Precalculus Examples

$\sin (50 °) \cos (170 °) - \cos (50 °) \sin (170 °)$

Step 1

Simplify each term.

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Step 1.1

Evaluate $\sin (50 °)$ .

$0.76604444 \cos (170 °) - \cos (50 °) \sin (170 °)$

Step 1.2

Evaluate $\cos (170 °)$ .

$0.76604444 \cdot - 0.98480775 - \cos (50 °) \sin (170 °)$

Step 1.3

Multiply $0.76604444$ by $- 0.98480775$ .

$- 0.7544065 - \cos (50 °) \sin (170 °)$

Step 1.4

Evaluate $\cos (50 °)$ .

$- 0.7544065 - 1 \cdot 0.6427876 \sin (170 °)$

Step 1.5

Multiply $- 1$ by $0.6427876$ .

$- 0.7544065 - 0.6427876 \sin (170 °)$

Step 1.6

Evaluate $\sin (170 °)$ .

$- 0.7544065 - 0.6427876 \cdot 0.17364817$

Step 1.7

Multiply $- 0.6427876$ by $0.17364817$ .

$- 0.7544065 - 0.11161889$

Step 2

Subtract $0.11161889$ from $- 0.7544065$ .

$- 0.8660254$

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 $a = - 0.8660254$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 0.8660254)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

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

$| z | = \sqrt{0 + {(- 0.8660254)}^{2}}$

Step 6.2

Raise $- 0.8660254$ to the power of $2$ .

$| z | = \sqrt{0 + 0.75}$

Step 6.3

Add $0$ and $0.75$ .

$| z | = \sqrt{0.75}$

Step 7

Evaluate the root.

$| z | = 0.8660254$

Step 8

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

$θ = \arctan (\frac{0}{- 0.8660254})$

Step 9

Since inverse tangent of $\frac{0}{- 0.8660254}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 10

Substitute the values of $θ = π$ and $| z | = 0.8660254$ .

$0.8660254 (\cos (π) + i \sin (π))$