Trigonometry Examples

$\cot (x) = - \sqrt{35}$

Step 1

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

$\cot (x) = \frac{adjacent}{opposite}$

Step 2

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(1)}^{2} + {(- \sqrt{35})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

One to any power is one.

Hypotenuse $= \sqrt{1 + {(- \sqrt{35})}^{2}}$

Step 4.2

Apply the product rule to $- \sqrt{35}$ .

Hypotenuse $= \sqrt{1 + {(- 1)}^{2} {\sqrt{35}}^{2}}$

Step 4.3

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{1 + 1 {\sqrt{35}}^{2}}$

Step 4.4

Multiply ${\sqrt{35}}^{2}$ by $1$ .

Hypotenuse $= \sqrt{1 + {\sqrt{35}}^{2}}$

Step 4.5

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

Hypotenuse $= \sqrt{1 + {(35^{\frac{1}{2}})}^{2}}$

Step 4.5.2

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

Hypotenuse $= \sqrt{1 + 35^{\frac{1}{2} \cdot 2}}$

Step 4.5.3

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

Hypotenuse $= \sqrt{1 + 35^{\frac{2}{2}}}$

Step 4.5.4

Cancel the common factor of $2$ .

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Step 4.5.4.1

Cancel the common factor.

Hypotenuse $= \sqrt{1 + 35^{\frac{2}{2}}}$

Step 4.5.4.2

Rewrite the expression.

Hypotenuse $= \sqrt{1 + 35}$

Step 4.5.5

Evaluate the exponent.

Hypotenuse $= \sqrt{1 + 35}$

Step 4.6

Add $1$ and $35$ .

Hypotenuse $= \sqrt{36}$

Step 4.7

Rewrite $36$ as $6^{2}$ .

Hypotenuse $= \sqrt{6^{2}}$

Step 4.8

Pull terms out from under the radical, assuming positive real numbers.

Hypotenuse $= 6$

Step 5

Find the value of sine.

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Step 5.1

Use the definition of sine to find the value of $\sin (x)$ .

$\sin (x) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{1}{6}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of $\cos (x)$ .

$\cos (x) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{- \sqrt{35}}{6}$

Step 6.3

Move the negative in front of the fraction.

$\cos (x) = - \frac{\sqrt{35}}{6}$

Step 7

Find the value of tangent.

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Step 7.1

Use the definition of tangent to find the value of $\tan (x)$ .

$\tan (x) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (x) = \frac{1}{- \sqrt{35}}$

Step 7.3

Simplify the value of $\tan (x)$ .

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Step 7.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 7.3.1.1

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

$\tan (x) = \frac{- 1 \cdot - 1}{- \sqrt{35}}$

Step 7.3.1.2

Move the negative in front of the fraction.

$\tan (x) = - \frac{1}{\sqrt{35}}$

Step 7.3.2

Multiply $\frac{1}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\tan (x) = - (\frac{1}{\sqrt{35}} \cdot \frac{\sqrt{35}}{\sqrt{35}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply $\frac{1}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\tan (x) = - \frac{\sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 7.3.3.2

Raise $\sqrt{35}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 7.3.3.3

Raise $\sqrt{35}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 7.3.3.4

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

$\tan (x) = - \frac{\sqrt{35}}{{\sqrt{35}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\tan (x) = - \frac{\sqrt{35}}{{\sqrt{35}}^{2}}$

Step 7.3.3.6

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 7.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\tan (x) = - \frac{\sqrt{35}}{{(35^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\tan (x) = - \frac{\sqrt{35}}{35^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\tan (x) = - \frac{\sqrt{35}}{35^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{\sqrt{35}}{35^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\tan (x) = - \frac{\sqrt{35}}{35}$

Step 7.3.3.6.5

Evaluate the exponent.

$\tan (x) = - \frac{\sqrt{35}}{35}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of $\sec (x)$ .

$\sec (x) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (x) = \frac{6}{- \sqrt{35}}$

Step 8.3

Simplify the value of $\sec (x)$ .

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Step 8.3.1

Move the negative in front of the fraction.

$\sec (x) = - \frac{6}{\sqrt{35}}$

Step 8.3.2

Multiply $\frac{6}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\sec (x) = - (\frac{6}{\sqrt{35}} \cdot \frac{\sqrt{35}}{\sqrt{35}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{6}{\sqrt{35}}$ by $\frac{\sqrt{35}}{\sqrt{35}}$ .

$\sec (x) = - \frac{6 \sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 8.3.3.2

Raise $\sqrt{35}$ to the power of $1$ .

$\sec (x) = - \frac{6 \sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 8.3.3.3

Raise $\sqrt{35}$ to the power of $1$ .

$\sec (x) = - \frac{6 \sqrt{35}}{\sqrt{35} \sqrt{35}}$

Step 8.3.3.4

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

$\sec (x) = - \frac{6 \sqrt{35}}{{\sqrt{35}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\sec (x) = - \frac{6 \sqrt{35}}{{\sqrt{35}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{35}}^{2}$ as $35$ .

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Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{35}$ as $35^{\frac{1}{2}}$ .

$\sec (x) = - \frac{6 \sqrt{35}}{{(35^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

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

$\sec (x) = - \frac{6 \sqrt{35}}{35^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

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

$\sec (x) = - \frac{6 \sqrt{35}}{35^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\sec (x) = - \frac{6 \sqrt{35}}{35^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\sec (x) = - \frac{6 \sqrt{35}}{35}$

Step 8.3.3.6.5

Evaluate the exponent.

$\sec (x) = - \frac{6 \sqrt{35}}{35}$

Step 9

Find the value of cosecant.

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Step 9.1

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

$\csc (x) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (x) = \frac{6}{1}$

Step 9.3

Divide $6$ by $1$ .

$\csc (x) = 6$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{1}{6}$

$\cos (x) = - \frac{\sqrt{35}}{6}$

$\tan (x) = - \frac{\sqrt{35}}{35}$

$\cot (x) = - \sqrt{35}$

$\sec (x) = - \frac{6 \sqrt{35}}{35}$

$\csc (x) = 6$