Triangle Solver

A triangle solver represents the practical application of trigonometry to determine all unknown side lengths and angle measures of a triangle when given a specific set of known values. This mathematical process is the foundational bedrock of fields ranging from civil engineering and satellite navigation to 3D computer graphics, enabling professionals to translate limited spatial data into complete geometric models. By mastering the principles of triangle resolution, you will learn how to leverage the Law of Sines, the Law of Cosines, and fundamental geometric rules to decode any triangular configuration with absolute precision.

What It Is and Why It Matters

Solving a triangle is the mathematical process of finding all six fundamental characteristics of a triangle—its three side lengths and its three interior angles—when only a subset of these measurements is known. In Euclidean geometry, a triangle is a two-dimensional polygon with three edges and three vertices. Because of the rigid nature of triangles, you do not need to measure all six components to define the shape completely. If you know exactly three of these measurements (provided at least one of them is a side length), the remaining three are mathematically locked into place and can be calculated with absolute certainty. This concept is known as the determination of a triangle.

The ability to solve a triangle matters because the physical world is rarely easy to measure directly. You cannot always stretch a tape measure across a rushing river, through a solid mountain, or between two orbiting satellites. However, by measuring accessible distances and angles, you can construct a conceptual triangle and calculate the inaccessible dimensions. Triangles are the only polygons that are structurally rigid; a square built of hinged sides will collapse into a rhombus if pushed, but a triangle built of hinged sides will hold its shape perfectly. This unique rigidity makes the triangle the fundamental building block of all structural architecture, land surveying, computer graphics rendering, and global positioning systems (GPS). Without the mathematical framework to solve triangles, modern engineering, cartography, and digital navigation would be fundamentally impossible.

History and Origin of Trigonometry and Triangle Solving

The mathematics required to solve triangles, known as trigonometry (from the Greek trigōnon "triangle" and metron "measure"), was not invented in a single day but evolved over thousands of years across multiple civilizations. The foundational concepts began in ancient Egypt and Babylon around the 2nd millennium BC. Babylonian astronomers recorded angular measurements to track celestial bodies, while Egyptian engineers utilized a primitive form of triangle solving—relying on the 3-4-5 right triangle—to re-survey property lines after the annual flooding of the Nile River and to construct the pyramids with perfect right angles. However, these early civilizations lacked a formalized system of trigonometric functions.

The transition to formal triangle solving began with the Greek mathematician Hipparchus of Nicaea (190–120 BC), who is universally recognized as the father of trigonometry. Hipparchus compiled the first trigonometric table, known as a table of chords, which allowed mathematicians to relate the lengths of a triangle's sides to its angles. Around 150 AD, Claudius Ptolemy expanded on this in his seminal work, the Almagest, creating highly accurate chord tables used for astronomical calculations. The modern approach to triangle solving, however, comes from the Indian golden age of mathematics. In the 5th century AD, the mathematician Aryabhata introduced the concept of the sine function (which he called jya or ardha-jya, meaning "half-chord").

During the Islamic Golden Age, mathematicians like Al-Khwarizmi and Al-Battani (9th century) refined these concepts, developing the six modern trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) and establishing the first rules for spherical trigonometry. The definitive codification of plane triangle solving arrived in Europe during the Renaissance. In 1464, the German astronomer Regiomontanus wrote De triangulis omnimodis (On Triangles of All Kinds), the first textbook completely devoted to trigonometry. The modern notation we use today—using capital letters $A$, $B$, $C$ for angles and lowercase $a$, $b$, $c$ for the opposite sides—was standardized by the legendary Swiss mathematician Leonhard Euler in the 18th century, finalizing the mathematical language that allows anyone to solve a triangle today.

Key Concepts and Terminology

To solve a triangle, you must first speak the language of geometry and trigonometry fluently. A triangle consists of three vertices (the points where the lines meet), three edges or sides (the line segments connecting the vertices), and three interior angles (the measure of rotation between two sides at a vertex). By universal convention, the vertices and their corresponding interior angles are labeled with uppercase letters, typically $A$, $B$, and $C$. The side directly opposite each angle is labeled with the corresponding lowercase letter: side $a$ is opposite angle $A$, side $b$ is opposite angle $B$, and side $c$ is opposite angle $C$.

Angles can be measured in degrees or radians. A full circle contains 360 degrees ($360^\circ$) or $2\pi$ radians. In plane geometry, the Interior Angle Sum Theorem dictates that the sum of the three interior angles of any flat triangle will always equal exactly $180^\circ$ (or $\pi$ radians). This is a critical rule: if you know two angles, you can instantly find the third by subtracting their sum from 180. Triangles are classified by their angles into three categories: Acute (all three angles are less than $90^\circ$), Right (one angle is exactly $90^\circ$), and Obtuse (one angle is strictly greater than $90^\circ$).

Triangles are also classified by their side lengths. An Equilateral triangle has three equal sides and three equal angles (always $60^\circ$). An Isosceles triangle has two equal sides and two equal angles opposite those sides. A Scalene triangle has no equal sides and no equal angles. Another vital concept is the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining third side ($a + b > c$). If this condition is not met, the sides will never meet to form a closed shape, and the triangle is physically impossible.

The Core Mathematical Laws: Sines and Cosines

The entire discipline of solving oblique triangles (triangles without a $90^\circ$ right angle) rests upon two monumental pillars of mathematics: the Law of Sines and the Law of Cosines. These two formulas are the master keys that unlock any triangular configuration.

The Law of Sines establishes a beautiful and elegant proportion: the ratio of a side's length to the sine of its opposite angle is constant for all three sides of a given triangle. Mathematically, it is written as: $$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$ This law is incredibly powerful but requires a specific set of knowns to function. To use the Law of Sines, you must know at least one "matching pair" (a side and its opposite angle, such as $a$ and $A$) plus one other measurement. Because equations require only one unknown to be solvable, you can isolate the unknown variable. For example, if you know $a$, $A$, and $B$, you can find $b$ using $b = \frac{a \cdot \sin(B)}{\sin(A)}$.

The Law of Cosines is a generalized version of the Pythagorean theorem ($a^2 + b^2 = c^2$), expanded to apply to all triangles, not just right triangles. It incorporates a corrective term to account for angles that are greater or less than $90^\circ$. The law states that the square of any side is equal to the sum of the squares of the other two sides, minus twice the product of those sides multiplied by the cosine of the included angle. It can be written in three forms depending on which side you are solving for: $$a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$$ $$b^2 = a^2 + c^2 - 2ac \cdot \cos(B)$$ $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$ The Law of Cosines is your primary tool when you do not have a matching pair of a side and its opposite angle. It allows you to calculate a third side when you know two sides and the angle between them, or to calculate any angle when you know all three side lengths.

Types, Variations, and Methods (The 5 Configurations)

In geometry, the specific combination of sides and angles you are initially given dictates the method you must use to solve the triangle. There are exactly five valid configurations, categorized by standard three-letter acronyms where 'S' stands for Side and 'A' stands for Angle.

1. SSS (Side-Side-Side): You know the lengths of all three sides, but no angles. To solve an SSS triangle, you must use the Law of Cosines to find the first angle. Once you have one angle, you can use the Law of Sines (or the Law of Cosines again) to find a second angle, and subtract their sum from $180^\circ$ to find the third.
2. SAS (Side-Angle-Side): You know two sides and the angle perfectly wedged between them. Here, you must use the Law of Cosines to find the missing third side. Once that side is known, you have a matching side-angle pair, allowing you to switch to the Law of Sines for the remaining angles.
3. ASA (Angle-Side-Angle): You know two angles and the side connecting them. This is the easiest configuration. First, find the third angle by subtracting the two known angles from $180^\circ$. You now have a matching side-angle pair, so you can use the Law of Sines to find the remaining two sides.
4. AAS (Angle-Angle-Side): You know two angles and a side that is not between them. Similar to ASA, you immediately find the third angle using the $180^\circ$ rule. Then, apply the Law of Sines to find the missing sides.
5. SSA (Side-Side-Angle): You know two sides and an angle that is not between them. This is notoriously known as the "Ambiguous Case" because the given information might result in zero, one, or two completely different valid triangles. It requires careful application of the Law of Sines and rigorous boundary checking.

Notice that AAA (Angle-Angle-Angle) is missing from this list. Knowing three angles does not give you enough information to solve a triangle's side lengths. It only guarantees the triangle's shape, not its size. An equilateral triangle with angles of $60^\circ$ could have side lengths of 5 centimeters or 5,000 miles. Without at least one side length to anchor the scale, an AAA configuration cannot be solved.

How It Works — Step by Step

To truly understand how to solve a triangle, we must walk through a complete, realistic calculation. Let us solve an SAS (Side-Angle-Side) triangle. Imagine a civil engineer is surveying a plot of land. They stand at vertex $C$. They measure the distance to property marker $A$ as $85.0$ meters (this is side $b$). They measure the distance to property marker $B$ as $120.0$ meters (this is side $a$). The angle between their lines of sight to $A$ and $B$ is measured as $42.0^\circ$ (this is angle $C$). The engineer needs to know the distance between markers $A$ and $B$ (side $c$) and the other two angles.

Step 1: Identify the given information and choose the correct law. We are given $a = 120.0$, $b = 85.0$, and $C = 42.0^\circ$. Because we do not have a matching side-angle pair (we have angle $C$ but not side $c$), we cannot use the Law of Sines. We must start with the Law of Cosines to find side $c$: $$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$

Step 2: Calculate the missing side. Plug in the known values: $$c^2 = 120^2 + 85^2 - 2(120)(85) \cdot \cos(42^\circ)$$ $$c^2 = 14400 + 7225 - 20400 \cdot 0.7431448$$ $$c^2 = 21625 - 15160.15$$ $$c^2 = 6464.85$$ Take the square root of both sides: $$c = \sqrt{6464.85} = 80.404 \text{ meters}$$

Step 3: Calculate the second angle. Now we know side $c = 80.404$ and angle $C = 42^\circ$. We have a matching pair, so we can use the Law of Sines to find angle $B$. $$\frac{\sin(B)}{b} = \frac{\sin(C)}{c}$$ $$\frac{\sin(B)}{85} = \frac{\sin(42^\circ)}{80.404}$$ $$\sin(B) = \frac{85 \cdot 0.66913}{80.404} = \frac{56.876}{80.404} = 0.70738$$ To find the angle, take the inverse sine (arcsin) of $0.70738$: $$B = \arcsin(0.70738) = 45.02^\circ$$

Step 4: Calculate the final angle. We now know angle $C = 42^\circ$ and angle $B = 45.02^\circ$. The sum of all angles is $180^\circ$. $$A = 180^\circ - 42^\circ - 45.02^\circ = 92.98^\circ$$

The triangle is now fully solved: Sides are $a=120$, $b=85$, $c=80.404$. Angles are $A=92.98^\circ$, $B=45.02^\circ$, $C=42^\circ$.

Area Calculations: Beyond Sides and Angles

Solving a triangle often involves finding its total area, which is critical for determining material costs, land acreage, or surface coverage. The traditional formula taught in elementary geometry is $Area = \frac{1}{2} \cdot \text{base} \cdot \text{height}$. However, in real-world scenarios, you rarely know the perpendicular height of an oblique triangle. Fortunately, trigonometry provides two robust methods to calculate the area using only the sides and angles.

The first method is the Sine Area Formula, which is perfect when you know an SAS configuration. The formula is: $$Area = \frac{1}{2}ab \cdot \sin(C)$$ Using our previous example where $a=120$, $b=85$, and $C=42^\circ$: $$Area = 0.5 \cdot 120 \cdot 85 \cdot \sin(42^\circ)$$ $$Area = 5100 \cdot 0.66913 = 3412.56 \text{ square meters}$$

The second method is Heron's Formula, named after Hero of Alexandria (circa 60 AD). This formula is a masterpiece of classical geometry because it calculates the area of a triangle using only its three side lengths (SSS), requiring no angles whatsoever. First, you calculate the semi-perimeter ($s$), which is exactly half of the triangle's total perimeter: $$s = \frac{a + b + c}{2}$$ Then, you apply Heron's Formula: $$Area = \sqrt{s(s-a)(s-b)(s-c)}$$ If a triangle has sides $a=7$, $b=8$, and $c=9$: $$s = \frac{7+8+9}{2} = 12$$ $$Area = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12(5)(4)(3)} = \sqrt{720} = 26.83 \text{ square units}.$$ These two formulas guarantee that no matter what information you use to solve the triangle, you can always determine its exact area.

The Ambiguous Case (SSA): Edge Cases, Limitations, and Pitfalls

The SSA (Side-Side-Angle) configuration is the most treacherous territory in trigonometry. It is universally referred to as the "Ambiguous Case" because providing two sides and a non-included angle does not rigidly lock the triangle into a single shape. Depending on the specific measurements, the given data can result in zero valid triangles, exactly one valid triangle, or two completely distinct valid triangles. Understanding this edge case is what separates novices from experts.

Let us assume you are given side $a$, side $b$, and angle $A$. To determine how many solutions exist, you must calculate the hypothetical altitude (height) of the triangle, denoted as $h$. The altitude is found using right-triangle trigonometry: $h = b \cdot \sin(A)$. This represents the absolute shortest distance from vertex $C$ down to the line containing side $c$.

If angle $A$ is an acute angle (less than $90^\circ$), there are four possible scenarios:

1. Zero Solutions ($a < h$): If side $a$ is shorter than the altitude ($h$), it is physically impossible for side $a$ to reach the base line. The triangle cannot be closed. The calculator will throw a mathematical error if you attempt the Law of Sines.
2. One Right Triangle ($a = h$): If side $a$ is exactly equal to the altitude, it perfectly touches the base line at a $90^\circ$ angle, forming exactly one right triangle.
3. Two Solutions ($h < a < b$): If side $a$ is longer than the altitude but shorter than side $b$, it can swing like a pendulum and intersect the base line in two different places. This creates two valid triangles: one acute and one obtuse. When you use the Law of Sines to find angle $B$, you will get an acute angle $B_1$. The second valid angle is $B_2 = 180^\circ - B_1$. You must solve both parallel realities.
4. One Solution ($a \ge b$): If side $a$ is longer than or equal to side $b$, it can only intersect the base line once on the correct side of the angle. There is only one valid triangle.

If angle $A$ is obtuse (greater than $90^\circ$), the rules are simpler: if $a \le b$, there are zero solutions. If $a > b$, there is exactly one solution. Failing to check for the ambiguous case in an SSA scenario is the single most common catastrophic failure in computational geometry.

Real-World Examples and Applications

The mathematics of solving triangles is not confined to chalkboards; it is the unseen engine driving massive sectors of the global economy. Consider the field of Geodetic Land Surveying. A surveyor needs to determine the exact distance between two mountain peaks, Peak A and Peak B, separated by an impassable gorge. The surveyor establishes a baseline of exactly 1,000.00 meters on flat ground between Point C and Point D. By measuring the angles from C and D to both peaks using a highly precise total station theodolite, the surveyor creates a network of ASA triangles. By solving these triangles step-by-step, the surveyor calculates the distances to the peaks without ever setting foot on them, achieving sub-centimeter accuracy.

In Aviation and Maritime Navigation, solving triangles is a matter of life and death. An airline pilot flying a Boeing 777 from New York to London at a cruising speed of 550 knots (side $a$) is hit by a crosswind of 80 knots blowing from $45^\circ$ off the nose (side $b$ and included angle $C$). The flight management computer forms an SAS triangle representing the velocity vectors. By solving this triangle using the Law of Cosines, the computer calculates the aircraft's true ground speed and the precise crab angle required to maintain the correct flight path.

In Computer Graphics and Video Game Development, every 3D object—from a character's face to a sprawling landscape—is constructed of thousands or millions of tiny flat triangles, a process called tessellation. When a graphics processing unit (GPU) renders a frame, it must calculate the exact orientation, area, and lighting angle for every single triangle 60 times per second. Algorithms based on the Law of Cosines and vector cross-products solve these triangles to determine whether a pixel should be illuminated by a virtual light source or hidden in shadow.

Best Practices and Expert Strategies

Professionals who solve triangles daily rely on a set of strict heuristics to prevent errors and ensure maximum precision. The first and most unbreakable rule is: Always draw a diagram. Even a rough, unscaled sketch of the triangle with the known sides and angles labeled prevents cognitive overload and protects against fundamental setup errors, such as confusing a side for an angle.

The second critical strategy is the Largest Angle First Rule. When solving an SSS triangle, you must use the Law of Cosines to find an angle. Experts always choose to solve for the angle opposite the longest side first. Why? Because the longest side corresponds to the largest angle, and a triangle can only have at most one obtuse angle (greater than $90^\circ$). The inverse cosine function ($\arccos$) natively returns angles between $0^\circ$ and $180^\circ$, so it handles obtuse angles perfectly. If you try to use the Law of Sines later to find an obtuse angle, the inverse sine function ($\arcsin$) will erroneously return an acute angle, silently ruining your calculation. By finding the largest possible angle first using the Law of Cosines, you guarantee that the remaining two angles must be acute, making the Law of Sines perfectly safe to use for the rest of the problem.

A third best practice is Precision Carry-Through. Never round your numbers during intermediate steps. If you calculate an angle to be $42.58392^\circ$, do not write down $42.6^\circ$ and use that rounded number in your next equation. Trigonometric functions are highly non-linear; a rounding error of 0.1 degrees early in a calculation can result in a final distance error of hundreds of meters in a surveying context. Keep all values in your calculator's memory or use at least six decimal places until the absolute final answer.

Common Mistakes and Misconceptions

The path to mastering triangle resolution is littered with predictable traps. The most pervasive misconception among beginners is the assumption that the Pythagorean theorem ($a^2 + b^2 = c^2$) and basic trigonometric ratios (SOH-CAH-TOA) apply to all triangles. They do not. SOH-CAH-TOA and the standard Pythagorean theorem are strictly limited to right ($90^\circ$) triangles. Attempting to use $\sin(\theta) = \text{opposite}/\text{hypotenuse}$ on a scalene triangle with no right angle is a fundamental mathematical violation that will yield completely fictitious numbers.

Another frequent error is calculators being set to the wrong angular mode. The mathematical formulas do not care whether you use degrees or radians, but your calculator's processing chip certainly does. If a problem provides an angle of $45^\circ$, and you input $\sin(45)$ while the calculator is in Radian mode, the machine interprets it as 45 radians (which is roughly $2,578^\circ$) and outputs a wildly incorrect ratio. Always verify that your computational tool is set to the correct unit system before beginning.

Finally, individuals often fail to execute the basic Sanity Checks at the end of a problem. In every valid triangle, the longest side must always be exactly opposite the largest angle, and the shortest side must always be opposite the smallest angle. Furthermore, the sum of the angles must equal exactly $180^\circ$ (allowing for microscopic rounding differences). If you finish solving a triangle and find that your longest side is opposite an angle of $30^\circ$, while a shorter side is opposite an angle of $100^\circ$, you have made an algebraic error. The geometry of the universe does not allow for such a configuration.

Industry Standards and Benchmarks

In professional disciplines, the accuracy of triangle solving is governed by strict industry standards. In the United States, the Federal Geographic Data Committee (FGDC) establishes the benchmarks for geodetic surveying. For "First-Order" surveying work—such as establishing national boundary lines or monitoring tectonic plate movement—the allowable error in angular measurement is often less than 1 arc-second (1/3600th of a degree), and distance errors must not exceed 1 part in 100,000. Achieving this requires solving triangles with extremely high-precision floating-point mathematics, typically conforming to the IEEE 754 double-precision standard (64-bit), which allows for 15 to 17 significant decimal digits.

Furthermore, industry standards dictate when plane trigonometry (the math of flat triangles) must be abandoned in favor of spherical trigonometry. The Earth is not flat; it is an oblate spheroid. Over short distances, treating the ground as a flat Euclidean plane is perfectly acceptable. However, surveying standards dictate that for any triangle whose area exceeds 75 square kilometers (roughly 29 square miles), the curvature of the Earth introduces a measurable phenomenon called "spherical excess." In a spherical triangle, the sum of the interior angles is actually greater than $180^\circ$. Professionals must apply Legendre's Theorem on spherical triangles to correct this excess, distributing the curvature error equally among the three angles before applying the standard Laws of Sines and Cosines.

Comparisons with Alternatives

While trigonometric triangle solving using the Laws of Sines and Cosines is the most direct method for determining lengths and angles, it is not the only mathematical approach. The primary alternative is Coordinate Geometry (Analytic Geometry). In coordinate geometry, instead of dealing with raw lengths and angles, the triangle's vertices are placed on an X-Y Cartesian plane (e.g., $A=(0,0)$, $B=(5,12)$, $C=(14,0)$).

The advantage of coordinate geometry is that it turns geometric problems into algebraic ones. You can find side lengths using the distance formula ($d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$) and find angles using the dot product of vectors. Coordinate geometry is vastly superior for computer programming because it completely avoids the ambiguous cases of trigonometry and handles obtuse/acute angles automatically through positive and negative coordinates. However, the major disadvantage is that you must know the exact spatial coordinates of the vertices. If you are standing in a field with a laser rangefinder and a compass, you have distance and angle data (polar data), not X-Y coordinates. Converting polar data to Cartesian coordinates requires trigonometry anyway.

Therefore, the choice between the two depends on the input data. If you have lengths and angles, traditional trigonometric triangle solving is the most efficient and natural tool. If you have a grid system, pixels on a screen, or GPS latitude/longitude points, coordinate geometry and vector mathematics are the industry standard alternatives.

Frequently Asked Questions

Can you solve a triangle if you only know the three angles (AAA)? No, it is mathematically impossible to solve for the side lengths of a triangle using only three angles. Knowing the three angles defines the shape and proportions of the triangle, but it provides zero information about its scale. An equilateral triangle with angles of $60^\circ-60^\circ-60^\circ$ could have sides of 1 inch, 100 feet, or 50 miles. To determine the specific size of a triangle, you must be given at least one side length to anchor the measurements.

What is the difference between plane trigonometry and spherical trigonometry? Plane trigonometry deals with triangles drawn on a perfectly flat, two-dimensional surface, where the interior angles always sum to exactly $180^\circ$. Spherical trigonometry deals with triangles drawn on the surface of a sphere, like the Earth. On a sphere, the sides of a triangle are curved arcs (great circles), and the sum of the interior angles is always strictly greater than $180^\circ$ and less than $540^\circ$. Spherical trigonometry requires entirely different formulas and is used primarily in global navigation and astronomy.

How do I know if the triangle measurements I was given are impossible? There are two primary ways a set of given measurements can be physically impossible. First, they might violate the Triangle Inequality Theorem, which states that the sum of the two shorter sides must be strictly greater than the longest side. If you are given sides 2, 3, and 10, no triangle can be formed ($2+3$ is not greater than 10). Second, in an SSA configuration, if the given side opposite the given angle is shorter than the calculated altitude ($h = b \cdot \sin(A)$), the side cannot physically reach the base, making the triangle impossible.

Why do we need both the Law of Sines and the Law of Cosines? Each law has a specific mathematical limitation that the other law solves. The Law of Sines requires you to know at least one "matching pair" of a side and its opposite angle to establish the baseline ratio. If you are given three sides (SSS) or two sides and an included angle (SAS), you do not have a matching pair, rendering the Law of Sines useless. The Law of Cosines does not require a matching pair, allowing you to bridge the gap and find the first missing piece, after which you can use either law to finish the job.

Does Heron's Formula work for right triangles? Yes, Heron's Formula works perfectly for absolutely any type of triangle—right, acute, obtuse, scalene, or equilateral—as long as you know the exact lengths of all three sides. While it is faster to use $Area = 0.5 \cdot \text{base} \cdot \text{height}$ for a right triangle (since the two legs act as the base and height), Heron's formula will yield the exact same mathematical result. It is a universal geometric truth.

What happens if I use the Law of Sines to find an obtuse angle? If you use the Law of Sines to calculate an angle that is actually obtuse (greater than $90^\circ$), your calculator will give you the wrong answer. The inverse sine function ($\arcsin$) is mathematically restricted to returning values between $-90^\circ$ and $+90^\circ$. If the true angle is $120^\circ$, the calculator will output $60^\circ$ (because $\sin(120^\circ) = \sin(60^\circ)$). To avoid this dangerous trap, experts always use the Law of Cosines to find the largest angle in a triangle, because the inverse cosine function ($\arccos$) safely returns values up to $180^\circ$.