Trigonometry: Exact Values
Exact trigonometric values for 30°, 45°, and 60° allow precise calculations without a calculator, forming the basis for more advanced trigonometric work.
What You Need to Know
Key Concept Diagram
The exact values for sin, cos, and tan at 30° and 60° are derived from an equilateral triangle with side 2
The exact values for sin 45°, cos 45°, and tan 45° come from an isosceles right triangle with legs of length 1
sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
sin 60° = √3/2, cos 60° = ½, tan 60° = √3; sin 45° = cos 45° = 1/√2 = √2/2, tan 45° = 1
Key Vocabulary
Exact value
A trigonometric ratio expressed as a fraction or surd rather than a decimal approximation
Surd
An irrational number written using a root symbol, such as √2 or √3
Special triangle
A right-angled triangle with angles 30-60-90 or 45-45-90 used to derive exact trig values
Rationalise the denominator
Rewriting a fraction so that no surds appear in the denominator
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the exact value of tan 60°?
Question 2
Calculate the exact value of sin 45° × cos 45°.
Question 3
A right triangle has an angle of 30°. If the hypotenuse is 10 cm, what is the exact length of the side opposite 30°?
Key Concepts Summary
- ●The exact values for sin, cos, and tan at 30° and 60° are derived from an equilateral triangle with side 2
- ●The exact values for sin 45°, cos 45°, and tan 45° come from an isosceles right triangle with legs of length 1
- ●sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
- ●sin 60° = √3/2, cos 60° = ½, tan 60° = √3; sin 45° = cos 45° = 1/√2 = √2/2, tan 45° = 1