Exact trigonometric values
Section: Trigonometry | Syllabus: Cambridge IGCSE Mathematics (0580)
The Table of Exact Trigonometric Values
Certain angles - 0°, 30°, 45°, 60° and 90° - have sine, cosine and tangent values that can be written exactly, as fractions or surds, rather than as calculator decimal approximations. These exact values come from two special right-angled triangles, and are essential for non-calculator questions.
- An equilateral triangle with sides of length 2, split in half, gives a 30°-60°-90° triangle with sides 1, √3 and 2 - this is the source of the exact values for 30° and 60°
- A right-angled isosceles triangle with two sides of length 1 gives a 45°-45°-90° triangle with sides 1, 1 and √2 - this is the source of the exact values for 45°
- These exact values are worth memorising, since they appear frequently in non-calculator questions
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan | 0 | √3/3 | 1 | √3 | undefined |
Worked Example: Deriving the Exact Values for 30° and 60°
- Question: Use a 30°-60°-90° triangle with sides 1, √3 and 2 to find the exact value of sin(30°) and cos(30°).
- Step 1: In this triangle, the side opposite 30° is 1, the side opposite 60° is √3, and the hypotenuse is 2
- Step 2: sin(30°)=(opposite/hypotenuse)=(1/2)
- Step 3: cos(30°)=(adjacent/hypotenuse)=(√3/2)
- Answer: sin(30°) = 1/2, cos(30°) = √3/2
Common Mistakes
MistakeMixing up which side (1 or √3) belongs to the 30° angle and which belongs to the 60° angle
Fixthe side opposite the smaller angle is always the shorter side - opposite 30° is 1, opposite 60° is √3
MistakeTrying to memorise the table as a random list of numbers rather than understanding where each value comes from
Fixsketching the two special triangles from memory is often more reliable than memorising the table directly - the values can always be re-derived
Using Exact Values to Find Exact Answers
In non-calculator questions, exact trigonometric values are substituted directly into a calculation, and the final answer is left as a fraction or a surd rather than a rounded decimal.
- When a question asks for an answer "in the form a√b" or says "without using a calculator", exact trigonometric values must be used
- Substitute the exact fraction or surd for sin, cos or tan directly into the working, and simplify using surd rules
- Rationalise the denominator if a surd is left on the bottom of a fraction, since a fully simplified exact answer should not have a surd there
Worked Example: Finding an Exact Length
- Question: In a right-angled triangle, the side adjacent to a 30° angle is 9 cm. Find the length of the hypotenuse, giving your answer in the form a√b.
- Step 1: cos(30)=(9/hypotenuse)
- Step 2: hypotenuse=(9/cos(30))=(9/(√3/2))=(18/√3)
- Step 3: Rationalise: (18/√3)=(18√3/3)=6√3
- Answer: 6√3 cm
Worked Example: Finding an Exact Distance Using sin(30°)
- Question: In a right-angled triangle, the hypotenuse is 14 cm, and one angle is 30°. Find, without using a calculator, the length of the side opposite this angle.
- Step 1: sin(30)=(opposite/14)
- Step 2: opposite=14(30)=14×(1/2)
- Step 3: =7
- Answer: 7 cm
Common Mistakes
MistakeConverting the exact fraction or surd to a rounded decimal before finishing the calculation, then giving a decimal final answer when an exact one was required
Fixkeep exact values as fractions or surds throughout the working, only converting to a decimal if the question specifically asks for one
MistakeLeaving a surd in the denominator of the final answer, instead of rationalising it
Fixmultiply both the numerator and denominator by the surd to rationalise, leaving a fully simplified exact answer
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