Lesson 56: Finding Angles of Rotation

New Concept

An angle formed by rotating the terminal side and keeping the initial side in place is called the angle of rotation.

What’s next

Next, you'll practice drawing these angles, finding their equivalents (coterminal angles), and using them to define fundamental trigonometric functions.

Coterminal angles are angles in standard position that share the same terminal side. An angle with measurement $x°$ is coterminal with any angle that can be represented as $x° + 360n$, where $n$ is an integer representing the number of full rotations.

To find a positive coterminal angle for $75°$, add $360°$: $75° + 360° = 435°$. To find a negative one, subtract $360°$: $75° - 360° = -285°$. For an angle like $480°$, you can find a smaller positive coterminal angle by subtracting $360°$: $480° - 360° = 120°$.

Think of coterminal angles like a race car on a circular track. Finishing one lap, or even spinning out backward, puts you right back at the starting line. These angles land in the exact same spot, they just took a different number of spins to get there! To find an angle's buddy, just add or subtract a full 360° circle.

A reference angle is the acute angle formed by the terminal side of an angle in standard position and the $x$-axis. Its measure is always positive and less than $90°$.

For an angle of $160°$ (Quadrant II), the reference angle is $180° - 160° = 20°$. For an angle of $220°$ (Quadrant III), the reference angle is $220° - 180° = 40°$. For a negative angle like $-45°$, first find its positive coterminal angle, $315°$, then find its reference angle: $360° - 315° = 45°$.

No matter how far an angle has rotated, its reference angle is its 'shortcut' back to the nearest horizontal line (the x-axis). It's always a sharp, positive turn under $90°$. This little angle is a secret weapon that lets us use simple right-triangle rules to figure out trigonometry for much bigger or more complicated angles.

For a point $Q(x, y)$ on the terminal side of an angle $\theta$ in standard position, where $r = \sqrt{x^2 + y^2}$, the trigonometric functions are defined as:

For a point $Q$ at $(-3, 4)$: First, find the distance $r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. Then, use the definitions to find the trigonometric values: $\sin \theta = \frac{y}{r} = \frac{4}{5}$, $\cos \theta = \frac{x}{r} = -\frac{3}{5}$, and $\tan \theta = \frac{y}{x} = -\frac{4}{3}$.

Any point on an angle's terminal side can be the corner of a right triangle. Its coordinates $(x, y)$ give you the triangle's legs, and the distance from the origin, $r$, is the hypotenuse. With just these three numbers, you can instantly find the sine, cosine, and tangent of the angle, no matter how large it is!