Can I use Law of Sines if I only know two sides and one angle?

Only if that angle is opposite one of the known sides (the SSA case). If the angle is between the two sides, you have SAS, and you must use Law of Cosines instead, because Law of Sines requires the angle to be opposite a known side.

What is the difference between Law of Sines and Law of Cosines?

Law of Sines relates sides to their opposite angles and is best when you have an angle-side pair. Law of Cosines relates all three sides to one angle and works when you know two sides and their included angle, or all three sides. Try Law of Sines first if you have an opposite angle-side pair; otherwise use Law of Cosines.

Why do all three ratios in Law of Sines equal each other?

Each ratio \frac{a}{\sin A} equals 2R, where R is the radius of the circle passing through all three vertices (the circumcircle). This equality comes from the geometry of inscribed angles in circles, which is a deep result connecting triangles to their circumcircles.

Does Law of Sines work if one of the angles is obtuse?

Yes, Law of Sines works for any triangle, including those with an obtuse angle. The sine of an obtuse angle is positive and equals the sine of its supplement, so the formula applies correctly to all cases.

Law of Sines

Law of Sines connects side lengths to their opposite angles in a triangle, used to find unknown sides or angles when an angle-side pair is known.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Solve a problem with Law of Sines

Type the problem. The solver will use Law of Sines where Law of Sines is the right tool, and tell you when it is not.

What each symbol means

What Law of Sines takes

Symbol	Meaning
$a$	The side opposite angle A, measured in the same units as sides b and c; if you pair it with the wrong angle the formula fails.
$b$	The side opposite angle B, measured in the same units as sides a and c; always ensure it corresponds to angle B.
$c$	The side opposite angle C, measured in the same units as sides a and b; each side must be opposite its matching angle.
$A$	An angle of the triangle, measured in degrees or radians; must always be paired with its opposite side a, not another side.
$B$	An angle of the triangle, measured in degrees or radians; represents the angle opposite side b in the formula.
$C$	An angle of the triangle, measured in degrees or radians; corresponds to side c, the side opposite it.

When to use it

When you know one angle-side pair and need to find another side or angle in a triangle that is not a right triangle.

Level

Usually taught in: Pre-Calculus · Appears on: ACT

Worked examples

1. Find a missing side with two angles and one side

Problem

In a triangle, angle

A = 40°

, side

a = 5

cm, and angle

B = 60°

. Find side

b

$\frac{a}{\sin A} = \frac{b}{\sin B}$
Set up the Law of Sines with two angle-side pairs.
$\frac{5}{\sin 40°} = \frac{b}{\sin 60°}$
Substitute the known values: side $a = 5$ cm with angle $A = 40°$ , and side $b$ (unknown) with angle $B = 60°$ .
$b = \frac{5 \cdot \sin 60°}{\sin 40°}$
Multiply both sides by $\sin 60°$ to isolate $b$ .
$b = \frac{5 \cdot 0.866}{0.643} = \frac{4.33}{0.643} \approx 6.73$
Calculate using $\sin 60° \approx 0.866$ and $\sin 40° \approx 0.643$ .

Answer: $b \approx 6.73 \text{ cm}$

We used Law of Sines because we had one complete angle-side pair ( $a$ and $A$ ) and another angle ( $B$ ), which gives enough information to find the missing side. Pairing each angle with its opposite side is essential—this is where most mistakes occur.

2. Find a missing angle using two sides and an opposite angle

Problem

A triangle has sides

a = 8

m and

b = 12

m, with angle

A = 35°

. Find angle

B

$\frac{a}{\sin A} = \frac{b}{\sin B}$
Apply Law of Sines with the two known sides and one known angle.
$\frac{8}{\sin 35°} = \frac{12}{\sin B}$
Substitute the known values: $a = 8$ m and $A = 35°$ , and $b = 12$ m with $B$ unknown.
$8 \sin B = 12 \sin 35°$
Cross-multiply to isolate $\sin B$ on one side.
$\sin B = \frac{12 \cdot 0.574}{8} = \frac{6.888}{8} = 0.861$
Divide both sides by 8 and calculate using $\sin 35° \approx 0.574$ .
$B = \arcsin(0.861) \approx 59.4°$
Use the inverse sine function to find the angle from its sine value.

Answer: $B \approx 59.4°$

This example shows Law of Sines working in reverse: given two sides and the angle opposite one of them, we found the angle opposite the other. This side-side-angle (SSA) situation is where Law of Sines shines, since other methods like the Pythagorean theorem cannot handle it.

3. Find a distance using angles measured from two positions

Problem

A surveyor at point A observes a flagpole at point C. The surveyor's assistant stands at point B, which is 150 meters away. From A, the angle to C is

40°

. From B, the angle to C is

65°

. Find the distance from A to C.

$C = 180° - 40° - 65° = 75°$
Find the third angle using the angle sum property: all angles in a triangle sum to $180°$ .
$\frac{AB}{\sin C} = \frac{AC}{\sin B}$
Set up Law of Sines: side AB (the baseline) is opposite angle C, and side AC (what we seek) is opposite angle B.
$\frac{150}{\sin 75°} = \frac{AC}{\sin 65°}$
Substitute the known values: $AB = 150$ m opposite the $75°$ angle at C, and $AC$ opposite the $65°$ angle at B.
$AC = \frac{150 \cdot \sin 65°}{\sin 75°} = \frac{150 \cdot 0.906}{0.966} = \frac{135.9}{0.966} \approx 140.6$
Multiply both sides by $\sin 65°$ and calculate using $\sin 65° \approx 0.906$ and $\sin 75° \approx 0.966$ .

Answer: $AC \approx 140.6 \text{ m}$

This real-world surveying problem required finding the missing angle first, then using Law of Sines to relate the known baseline (opposite the unknown angle) to the unknown distance (opposite a known angle). This is why surveyors rely on this formula when they cannot measure one side directly.

Common mistakes

Where Law of Sines usually goes wrong

The mistake	Why it is wrong	The fix
Writing $\frac{a}{\sin B} = \frac{b}{\sin A}$ or pairing side $a$ with angle $B$ instead of angle $A$ .	Law of Sines only works when each side is paired with its opposite angle; mixing them up gives completely wrong answers.	Always verify that the angle is directly across from the side in the triangle before substituting: side $a$ pairs only with angle $A$ , side $b$ only with angle $B$ , and side $c$ only with angle $C$ .
Attempting to use Law of Sines when you know two sides and the angle between them (SAS), such as when $a$ , angle $C$ , and $b$ are given.	Law of Sines requires an angle-side pair where the angle is opposite the side; when the angle is between the sides, the formula does not apply.	If you have two sides and the included angle (SAS), use the Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ instead.
Trying to apply Law of Sines without finding all three angles first, when only two are initially given.	You cannot set up the formula if you do not know all the angles you need; attempting to use an unknown angle yields an equation you cannot solve.	Always find the third angle using $A + B + C = 180°$ before applying Law of Sines, especially when the third angle is opposite a side you are trying to find.

Tips and when to use something else

Remember the correspondence: side $a$ is always opposite angle $A$ , side $b$ opposite angle $B$ , and side $c$ opposite angle $C$ —this pairing is the entire foundation of the formula.
If two angles are known, find the third using $A + B + C = 180°$ before solving; you will often need the third angle to set up Law of Sines correctly.
Use Law of Cosines instead when you have two sides and the included angle (SAS) or all three sides (SSS); Law of Sines is not the right tool for those cases.
Law of Sines works for any triangle—acute, right, or obtuse—as long as you have an angle-side pair and at least one other angle or side.

Related formulas

Law of CosinesFinds unknown sides or angles in any triangle when you know two sides and the angle between them, or all three sides.Inverse Trigonometric FunctionsInverse trigonometric functions find angles when you know their sine, cosine, or tangent values; use them to solve equations where an angle is unknown.Sine Cosine TangentFind missing sides or angles in right triangles using sine, cosine, and tangent ratios—essential for SAT and ACT geometry.Area of a Triangle with SineFind the area of a triangle when you know two sides and the included angle—much faster than using just base and height.Solving Trigonometric EquationsFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.

Frequently asked questions

Can I use Law of Sines if I only know two sides and one angle?: Only if that angle is opposite one of the known sides (the SSA case). If the angle is between the two sides, you have SAS, and you must use Law of Cosines instead, because Law of Sines requires the angle to be opposite a known side.
What is the difference between Law of Sines and Law of Cosines?: Law of Sines relates sides to their opposite angles and is best when you have an angle-side pair. Law of Cosines relates all three sides to one angle and works when you know two sides and their included angle, or all three sides. Try Law of Sines first if you have an opposite angle-side pair; otherwise use Law of Cosines.
Why do all three ratios in Law of Sines equal each other?: Each ratio $\frac{a}{\sin A}$ equals $2R$ , where $R$ is the radius of the circle passing through all three vertices (the circumcircle). This equality comes from the geometry of inscribed angles in circles, which is a deep result connecting triangles to their circumcircles.
Does Law of Sines work if one of the angles is obtuse?: Yes, Law of Sines works for any triangle, including those with an obtuse angle. The sine of an obtuse angle is positive and equals the sine of its supplement, so the formula applies correctly to all cases.

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Reviewed 2026-09-18