Trigonometry Examples

$\csc (- 135) + \sin (225) - \tan (30) + \tan (150)$

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 cosecant is negative in the third quadrant.

$- \csc (45) + \sin (225) - \tan (30) + \tan (150)$

Step 1.2

The exact value of $\csc (45)$ is $\sqrt{2}$ .

$- \sqrt{2} + \sin (225) - \tan (30) + \tan (150)$

Step 1.3

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

$- \sqrt{2} - \sin (45) - \tan (30) + \tan (150)$

Step 1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- \sqrt{2} - \frac{\sqrt{2}}{2} - \tan (30) + \tan (150)$

Step 1.5

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$- \sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{3} + \tan (150)$

Step 1.6

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

$- \sqrt{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{3} - \tan (30)$

Step 1.7

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

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

Step 3

Simplify each term.

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Step 3.1

Move the negative in front of the fraction.

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

Step 3.2

Move the negative in front of the fraction.

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

Step 4

To write $- \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5

Combine fractions.

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Step 5.1

Combine $- \sqrt{2}$ and $\frac{2}{2}$ .

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $2$ by $- 1$ .

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

Step 6.1.2

Subtract $\sqrt{2}$ from $- 2 \sqrt{2}$ .

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

Step 6.2

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 3.27602088 \dots$