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This page sets out the method for exact trig values 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 Exact trig values topic page.
When you cannot remember a value in the exam, draw the triangle instead of guessing.
When a non-calculator question gives a , or angle in a right-angled triangle.
The exact value of sin 30 is 1/2. The exact value of sin 45 is √2/2. The exact value of sin 60 is √3/2.
Cosine runs the other way. The exact value of cos 60 is 1/2. The exact value of cos 30 is √3/2.
The exact value of tan 45 is 1. The exact value of tan 30 is √3/3. The exact value of tan 60 is √3.
sin 30 equals cos of which angle? The answer is 60, because 30 and 60 add to 90.
sin 45 and cos 45 are equal. The other two angles swap sine and cosine.
Leave the root in the answer. A decimal such as 0.866 is not the exact value.
Start with an equilateral triangle of side 2. Every angle is 60^° and every side is the same length. Drop a perpendicular from the top vertex to the base. It cuts the base in half, so each piece is 1, and it cuts the top angle into two angles of 30^°. Each half is a right-angled triangle with angles 30^°, 60^° and 90^°.
The hypotenuse of each half is still 2, and the side opposite 30^° is 1. The remaining side, opposite 60^°, is found by Pythagoras: 1^2 + h^2 = 2^2, so h^2 = 3 and h = √(3). The three sides are therefore 1, √(3) and 2. Any larger 30-60-90 triangle is a scale copy of this one. Double the sides and you get 2, 2√(3) and 4. Multiply by 6 and you get 6, 6√(3) and 12.
Method: 1. Split an equilateral triangle of side 2 with a perpendicular. 2. Each half has angles 30^°, 60^° and 90^°, and short side 1. 3. Pythagoras gives the third side: h=√(3), so the sides are 1, √(3) and 2.
A square has four right angles and four equal sides. Draw one diagonal and you split it into two congruent right-angled triangles. The diagonal is opposite the right angle, so it is the hypotenuse. The two acute angles are equal, and 180^° - 90^° = 90^° to share, so each is 45^°.
Give each leg length 1. Pythagoras then gives a hypotenuse of sqrt1^2 + 1^2 = √(2). The sides are 1, 1 and √(2). Because the legs match, the side opposite 45^° equals the side adjacent to 45^°. That single fact is why tangent of 45^° is exactly 1. A larger copy with legs of length 5 has hypotenuse 5√(2).
Method: 1. Split a square along a diagonal to get two 45^°-45^°-90^° triangles. 2. Legs of length 1 give hypotenuse sqrt1^2+1^2=√(2). 3. Equal legs mean tan 45^°=1.
Pick the acute angle, name the opposite, the adjacent and the hypotenuse, then write the ratio. In the 1, √(3), 2 triangle, sine of 30^° is 1/2, cosine of 30^° is frac√(3)2, and tangent of 30^° is frac1√(3). From the 60^° angle the opposite and adjacent swap, so sine of 60^° is frac√(3)2, cosine of 60^° is 1/2, and tangent of 60^° is √(3).
In the 1, 1, √(2) triangle, sine and cosine of 45^° are both frac√(2)2. (Higher) The same value can be written frac1√(2), and rationalising that form means multiplying the top and bottom by √(2). Both forms are exact. Tangent of 45^° is 1/1 = 1. At the ends of the list, sine of 0^° is 0 and cosine of 0^° is 1. Sine of 90^° is 1 and cosine of 90^° is 0.
Method: 1. Read sin 30^°=1/2, cos 30^°=dfrac√(3)2 and tan 30^°=dfrac1√(3). 2. Swap opposite and adjacent for 60^°, so sin 60^°=dfrac√(3)2 and tan 60^°=√(3). 3. In the 1, 1, √(2) triangle, sin 45^°=cos 45^°=dfrac√(2)2.
A missing side is a short rearrangement. If sine of the angle equals opposite over hypotenuse, then the opposite is the hypotenuse times the sine, and the hypotenuse is the opposite divided by the sine. Cosine works the same way with the adjacent side. Tangent links the two legs and does not use the hypotenuse at all. When the angle is 45^°, tangent equals 1, so the two legs are the same length.
Keep the surd in the arithmetic. Half of 10√(3) is 5√(3), not a decimal. If a height is 6√(3) and you need an area, multiply by the base and by a half, and simplify the whole numbers in front of the root. The root stays. Only a question that asks for a decimal, or that allows a calculator, is a place to round. This lesson does not round exact values.
Method: 1. Opposite equals hypotenuse times sine, and hypotenuse equals opposite divided by sine. 2. At 45^°, the two legs are equal because the tangent is 1. 3. Keep the surd: half of 10√(3) is 5√(3).
On a non-calculator paper you may be given a line such as cos 30^° × tan 60^° + sin 30^° and asked to work out its exact value or to show that it is an integer. Replace each ratio by its exact value first, then do the arithmetic. Brackets and multiplication come before addition, as always.
Many of these collapse neatly. Since √(3) × √(3) = 3 and frac1√(3) × √(3) = 1, the surds often cancel. For example cos 30^° × tan 60^° = frac√(3)2 × √(3) = 3/2. Write every substitution on its own line, because each correct value earns credit. Doing this for each term keeps the working short, and a fraction that looks messy often simplifies to a whole number at the end. Check by using a calculator for a quick decimal if you are practising, but never rely on it in the exam.
Method: 1. Write the exact value of each ratio underneath the expression. 2. Multiply or divide first, cancelling √(3) × √(3) = 3 and √(2) × √(2) = 2. 3. Then add or subtract, using a common denominator for fractions. 4. State the final exact value and, in a "show that" question, say the word integer or the form asked.
Tip: Learn the values for 30^°, 45^° and 60^° as a small table. They are not given in the exam.
The exact value of tan 30^° is frac1√(3). Many mark schemes also accept frac√(3)3, which has a whole-number denominator. To get from one to the other, multiply the top and bottom by √(3). This is called rationalising the denominator. You can also meet tan 30^° = sin 30^° ÷ cos 30^°. The two forms are equal, so either is correct unless the question asks for a particular one.
When you divide by a ratio, such as finding a side from tan 30^°, you usually divide by a surd. Turn the division into a multiplication by the reciprocal. For instance 3 ÷ frac1√(3) = 3√(3). Your final answer should be a surd with no square root left in the denominator. Rationalising is also a skill you will need in surd questions on the same paper.
Method: 1. Replace the ratio with its exact value. 2. If you are dividing by a fraction, multiply by its reciprocal instead. 3. If a surd is left in the denominator, multiply top and bottom by that surd. 4. Simplify and write the final answer in the form a√(b) or as a fraction.
A common exam question gives a right-angled triangle with an angle of 30^°, 45^° or 60^° and one side, and asks for another side "in exact form". Choose sine, cosine or tangent as normal. Then write the equation with the exact value, such as BC/12 = sin 60^° = frac√(3)2, and solve without a calculator.
Be careful about where the unknown sits. If the unknown is on top, multiply by the other side. If it is on the bottom, divide, which usually means multiplying by a reciprocal and sometimes rationalising. Check the size: in a right-angled triangle the hypotenuse is longest, so an answer of 6√(3) (about 10.4) must be smaller than the hypotenuse of 12.
Method: 1. Label the sides and choose sine, cosine or tangent. 2. Write the equation using the exact value of the ratio. 3. Rearrange to get the unknown by itself. 4. Simplify and check the side is shorter than the hypotenuse.
Some exam questions give you an equation such as cos x = sin 60^° × tan 30^° and ask for the angle x between 0^° and 90^°. Work out the right-hand side exactly. Here it is frac√(3)2 × frac1√(3) = 1/2. Then recall which angle has that cosine: x = 60^°. Showing this substitution is where the method marks are earned, so write each stage on its own line.
This is why the exact values must be learnt as pairs. The same number often appears for two different ratios, for example sin 30^° = cos 60^° = 1/2. Read the question carefully to see whether it asks for sine, cosine or tangent, and give the angle in degrees. If your value matches none in the table, go back and look for an arithmetic slip, because these questions are always designed to give a table value.
Method: 1. Work out the right-hand side exactly and simplify to one fraction or surd. 2. Match that value to the table for the ratio in the equation. 3. Read off the angle in degrees. 4. Substitute your angle back to confirm both sides agree.
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