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This page sets out the method for the sine rule in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the The sine rule topic page.
When you know two angles and one side, find the side opposite one of the known angles.
When you know two sides and an angle that is opposite one of them.
a / sin A = b / sin B. The angle sits opposite its side.
If the side opposite 30 degrees is 5, the side opposite 45 degrees is 5√2.
If the side opposite 45 degrees is 5√2, the side opposite 30 degrees is 5.
If the side opposite 30 degrees is 5 and the side opposite the unknown acute angle is 5√2, that angle is 45.
The angles in a triangle add up to 180. If two angles are 50 and 60, the third is 180 - 50 - 60 = 70.
The sine rule works in any triangle, not just a right-angled one. A triangle with angles 55 and 65 has a third angle of 60 and no right angle.
Write a capital letter in each corner and the matching small letter on the opposite side. In the triangle shown, the angles are 40^°, 60^° and 80^°. The longest side is opposite 80^°.
You need a side paired with the angle opposite it. Two angles are enough, because the third is 180^° minus their sum. Two sides and the angle between them are not a pair: that is a cosine-rule question. Set the calculator to degrees. sin 30^° should show 0.5. If sin 35^° shows about -0.428, it is in radians.
Method: 1. Mark angles with capitals and the opposite sides with small letters. 2. Find a matching pair: a side and the angle opposite it. 3. Check the calculator is in degrees. 4. Expect the largest angle to face the longest side.
Tip: If the diagram is not to scale, trust the labels, but still check that your finished side is on the right side of the size order.
Put each side over the sine of the opposite angle. If you know angle A, side a and angle B, then b = (a sin B)/(sin A).
Calculate in one go if you can, and do not replace a sine by a rounded decimal before the last step. Give a length to 3 significant figures unless told otherwise. If angle B is bigger than angle A, side b must be longer than side a.
Method: 1. Write the sine rule with the known pair and the pair that contains the unknown side. 2. Rearrange so the unknown side is the subject. 3. Calculate in degrees, keeping the full values. 4. Round the length to 3 significant figures and check the size order.
Flip the rule so the sines are on top, or start from the side form and rearrange. If you know side a, angle A and side b, then sin B = (b sin A)/a. The last key is sin^-1. A decimal such as 0.58 is a sine, not an angle.
Inverse sine returns the acute angle, from 0^° to 90^°. Use it when the question says acute. The Higher note below covers 180^° minus that angle. Round to 1 decimal place.
Method: 1. Write sin of the unknown angle as a fraction using the opposite pair. 2. Calculate that sine, keeping the full value. 3. Press inverse sine. 4. Round to 1 decimal place and decide whether the obtuse angle is also possible.
Tip: If the sine is bigger than 1, there is no triangle. Check the pairing.
When two angles are given, the third is 180^° minus their sum. That is ordinary angle sum, and it often creates the pair you need. Find it before you touch the sine rule, and write it on the diagram.
Then pick a known opposite pair and find the side you want. Do not round the third angle before you use it. An angle sum of 180^° is exact when the given angles are exact.
Method: 1. Add the two known angles and subtract from 180^°. 2. Label the third angle on the diagram. 3. Choose a known opposite pair and write the sine rule. 4. Rearrange for the missing side and round at the end.
Inverse sine gives one angle, the acute one. The sine of an obtuse angle equals the sine of its supplement, so 180^° - θ has the same sine. Both can be angles in a triangle, but only if the angle sum still leaves a positive third angle.
Both answers work when angle A is acute and b sin A < a < b. If a ≥ b, only the acute angle works. If a < b sin A, the side cannot reach. For example, A = 40^°, a = 6 and b = 10 gives sin B ≈ 1.07, which is impossible. When both triangles do fit, give both angles unless the question says acute or obtuse.
Method: 1. Find the acute angle with inverse sine. 2. The other candidate is 180^° minus that angle. 3. Add each candidate to the given angle. Both must be under 180^°. 4. If the question or the diagram fixes acute or obtuse, keep only that one.
Tip: Work the obtuse angle from the full inverse-sine value, then round. Rounding first and then subtracting can move the last decimal place.
The sine rule links each side to the angle opposite it: a/(sin A) = b/(sin B) = c/(sin C). To use it you need one complete pair, a side and the angle opposite, plus one other piece of information. Most questions give a pair in the first line, for example "angle A = 40^° and side a = 9.3 cm". Underline it. If no side has its opposite angle given, the sine rule is not yet available.
Sometimes the pair is hidden. If two angles are given, the third angle comes from 180^° and may complete the pair you need. Write the angle sum first, then the sine rule. Use only the two fractions that contain what you know and what you want. Putting all three fractions in the working just adds clutter, and each unused fraction hides which side matches which angle.
Method: 1. Label each side with the capital letter of the angle opposite it. 2. Underline the complete pair of a side and its opposite angle. 3. If only two angles are given, find the third from 180^°. 4. Write the two fractions you need and then substitute.
Tip: The largest side is always opposite the largest angle. Use this to check your answer.
In non-calculator questions the angles are 30^°, 45^° or 60^° and you use sin 30^° = tfrac12, sin 45^° = tfrac√(2)2 and sin 60^° = tfrac√(3)2. Suppose angle A = 30^°, angle B = 45^° and a = 8 cm. Then b = (8sin 45^°)/(sin 30^°) = dfrac8 × frac√(2)21/2 = 8√(2). The halves cancel and the surd remains.
Another style gives the sine as a fraction. If sin A = tfrac35, b = 7 and B = 30^°, then a = (bsin A)/(sin B) = dfrac7 × 3/51/2 = 21/5 × 2 = 8.4. Keep everything as fractions until the end, and cancel before you multiply. Give surds in simplest form, such as 8√(2), not √(128) and not a decimal.
Method: 1. Write the sine rule with the exact values in place. 2. Rearrange for the unknown side or sine. 3. Cancel halves and common factors before you multiply. 4. Leave the answer as a simplified surd or a fraction.
Real exam context questions use the sine rule when the triangle is not right-angled. An example is a tower seen from two points in line with its base. The angle of elevation is measured from the horizontal, so the interior angle at the nearer point is 180^° minus the elevation. The angle at the top is the difference of the two elevations, which you can find with the exterior angle theorem or the angle sum.
Bearings give interior angles through back-bearings, which differ by 180^°. After the sine rule gives a sloping length, a final step usually finds a vertical height with height = length × sin(angle). Do not round in the middle. Label every angle and length in a clear sketch and keep the full calculator value.
Method: 1. Draw the triangle with all given angles, using 180^° minus elevation for the near point. 2. Find the missing interior angle from the angle sum. 3. Use the sine rule to find the sloping side. 4. Use right-angled trigonometry for a height, if the question asks for one.
Three quick checks catch most mistakes. First, the longest side must be opposite the largest angle. Second, an angle found from the sine rule must be possible, because sin cannot be more than 1. If your calculator shows an error, you have probably swapped the numerator and the denominator. Third, the sum of your three angles must be 180^°.
Present the working in three lines: the rule with numbers, the rearrangement, and the rounded answer with units. Marks are given for the substitution even if the arithmetic goes wrong. If the question says "3 significant figures", do not give extra digits. Remember that for an angle the calculator returns an acute value; for an obtuse possibility, subtract it from 180^° and check that the triangle still works.
Method: 1. Check that the largest side faces the largest angle. 2. Check that every angle is positive and the three add to 180^°. 3. Write the sine rule, the rearrangement and the rounded answer on separate lines. 4. Add the unit: cm, m or degrees.
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