Trigonometry Examples

$2 \sin (10 °) \cos (10 °)$

Step 1

Apply the sine double-angle identity.

$\sin (2 \cdot 10 °)$

Step 2

Multiply $2$ by $10 °$ .

$\sin (20)$

Step 3

Evaluate $\sin (20)$ .

$0.34202014$

Step 4

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 5

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 6

Substitute the actual values of $a = 0.34202014$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {0.34202014}^{2}}$

Step 7

Find $| z |$ .

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Step 7.1

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

$| z | = \sqrt{0 + {0.34202014}^{2}}$

Step 7.2

Raise $0.34202014$ to the power of $2$ .

$| z | = \sqrt{0 + 0.11697777}$

Step 7.3

Add $0$ and $0.11697777$ .

$| z | = \sqrt{0.11697777}$

Step 8

Evaluate the root.

$| z | = 0.34202014$

Step 9

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

$θ = \arctan (\frac{0}{0.34202014})$

Step 10

Since inverse tangent of $\frac{0}{0.34202014}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 11

Substitute the values of $θ = 0$ and $| z | = 0.34202014$ .

$0.34202014 (\cos (0) + i \sin (0))$