Question
Find the values of:
- sin 30°
- cos 60°
- tan 45°
- Evaluate: sin 30° + cos 60° + tan 45°
Solution — Step by Step
Standard angle values are the bedrock of trigonometry. Every exam uses them. We derive them from first principles so you understand where they come from — then the memory trick makes them instant recall.
Step 1: Understand why these angles are “standard.”
The angles 0°, 30°, 45°, 60°, 90° come from two special right triangles:
- 45-45-90 triangle: An isosceles right triangle. Sides in ratio 1 : 1 : √2.
- 30-60-90 triangle: Half of an equilateral triangle. Sides in ratio 1 : √3 : 2.
These triangles have exact, clean side ratios — which is why their trig values are exact fractions, not messy decimals.
Step 2: The 30-60-90 triangle.
Take an equilateral triangle with side 2. All angles = 60°. Drop a perpendicular from one vertex to the opposite side — it bisects the base and the angle.
Now we have a right triangle with:
- Hypotenuse = 2
- Base = 1 (half of 2)
- Height = √(4 − 1) = √3
For the 30° angle (at the top): opposite = 1, adjacent = √3, hypotenuse = 2. For the 60° angle (at the base): opposite = √3, adjacent = 1, hypotenuse = 2.
Step 3: The 45-45-90 triangle.
An isosceles right triangle with legs = 1 and hypotenuse = √2. For the 45° angle: opposite = 1, adjacent = 1, hypotenuse = √2.
Step 4: Read off the values.
sin 30° = opposite/hypotenuse = 1/2
cos 60° = adjacent/hypotenuse = 1/2 (same triangle, cos of 60° uses the adjacent side = 1, hypotenuse = 2) = 1/2
tan 45° = opposite/adjacent = 1/1 = 1
Step 5: Evaluate the expression.
sin 30° + cos 60° + tan 45° = 1/2 + 1/2 + 1 = 1 + 1 = 2
sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = undefined
Why This Works
Notice that sin and cos are complementary: sin θ = cos(90° − θ). So sin 30° = cos 60°, sin 45° = cos 45°, sin 60° = cos 30°. You only need to remember one row, and you get the other for free.
Memory Trick for the sin Row
Write the values as √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°:
- √0/2 = 0
- √1/2 = 1/2
- √2/2 = 1/√2
- √3/2 = √3/2
- √4/2 = 1
For cos, read the same sequence in reverse. For tan, divide the corresponding sin by cos.
In CBSE Class 10, questions that ask you to “evaluate without using tables” are asking you to substitute standard angle values. If you see 30°, 45°, or 60° in an expression, you should immediately know the values without thinking. Drill this table until it’s automatic.
Common Mistake
Mistake: Confusing sin and cos values for 30° and 60°.
Students often swap sin 30° and sin 60°. Remember: sin increases from 0° to 90°. So sin 30° = 1/2 (smaller) and sin 60° = √3/2 (larger). For cos, it decreases: cos 30° = √3/2 (larger) and cos 60° = 1/2 (smaller). The complementary relationship makes this automatic: sin 30° = cos 60° and sin 60° = cos 30°.