# Worksheet: Euler’s Formula for Trigonometric Identities

In this worksheet, we will practice using Euler’s formula to prove trigonometric identities like double angle and half angle.

Q1:

Using Euler’s formula, express in terms of .

• A
• B
• C
• D
• E

Q2:

Using Euler’s formula, express in the form , where , , and are constants to be found.

• A
• B
• C
• D
• E

Q3:

Using Euler’s formula, derive a formula for and in terms of and .

• A,
• B,
• C,
• D,
• E,

Q4:

Using Euler’s formula, express in the form , where , , and are constants to be found.

• A
• B
• C
• D
• E

Q5:

Express and in terms of and .

• A,
• B,
• C,
• D,
• E,

Q6:

Use Euler’s formula to express in terms of .

Hint: First write and in terms of and .

• A
• B
• C
• D
• E

Q7:

Using Euler’s formula, express in terms of sine and cosine.

• A
• B
• C
• D
• E

Q8:

What trigonometric identities can be derived by applying Euler’s identity to ?

• A,
• B,
• C,
• D,
• E,

Q9:

Use Euler’s formula to express in terms of sine and cosine.

• A
• B
• C
• D
• E

Given that , what trigonometric identity can be derived by expanding the exponentials in terms of trigonometric functions?

• A
• B
• C
• D
• E

Q10:

Derive two trigonometric identities by considering the real and imaginary parts of .

• A,
• B,
• C,
• D,
• E,

Similarly, what two trigonometric identities can be derived by considering the real and imaginary parts of ?

• A,
• B,
• C,
• D
• E,

Q11:

Use Euler’s formula to derive a formula for and in terms of and .

• A,
• B,
• C,
• D,
• E,

Hence, express in terms of .

• A
• B
• C
• D
• E

Q12:

Use Euler’s formula to express in the form , where , , and are constants to be found.

• A
• B
• C
• D
• E

Hence, find the solutions of in the interval . Give your answer in exact form.

• A, , , ,
• B, ,
• C, ,
• D, ,
• E, , , ,

Q13:

Using Euler’s formula, express in the form , where , , and are constants to be found.

• A
• B
• C
• D
• E

Q14:

Use Euler’s formula to derive a formula for and in terms of and .

• A,
• B,
• C,
• D,
• E,

Q15:

Use Euler’s formula to derive a formula for in terms of .

• A
• B
• C
• D
• E

Use Euler’s formula to drive a formula for in terms of and .

• A
• B
• C
• D
• E

Q16:

Use Euler’s formula to express in terms of sine and cosine.

• A
• B
• C
• D
• E

Given that , what trigonometric identity can be derived by expanding the exponential in terms of trigonometric functions?

• A
• B
• C
• D
• E

Q17:

Use Euler’s formula to drive a formula for and in terms of and .

• A,
• B,
• C,
• D,
• E,