Trigonometry Table: Standard Values You Need to Know by Heart

There are five angles that appear in trigonometry questions far more often than any others: 0, 30, 45, 60 and 90 degrees. If you know the sine, cosine and tangent values for those five angles, you can answer a large share of trigonometry questions without a calculator.

This guide gives the full table, a reliable way to remember it, the extension to 360 degrees and worked examples showing how the values are actually used.

The three basic ratios

In a right angled triangle, relative to one chosen angle, the sides are named as follows. The hypotenuse is the longest side, opposite the right angle. The opposite is the side facing the chosen angle. The adjacent is the remaining side, next to the chosen angle.

•      Sine equals opposite divided by hypotenuse

•      Cosine equals adjacent divided by hypotenuse

•      Tangent equals opposite divided by adjacent

The traditional memory aid is SOH CAH TOA, taking the first letter of each word in those three definitions.

The trigonometry table for standard angles

Angle	sin	cos	tan
0 degrees	0	1	0
30 degrees	1/2	root 3 over 2	1 over root 3
45 degrees	1 over root 2	1 over root 2	1
60 degrees	root 3 over 2	1/2	root 3
90 degrees	1	0	Undefined

As decimals, for checking your work: sin 30 is 0.5, sin 45 is about 0.707, sin 60 is about 0.866, tan 30 is about 0.577 and tan 60 is about 1.732.

Tan 90 is undefined, not infinity and not zero. At 90 degrees the adjacent side has length zero, and division by zero has no value. Writing infinity in an exam will usually lose the mark.

The easiest way to remember the table

Do not memorise fifteen separate values. Build the row instead.

1.      Write the numbers 0, 1, 2, 3, 4 in a row.

2.      Divide each by 4, giving 0, 1/4, 2/4, 3/4 and 1.

3.      Take the square root of each, giving 0, 1/2, 1 over root 2, root 3 over 2 and 1.

4.      That sequence is the sine row, for 0, 30, 45, 60 and 90 degrees in order.

5.      The cosine row is the same sequence read backwards, since cos of an angle equals sin of 90 minus that angle.

6.      The tangent row is simply sine divided by cosine for each column.

Rebuilding the table this way takes about twenty seconds and is far more reliable under exam pressure than recalling memorised fractions.

Extending the table beyond 90 degrees

Beyond 90 degrees, the size of each value repeats but the sign changes depending on the quadrant.

Quadrant	Angle range	Positive ratios
First	0 to 90 degrees	All three are positive
Second	90 to 180 degrees	Sine only
Third	180 to 270 degrees	Tangent only
Fourth	270 to 360 degrees	Cosine only

The memory aid is the phrase All Students Take Calculus, reading the quadrants anticlockwise from the first: All, Sine, Tangent, Cosine.

Some values worth knowing directly: sin 180 is 0, cos 180 is negative 1, tan 180 is 0, sin 270 is negative 1, cos 270 is 0, sin 360 is 0 and cos 360 is 1.

Worked example 1: finding a missing side

A right angled triangle has a hypotenuse of 10 cm and an angle of 30 degrees. Find the length of the side opposite that angle.

Sine links the opposite and the hypotenuse, so use sin 30 equals opposite divided by 10. From the table, sin 30 is 1/2. Therefore the opposite equals 10 multiplied by 1/2, which gives 5 cm.

Worked example 2: finding a missing angle

In a right angled triangle the opposite side is 7 cm and the adjacent side is 7 cm. Find the angle.

Tangent links the opposite and the adjacent, so tan of the angle equals 7 divided by 7, which is 1. From the table, tan 45 equals 1, so the angle is 45 degrees. This also makes sense, because a right angled triangle with two equal shorter sides must have two angles of 45 degrees.

Worked example 3: using exact values

Find the exact value of sin 60 multiplied by cos 30.

From the table, sin 60 is root 3 over 2 and cos 30 is root 3 over 2. Multiplying gives 3 over 4. Leaving the answer as a fraction rather than a decimal is what exact value questions are asking for, so 3/4 is the correct form and 0.75 may not earn full credit.

Mistakes that cost marks

•      Mixing up sin 30 and cos 30. Sine of the smaller angle gives the smaller value, so sin 30 is 1/2 while cos 30 is root 3 over 2.

•      Writing tan 90 as infinity instead of undefined.

•      Having the calculator in radians when the question is in degrees. Check the mode before you start.

•      Rounding too early in a multi step calculation, which shifts the final answer.

•      Giving a decimal when the question asks for an exact value.

•      Labelling opposite and adjacent relative to the right angle rather than relative to the angle in question.

Quick practice

7.      Find the exact value of sin 45 multiplied by cos 45.

8.      Evaluate tan 60 divided by tan 30.

9.      A right angled triangle has an angle of 60 degrees and an adjacent side of 8 cm. Find the hypotenuse.

10.   State the exact value of cos 0 plus sin 90.

Answers

11.   1/2, since 1 over root 2 multiplied by 1 over root 2 gives 1/2

12.   3, since root 3 divided by 1 over root 3 gives 3

13.   16 cm, since cos 60 equals 8 divided by the hypotenuse, and cos 60 is 1/2

14.   2, since cos 0 is 1 and sin 90 is 1

Frequently Asked Questions

Q. Which trigonometry values do I need to memorise?

A. The sine, cosine and tangent of 0, 30, 45, 60 and 90 degrees cover the vast majority of questions. Rather than memorising fifteen values, build the sine row from the sequence 0, 1, 2, 3, 4 and derive the rest.

Q. Why is tan 90 undefined?

A. Tangent equals sine divided by cosine. At 90 degrees cosine is zero, and division by zero has no defined value. This is why the answer is undefined rather than infinity.

Q. Is the trigonometry table the same in degrees and radians?

A. The values are identical, only the labels change. 30 degrees is pi over 6, 45 degrees is pi over 4, 60 degrees is pi over 3 and 90 degrees is pi over 2.

Q. What does SOH CAH TOA mean?

A. It records the three ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse and Tangent is Opposite over Adjacent.

Q. How do I remember which ratios are positive in each quadrant?

A. Use the phrase All Students Take Calculus, moving anticlockwise from the first quadrant. All three are positive in the first, only sine in the second, only tangent in the third and only cosine in the fourth.

Disclaimer

This article is for general educational guidance. Always follow the formats and rounding rules required by your own exam board.