Trigonometry Examples

$\sin (θ) = \frac{1}{2}$ , $\csc (θ)$

Step 1

Use the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sin (θ) = \frac{opposite}{hypotenuse}$

Step 2

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(2)}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

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

Adjacent $= \sqrt{4 - {(1)}^{2}}$

Step 4.2

One to any power is one.

Adjacent $= \sqrt{4 - 1 \cdot 1}$

Step 4.3

Multiply $- 1$ by $1$ .

Adjacent $= \sqrt{4 - 1}$

Step 4.4

Subtract $1$ from $4$ .

Adjacent $= \sqrt{3}$

Step 5

Use the definition of cosecant to find the value of $\csc (θ)$ .

$\csc (θ) = \frac{hypotenuse}{opposite}$

Step 6

Substitute in the known values.

$\csc (θ) = \frac{2}{1}$

Step 7

Divide $2$ by $1$ .

$\csc (θ) = 2$