Small angle approximation
The approximations
As
\sin\theta\approx\theta \cos\theta\approx1-\frac12\theta^2 \tan\theta\approx\theta
These only work when working with radians - not degrees.
What is a small angle?
A ‘small’ angle is considered anything under half a radian.
Sine approximation
For a small angle
Tangent approximation
For a small angle
Cosine approximation
For a small angle
Approximate the value of \cos0.4 , then find the percentage error
\cos(0.4) \approx1-\frac12(0.4)^2 =1-\frac{0.16}2 =1-0.08 =0.92
The actual value is around
- Difference:
0.92106-0.92=0.00106 \frac{0.00106}{0.92106} =0.00115 =0.115\%
flashcards
| Question | Answer |
|---|---|
| sinθ ≈ θ | For a small angle θ in radians, sin θ is approximately equal to θ. |
| cosθ ≈ 1 - ½θ² | For a small angle θ in radians, cos θ is approximately equal to 1 minus one-half θ squared. |
| tanθ ≈ θ | For a small angle θ in radians, tan θ is approximately equal to θ. |
| What units must be used for the small angle approximations? | Radians. |
| What is considered a ‘small’ angle for these approximations? | Any angle under half a radian. |
| Approximate cos(0.4) using the small angle approximation. | 1 - ½(0.4)² = 1 - 0.08 = 0.92. |
| Calculate the percentage error when approximating cos(0.4) as 0.92 (actual value ≈ 0.92106). | Difference = 0.00106; Percentage error = (0.00106 / 0.92106) × 100% ≈ 0.115%. |