Trigonometry Examples

$\frac{1 - \sin (\frac{2 π}{3})}{\cos (\frac{2 π}{3})}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

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

$\frac{1 - \sin (\frac{π}{3})}{\cos (\frac{2 π}{3})}$

Step 1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1 - \frac{\sqrt{3}}{2}}{\cos (\frac{2 π}{3})}$

Step 1.3

Write $1$ as a fraction with a common denominator.

$\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{\cos (\frac{2 π}{3})}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{2 - \sqrt{3}}{2}}{\cos (\frac{2 π}{3})}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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.

$\frac{\frac{2 - \sqrt{3}}{2}}{- \cos (\frac{π}{3})}$

Step 2.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{\frac{2 - \sqrt{3}}{2}}{- \frac{1}{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 - \sqrt{3}}{2} (- 1 \cdot 2)$

Step 4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $- 1 \cdot 2$ .

$\frac{2 - \sqrt{3}}{2} (2 \cdot - 1)$

Step 4.2

Cancel the common factor.

$\frac{2 - \sqrt{3}}{2} (2 \cdot - 1)$

Step 4.3

Rewrite the expression.

$(2 - \sqrt{3}) \cdot - 1$

Step 5

Apply the distributive property.

$2 \cdot - 1 - \sqrt{3} \cdot - 1$

Step 6

Multiply $2$ by $- 1$ .

$- 2 - \sqrt{3} \cdot - 1$

Step 7

Multiply $- \sqrt{3} \cdot - 1$ .

Tap for more steps...

Step 7.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

Step 7.2

Multiply $\sqrt{3}$ by $1$ .

$- 2 + \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$