered at the origin. We can use the unit circle to help define the trigonometric functions and visualize their values.

Learn more with our Unit circle video.

Practice with our Unit circle exercise.

Radians are a unit of measurement for angles.

One radian is the angle measure that we turn to travel one radius length around the circumference of a circle.

We often use radians in trigonometry because they make working with trigonometric functions easier.

Learn more with our Intro to radians video.

Practice with our Radians & degrees exercise.

The Pythagorean identity is an equation that connects sine and cosine. It states that ${\sin }^2(x)+{\cos }^2(x)=1$‍ for any angle measure $x$‍.

Learn more with our Proof of the Pythagorean trig identity video.

Practice with our Use the Pythagorean identity exercise.

The graphs of sin(x) and cos(x) are both sinusoidal, meaning they have a wavy shape that repeats periodically. Tan(x) also repeats periodically, but its graph has discontinuities (places where the function is not defined) and vertical asymptotes.

Learn more with our Graph of y=sin(x) video.

Learn more with our Intersection points of y=sin(x) and y=cos(x) video.

Learn more with our Graph of y=tan(x) video.

Amplitude, midline, and period are all terms we use to describe sinusoidal graphs. Amplitude is the height of the wave from the midline, midline is the horizontal line around which the wave oscillates, and period is the length of one complete cycle.

Learn more with our Features of sinusoidal functions video.

We can change the amplitude, midline, and period of a sinusoidal graph by modifying the equation of the function. For example, if we multiply the function by a constant, we change the amplitude. If we add or subtract a constant, we change the midline. And if we multiply the input by a constant, we change the period.

Learn more with our Transforming sinusoidal graphs: vertical stretch & horizontal reflection video.

Learn more with our Transforming sinusoidal graphs: vertical & horizontal stretches video.

Both tau ($\tau$‍) and pi ($\pi$‍) are constants related to circles. Tau is equal to two times pi, or roughly $6.28$‍. Some mathematicians argue that using tau instead of pi makes certain equations and formulas simpler.

Learn more with our Tau versus pi video.

Learn more with our Pi is (still) wrong video.