The exact trig values come from two triangles: a right isosceles triangle with sides 1, 1 and √2 gives the 45° values, and half an equilateral triangle with sides 1, √3 and 2 gives the 30° and 60° values. Combined with the symmetry of the sine, cosine and tangent graphs, they give exact values in every quadrant and let you solve simple equations without a calculator.
Exact values to know
- sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
- sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 60° = √3/2, cos 60° = ½, tan 60° = √3
- sin 0° = 0, cos 0° = 1; sin 90° = 1, cos 90° = 0
Symmetry results
- sin(180° − θ) = sin θ
- cos(−θ) = cos θ and sin(−θ) = −sin θ
- sin and cos repeat every 360°; tan repeats every 180°
Where do the exact values come from?
Two reference triangles give every special value. A right-angled isosceles triangle with shorter sides 1 has hypotenuse √2, which gives sin 45° = cos 45° = 1/√2 = √2/2 and tan 45° = 1. Cut an equilateral triangle of side 2 in half and you get a 30°, 60°, 90° triangle with sides 1, √3 and 2, which gives all the 30° and 60° values.
Sketch both triangles in the margin when you need them. It takes seconds and stops you swapping sin and cos under exam pressure.
How do you find the sign of sin, cos or tan for any angle?
The sign is simply whether the graph is above or below the horizontal axis at that angle. y = sin θ is positive from 0° to 180° and negative from 180° to 360°. y = cos θ is positive near both ends of that interval and negative in the middle. y = tan θ follows its own pattern, because it repeats every 180°.
There is no need for a mnemonic: picture the curve and ask whether it is above or below the axis.
Exact values in every quadrant
The symmetry of each curve turns one special value into its partners. The sine curve is symmetric about 90°, so sin 150° = sin 30° = ½. Its dip below the axis is a mirror image, so sin 210° = sin 330° = −½.
Read the matching height off the graph, then attach the sign from whether the curve is above or below the axis at that angle. Same height means the same value; the mirror image below the axis means the negative of it.
Symmetry and periodicity
Each of these relationships is just a feature of a graph's shape, not an extra fact to memorise:
- sin(180° − θ) = sin θ, because the sine curve is symmetric about 90°
- cos(−θ) = cos θ, because the cosine curve is symmetric about the vertical axis
- sin(−θ) = −sin θ, because the sine curve has rotational symmetry about the origin
- sine and cosine repeat every 360°, while tangent repeats every 180°
These let you relate the value at any angle to an angle between 0° and 90°.
Solving simple equations from the graph
To solve f(θ) = k over an interval, imagine the graph and the horizontal line y = k, and read off every point where they cross. The graph shows how many solutions there are and roughly where they sit; the special values and symmetry give their exact positions. Always check the interval: a wider or shifted interval can add or remove solutions.
Worked examples
Example 1. Use the 30°, 60°, 90° triangle (sides 1, √3, 2) to write down sin 60°, cos 60° and tan 60°.
- Opposite 60°: √3. Adjacent: 1. Hypotenuse: 2.
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3/1 = √3.
Answer: sin 60° = √3/2, cos 60° = ½, tan 60° = √3.
Marks: B1 for the exact values.
Example 2. Find the exact value of sin 150°.
- The sine curve is symmetric about 90°, so 150° mirrors 30°.
- At 150° the curve is above the axis, so the value is positive.
- sin 150° = sin 30° = ½.
Answer: sin 150° = ½.
Marks: B1 for ½.
Example 3. Given that sin 40° ≈ 0.643, write down sin 140° and sin(−40°).
- sin 140° = sin(180° − 40°) = sin 40° ≈ 0.643, by symmetry about 90°.
- sin(−40°) = −sin 40° ≈ −0.643, by rotational symmetry about the origin.
Answer: sin 140° ≈ 0.643, sin(−40°) ≈ −0.643.
Marks: B1 for both.
Example 4. Solve cos θ = ½ for 0° ≤ θ ≤ 360°.
- The line y = ½ first meets the cosine curve at the special value cos 60° = ½.
- The cosine curve is symmetric about 180°, so the second crossing is at 360° − 60° = 300°.
Answer: θ = 60° or θ = 300°.
Marks: M1 for one value, A1 for both.
Common mistakes
- Swapping sin and cos values. sin 30° = ½ but cos 30° = √3/2. Sketch the reference triangle rather than relying on memory.
- Getting the sign wrong in other quadrants. Picture the graph: cos 200° is negative because the cosine curve is below the axis between 90° and 270°.
- Stopping at one solution. Sine and cosine usually cross a horizontal line twice in each 360°. Use symmetry to find the second solution.
- Using tan's period as 360°. tan repeats every 180°, so solutions to tan θ = k are 180° apart.
- Leaving 1/√2 unsimplified when a form is asked for. If the question wants a rationalised answer, write √2/2 rather than 1/√2.
How exact values are examined
In Edexcel International A-Level this is Pure 1 content (paper WMA11), in the trigonometric ratios chapter. In Edexcel A-Level (9MA0) it is AS (Year 1) Pure. Non-calculator style questions ask you to:
- write down exact values at angles such as 120°, 225° or 300°
- combine exact values into a single surd expression
- solve simple equations such as sin θ = √2/2 over a given interval
- state how many solutions an equation has, with a reason from the graph
The Pure 1 trigonometric ratios exam questions come with full mark schemes, and the trigonometric ratios worksheet gives more practice. You can also generate unlimited questions in the A-Level trigonometry revision engine.
Exact values feed straight into the sine and cosine rules, trig graphs and transformations and, later, the addition formulae.
Frequently asked questions
How do I remember the exact trig values?
Draw the two reference triangles: sides 1, 1, √2 for 45°, and sides 1, √3, 2 for 30° and 60°. Every value can be read off them in seconds.
Is tan 30° equal to 1/√3 or √3/3?
Both. They are the same number; √3/3 is the rationalised form, which some questions ask for.
Why is cos 135° negative?
135° is between 90° and 270°, where the cosine curve is below the axis. Its size matches cos 45°, so cos 135° = −√2/2.