# Lesson Worksheet: Small-Angle Approximations Mathematics

In this worksheet, we will practice approximating trigonometric functions when the angle of the function is close to zero.

Q1:

What is the approximate value of for small values of measured in radians?

Q2:

What is the approximate value of for small values of measured in radians?

• A
• B
• C
• D
• E

Q3:

By first showing that when is a small angle measured in radians, state the approximate value of for small values of measured in radians.

Q4:

What is the approximate value of for small values of measured in radians?

Q5:

Find an expression for the approximate value of , given that is a small angle measured in radians.

• A
• B
• C
• D
• E

Q6:

Given that is a small angle measured in radians, find the values of and such that .

• A,
• B,
• C,
• D,
• E,

Use your approximation to give an approximate numerical value of .

Q7:

By first calculating and then approximating using the approximation for for an angle measured in radians, calculate the percentage error in the approximation for to two significant figures.

Q8:

The percentage error in the approximation for is approximately .

By using the exact value of , find the percentage error in approximating this value using the small angle approximation. Give the answer to two significant figures. Assume all angles are in radians.

Which of the following is the reason for the difference between the two percentage errors?

• AThe larger the angle, the smaller the error will be.
• BThe smaller the angle, the smaller the error will be.
• CThere is no relationship between the errors.

Q9:

Given that the percentage error for an approximation of for a small positive angle measured in radians is , which of the following is true?

• A
• B
• C
• D
• E

Q10:

Find an expression for the approximate value of , given that is a small angle measured in radians.

• A
• B
• C
• D
• E

By setting this expression equal to 0 and solving, find the approximate solutions to and comment on the validity of the solutions.

• A and are approximate valid solutions.
• B and are approximate valid solutions.
• C is the only approximate valid solution since the small angle approximations are only valid for small angles.
• D is the only approximate valid solution since the small angle approximations are only valid for small angles.
• EThere are no solutions.