Trigonometry Examples

$3 (\cos (π) + i \sin (π))$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$3 (- \cos (0) + i \sin (π))$

Step 1.2

The exact value of $\cos (0)$ is $1$ .

$3 (- 1 \cdot 1 + i \sin (π))$

Step 1.3

Multiply $- 1$ by $1$ .

$3 (- 1 + i \sin (π))$

Step 1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$3 (- 1 + i \sin (0))$

Step 1.5

The exact value of $\sin (0)$ is $0$ .

$3 (- 1 + i \cdot 0)$

Step 1.6

Multiply $i$ by $0$ .

$3 (- 1 + 0)$

Step 2

Simplify the expression.

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Step 2.1

Add $- 1$ and $0$ .

$3 \cdot - 1$

Step 2.2

Multiply $3$ by $- 1$ .

$- 3$

Step 3

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 4

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 5

Substitute the actual values of $a = - 3$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 3)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(- 3)}^{2}}$

Step 6.2

Raise $- 3$ to the power of $2$ .

$| z | = \sqrt{0 + 9}$

Step 6.3

Add $0$ and $9$ .

$| z | = \sqrt{9}$

Step 6.4

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2}}$

Step 6.5

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 3$

Step 7

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{0}{- 3})$

Step 8

Since inverse tangent of $\frac{0}{- 3}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 9

Substitute the values of $θ = π$ and $| z | = 3$ .

$3 (\cos (π) + i \sin (π))$