What are Standard Values?
Standard values are fixed, exact numerical values of trigonometric ratios for five specific angles:0°, 30°, 45°, 60°, and 90°.
In Class 10 Mathematics, these five angles are fundamental because their trigonometric ratios can be derived using simple geometry without needing a calculator.
Memorising standard values makes solving trigonometric expressions, heights and distances, and algebraic simplification much faster.
Trigonometric Values Table
Here is the complete reference table for all 6 trigonometric ratios across the standard angles:
How to Remember the Table Easily
You do not need to memorize all 30 values! Follow these 3 simple tricks:
The Square Root Pattern for Sin θ
Write numbers 0 to 4 under each angle, divide by 4, and take the square root:
Just copy the sin θ values from right to left (90° down to 0°) to get all cos θ values.
Divide the top number of sin θ by the top number of cos θ to get tan θ instantly!
Where do these values come from?
Let's derive the values for 45°, 30°, and 60° using simple right triangles:
Trigonometric Ratios of 45°
Consider an isosceles right-angled triangle with two equal sides of length a:
- Perpendicular = a
- Base = a
- By Pythagoras theorem: Hypotenuse = √(a² + a²) = a√2
sin 45° = Perpendicular / Hypotenuse = a / (a√2) = 1/√2
cos 45° = Base / Hypotenuse = a / (a√2) = 1/√2
tan 45° = Perpendicular / Base = a / a = 1
Trigonometric Ratios of 30° and 60°
Consider an equilateral triangle of side 2a split by an altitude:
- Base = a
- Hypotenuse = 2a
- By Pythagoras theorem: Altitude = √((2a)² - a²) = a√3
sin 30° = Base / Hypotenuse = a / 2a = 1/2
sin 60° = Altitude / Hypotenuse = (a√3) / 2a = √3/2
tan 60° = Altitude / Base = (a√3) / a = √3
Practice Problems
Evaluate: sin 60° cos 30° + sin 30° cos 60°
Solution:
Substitute the standard values from the table:
- sin 60° = √3/2
- cos 30° = √3/2
- sin 30° = 1/2
- cos 60° = 1/2
= (3/4) + (1/4)
= 4/4 = 1
Evaluate: 2 tan² 45° + cos² 30° - sin² 60°
Solution:
- tan 45° = 1 → tan² 45° = 1² = 1
- cos 30° = √3/2 → cos² 30° = 3/4
- sin 60° = √3/2 → sin² 60° = 3/4
= 2 + 0 = 2
Quick Revision
Standard values apply to 0°, 30°, 45°, 60°, and 90°.
Use the pattern √(n)/2 for n = 0, 1, 2, 3, 4 to get sin values.
cos θ values are simply the reverse order of sin θ values.
tan 90°, cosec 0°, sec 90°, and cot 0° are Not Defined (division by 0).