Trigonometry Examples

$\cot (105)$

Step 1

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

$- \cot (75)$

Step 2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$- \cot (30 + 45)$

Step 3

Apply the sum of angles identity.

$- \frac{\cot (30) \cot (45) - 1}{\cot (45) + \cot (30)}$

Step 4

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

$- \frac{\sqrt{3} \cot (45) - 1}{\cot (45) + \cot (30)}$

Step 5

The exact value of $\cot (45)$ is $1$ .

$- \frac{\sqrt{3} \cdot 1 - 1}{\cot (45) + \cot (30)}$

Step 6

The exact value of $\cot (45)$ is $1$ .

$- \frac{\sqrt{3} \cdot 1 - 1}{1 + \cot (30)}$

Step 7

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

$- \frac{\sqrt{3} \cdot 1 - 1}{1 + \sqrt{3}}$

Step 8

Simplify $- \frac{\sqrt{3} \cdot 1 - 1}{1 + \sqrt{3}}$ .

Tap for more steps...

Step 8.1

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

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

Step 8.2

Multiply $\frac{\sqrt{3} - 1}{1 + \sqrt{3}}$ by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$ .

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

Step 8.3

Multiply $\frac{\sqrt{3} - 1}{1 + \sqrt{3}}$ by $\frac{1 - \sqrt{3}}{1 - \sqrt{3}}$ .

$- \frac{(\sqrt{3} - 1) (1 - \sqrt{3})}{(1 + \sqrt{3}) (1 - \sqrt{3})}$

Step 8.4

Expand the denominator using the FOIL method.

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

Step 8.5

Simplify.

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

Step 8.6

Simplify the numerator.

Tap for more steps...

Step 8.6.1

Factor $- 1$ out of $\sqrt{3}$ .

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

Step 8.6.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 8.6.3

Factor $- 1$ out of $- 1 (- \sqrt{3}) - 1 (1)$ .

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

Step 8.6.4

Reorder terms.

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

Step 8.6.5

Raise $1 - \sqrt{3}$ to the power of $1$ .

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

Step 8.6.6

Raise $1 - \sqrt{3}$ to the power of $1$ .

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

Step 8.6.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.6.8

Add $1$ and $1$ .

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

Step 8.7

Rewrite ${(1 - \sqrt{3})}^{2}$ as $(1 - \sqrt{3}) (1 - \sqrt{3})$ .

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

Step 8.8

Expand $(1 - \sqrt{3}) (1 - \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 8.8.1

Apply the distributive property.

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

Step 8.8.2

Apply the distributive property.

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

Step 8.8.3

Apply the distributive property.

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

Step 8.9

Simplify and combine like terms.

Tap for more steps...

Step 8.9.1

Simplify each term.

Tap for more steps...

Step 8.9.1.1

Multiply $1$ by $1$ .

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

Step 8.9.1.2

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

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

Step 8.9.1.3

Multiply $- 1$ by $1$ .

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

Step 8.9.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 8.9.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 8.9.1.4.2

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

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

Step 8.9.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 8.9.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 8.9.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8.9.1.4.6

Add $1$ and $1$ .

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

Step 8.9.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 8.9.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 8.9.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.9.1.5.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 8.9.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.9.1.5.4.1

Cancel the common factor.

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

Step 8.9.1.5.4.2

Rewrite the expression.

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

Step 8.9.1.5.5

Evaluate the exponent.

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

Step 8.9.2

Add $1$ and $3$ .

$- \frac{- 1 (4 - \sqrt{3} - \sqrt{3})}{- 2}$

Step 8.9.3

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

$- \frac{- 1 (4 - 2 \sqrt{3})}{- 2}$

Step 8.10

Cancel the common factor of $4 - 2 \sqrt{3}$ and $- 2$ .

Tap for more steps...

Step 8.10.1

Factor $2$ out of $- 1 (4 - 2 \sqrt{3})$ .

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

Step 8.10.2

Move the negative one from the denominator of $\frac{- 1 (2 - \sqrt{3})}{- 1}$ .

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

Step 8.11

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

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

Step 8.12

Apply the distributive property.

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

Step 8.13

Multiply $- 1$ by $2$ .

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

Step 8.14

Multiply $- 1 (- \sqrt{3})$ .

Tap for more steps...

Step 8.14.1

Multiply $- 1$ by $- 1$ .

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

Step 8.14.2

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

$- - (- 2 + \sqrt{3})$

Step 8.15

Apply the distributive property.

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

Step 8.16

Multiply $- 1$ by $- 2$ .

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

Step 8.17

Apply the distributive property.

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

Step 8.18

Multiply $- 1$ by $2$ .

$- 2 - - \sqrt{3}$

Step 8.19

Multiply $- - \sqrt{3}$ .

Tap for more steps...

Step 8.19.1

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

Step 8.19.2

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

$- 2 + \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$