## Coterminal Angles: Definition, Formula, and Examples Coterminal angles are angles in standard position that share the same terminal side — they land in the same place after differing by a whole number of full turns. You find them by adding or subtracting 360° (or 2π radians). This article covers the definition, the formula in degrees and radians, positive and negative coterminal angles, the unit-circle picture, and worked examples. ## What Are Coterminal Angles? To define coterminal angles cleanly, picture an angle in **standard position**: its vertex sits at the origin of a coordinate plane, and its **initial side** lies along the positive x-axis. The angle opens to its **terminal side** — the ray you end on after rotating. **Coterminal angles are two or more angles in standard position that share the same terminal side.** The initial side and the vertex are the same; only the _amount_ of rotation differs, and it differs by a whole number of full turns. Because a full turn is 360° (or 2π radians), coterminal angles always differ by a multiple of 360° (or 2π). A direct consequence — **do coterminal angles have the same trig values?** Yes. Because 30°, 390°, and −330° all end on the same ray, their sine, cosine, and tangent are identical: sin⁡30°=sin⁡390°=sin⁡(−330°)=0.5. The terminal side is what the trig functions "see," so coterminal angles are interchangeable inside sin⁡, cos⁡, and tan⁡. ## The Coterminal Angles Formula Finding a coterminal angle is one operation: add or remove full turns. The formula has two forms, one per unit. **In degrees,** the coterminal angles of an angle θ are: θ + 360°n, n = ±1, ±2, ±3,… **In radians,** they are: θ + 2πn, n = ±1, ±2, ±3,… Here n is any nonzero integer — it counts how many full turns you add (n > 0) or remove (n < 0). Each variable earns its place: θ is your starting angle, 360° (or 2π) is one complete revolution, and n is the number of revolutions. Since n can be any integer, **every angle has infinitely many coterminal angles** — you can keep adding turns forever. ## Positive and Negative Coterminal Angles The sign of n chooses the direction of the extra turns. - A **positive coterminal angle** comes from **adding** 360° (or 2π) one or more times — rotating further counter-clockwise. - A **negative coterminal angle** comes from **subtracting** 360° (or 2π) one or more times — rotating clockwise. For an angle of 45° (π/4 radians): | Operation | Degrees | Radians | |----------------|----------------------------|------------------------------| | Given angle | 45° | π/4 | | + one turn | 45°+360°=405° | π/4+2π=9π/4 | | − one turn | 45°−360°=−315° | π/4−2π=−7π/4 | ## Coterminal Angles and Reference Angles Coterminal angles are easy to confuse with **reference angles**, but they answer different questions. Coterminal angles ask _which angles end on the same ray_; the reference angle asks _how far that ray sits from the nearest part of the x-axis_, always as an acute angle between 0° and 90°. You often use them together: reduce a large angle to its smallest positive coterminal partner first, then read off the reference angle. ## Examples of Coterminal Angles ### Example 1 **Find a positive coterminal angle of 50°.** Add one full turn: 50°+360°=410°. So 410° is coterminal with 50°. ### Example 2 **Find a negative coterminal angle of 100° by adding 360°.** _Correct._ A negative coterminal angle is reached by _subtracting_ a full turn from the original: 100°−360°=−260°. Check: −260° and 100° differ by exactly 360°, so they share a terminal side. ### Example 3 **Find the smallest positive coterminal angle of −200°.** Add a full turn: −200°+360°=160°. Since 160° is already between 0° and 360°, the smallest positive coterminal angle is 160°. ### Example 4 **Find a positive and a negative coterminal angle of π/3 radians.** Add 2π: π/3+2π=7π/3 (positive). Subtract 2π: π/3−2π=−5π/3 (negative). ### Example 5 **Find the smallest positive coterminal angle of 480°.** Subtract one full turn: 480°−360°=120°. The smallest positive coterminal angle is 120°. ### Example 6 **Are 765° and 45° coterminal?** Subtract: 765°−45°=720°=2×360°. The difference is a whole multiple of 360°, so yes, they are coterminal. ## Why Coterminal Angles Matter Beyond the Classroom Coterminal angles are how rotation gets bookkept — anywhere a thing turns past a full circle, the idea is doing quiet work. - **Circular motion and engineering.** A spinning wheel or turbine blade passes through 360° over and over; describing its position needs the coterminal idea. - **Navigation and bearings.** A heading of 400° is meaningless on a compass until you reduce it to its coterminal partner 40° — the actual direction. - **Trigonometry and waves.** Because sin and cos repeat every 360° (or 2π), coterminal angles explain _why_ these functions are periodic. - **Animation and graphics.** Rotating an object by 725° in a game engine is stored and rendered as its coterminal 5°. ## Where Students Trip Up on Coterminal Angles ### Mistake 1 Confusing a negative angle with a negative coterminal angle. **The correct way:** A negative coterminal angle is found by _subtracting_ a full turn. ### Mistake 2 Mixing units — adding 360 to an angle in radians. **The correct way:** In radians, one full turn is 2π, so add 2π. ### Mistake 3 Forgetting there are infinitely many. ## Key Takeaways - **Coterminal angles** are angles in standard position that share the same terminal side, differing by whole turns. - Find them with θ + 360°n in degrees or θ + 2πn in radians, where n is any nonzero integer. - Adding turns gives positive coterminal angles; subtracting turns gives negative ones. - Every angle has infinitely many coterminal partners; the smallest positive one is usually what a problem wants. - Coterminal angles share all trig values, which is why sine and cosine are periodic. ## Practice These Problems to Solidify Your Understanding 1. Find the smallest positive coterminal angle of −120°. 2. Find a positive and a negative coterminal angle of 2π/5 radians. 3. Are −150° and 570° coterminal? Justify your answer.