Limits and Small Angle Approximations

The video tutorial explains the concept of small angle approximation in trigonometry, focusing on using radians in calculations. It demonstrates how sine and tangent values for small angles are approximately equal to the angle itself when measured in radians. The tutorial further explores this concept through limits and applies it to a practical problem involving a car traveling up an inclined road. The video emphasizes understanding the relationship between angles and trigonometric functions, providing a foundation for calculus concepts.

What should you ensure about your calculator before starting calculations with small angles?

• It is set to radians.

• It is in basic mode.

• It is in scientific mode.

• It is set to degrees.

What happens to the sine of a very small angle in radians?

• It becomes significantly larger than the angle.

• It becomes significantly smaller than the angle.

• It becomes zero.

• It is approximately equal to the angle.

What is the limit of sine theta over theta as theta approaches zero?

• 0

• Infinity

• 1

• Undefined

How does the limit of tan theta over theta compare to that of sine theta over theta as theta approaches zero?

• It is equal to 1.

• It is less than 1.

• It is undefined.

• It is greater than 1.

In the context of the car traveling up an inclined road, what does the small angle approximation help determine?

• The total distance traveled.

• The horizontal distance traveled.

• The vertical distance climbed.

• The speed of the car.

Where is the angle of inclination placed in the diagram of the car on the inclined road?

• At the vertical side of the triangle.

• At the hypotenuse of the triangle.

• At the base of the triangle.

• At the top of the triangle.

Why is it necessary to convert the angle from degrees to radians in the problem involving the car?

• Radians are easier to calculate.

• Degrees are not accurate.

• Radians are the standard unit for trigonometric calculations.

• Degrees are not used in physics.

What is the conversion factor from degrees to radians?

• Multiply by pi/360

• Multiply by 360/pi

• Multiply by pi/180

• Multiply by 180/pi

What is the approximate value of pi/36 in decimal form?

• 0.087

• 0.174

• 0.349

• 0.262

What is the final vertical distance climbed by the car, rounded to the nearest meter?

• 87 meters

• 86 meters

• 88 meters

• 85 meters