# 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

**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

**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,