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
Complete Standard Values Table
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.
💡 Expert Tip
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
⚠️ 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°.