Trigonometry Examples

$\cos (θ) = \frac{1}{3}$ , $\sin (θ) < 0$

Step 1

The sine function is negative in the third and fourth quadrants. The cosine function is positive in the first and fourth quadrants. The set of solutions for $θ$ are limited to the fourth quadrant since that is the only quadrant found in both sets.

Solution is in the fourth quadrant.

Step 2

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

$\cos (θ) = \frac{adjacent}{hypotenuse}$

Step 3

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

$Opposite = - \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 4

Replace the known values in the equation.

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

Step 5

Simplify inside the radical.

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Step 5.1

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

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

Step 5.2

Raise $3$ to the power of $2$ .

Opposite $= - \sqrt{9 - {(1)}^{2}}$

Step 5.3

One to any power is one.

Opposite $= - \sqrt{9 - 1 \cdot 1}$

Step 5.4

Multiply $- 1$ by $1$ .

Opposite $= - \sqrt{9 - 1}$

Step 5.5

Subtract $1$ from $9$ .

Opposite $= - \sqrt{8}$

Step 5.6

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.6.1

Factor $4$ out of $8$ .

Opposite $= - \sqrt{4 (2)}$

Step 5.6.2

Rewrite $4$ as $2^{2}$ .

Opposite $= - \sqrt{2^{2} \cdot 2}$

Step 5.7

Pull terms out from under the radical.

Opposite $= - (2 \sqrt{2})$

Step 5.8

Multiply $2$ by $- 1$ .

Opposite $= - 2 \sqrt{2}$

Step 6

Find the value of sine.

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Step 6.1

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

$\sin (θ) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

$\sin (θ) = \frac{- 2 \sqrt{2}}{3}$

Step 6.3

Move the negative in front of the fraction.

$\sin (θ) = - \frac{2 \sqrt{2}}{3}$

Step 7

Find the value of tangent.

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Step 7.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (θ) = \frac{- 2 \sqrt{2}}{1}$

Step 7.3

Divide $- 2 \sqrt{2}$ by $1$ .

$\tan (θ) = - 2 \sqrt{2}$

Step 8

Find the value of cotangent.

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Step 8.1

Use the definition of cotangent to find the value of $\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (θ) = \frac{1}{- 2 \sqrt{2}}$

Step 8.3

Simplify the value of $\cot (θ)$ .

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Step 8.3.1

Move the negative in front of the fraction.

$\cot (θ) = - \frac{1}{2 \sqrt{2}}$

Step 8.3.2

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cot (θ) = - (\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cot (θ) = - \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 8.3.3.2

Move $\sqrt{2}$ .

$\cot (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$\cot (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.3.3.5

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

$\cot (θ) = - \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 8.3.3.6

Add $1$ and $1$ .

$\cot (θ) = - \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 8.3.3.7

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

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Step 8.3.3.7.1

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

$\cot (θ) = - \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 8.3.3.7.2

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

$\cot (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.7.3

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

$\cot (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.3.3.7.4

Cancel the common factor of $2$ .

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Step 8.3.3.7.4.1

Cancel the common factor.

$\cot (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.3.3.7.4.2

Rewrite the expression.

$\cot (θ) = - \frac{\sqrt{2}}{2 \cdot 2}$

Step 8.3.3.7.5

Evaluate the exponent.

$\cot (θ) = - \frac{\sqrt{2}}{2 \cdot 2}$

Step 8.3.4

Multiply $2$ by $2$ .

$\cot (θ) = - \frac{\sqrt{2}}{4}$

Step 9

Find the value of secant.

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Step 9.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (θ) = \frac{3}{1}$

Step 9.3

Divide $3$ by $1$ .

$\sec (θ) = 3$

Step 10

Find the value of cosecant.

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Step 10.1

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

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (θ) = \frac{3}{- 2 \sqrt{2}}$

Step 10.3

Simplify the value of $\csc (θ)$ .

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Step 10.3.1

Move the negative in front of the fraction.

$\csc (θ) = - \frac{3}{2 \sqrt{2}}$

Step 10.3.2

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\csc (θ) = - (\frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 10.3.3

Combine and simplify the denominator.

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Step 10.3.3.1

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 10.3.3.2

Move $\sqrt{2}$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.3.5

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

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

Step 10.3.3.6

Add $1$ and $1$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 10.3.3.7

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

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Step 10.3.3.7.1

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

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

Step 10.3.3.7.2

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

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.3.3.7.3

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

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.3.7.4

Cancel the common factor of $2$ .

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Step 10.3.3.7.4.1

Cancel the common factor.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.3.7.4.2

Rewrite the expression.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 10.3.3.7.5

Evaluate the exponent.

$\csc (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 10.3.4

Multiply $2$ by $2$ .

$\csc (θ) = - \frac{3 \sqrt{2}}{4}$

Step 11

This is the solution to each trig value.

$\sin (θ) = - \frac{2 \sqrt{2}}{3}$

$\cos (θ) = \frac{1}{3}$

$\tan (θ) = - 2 \sqrt{2}$

$\cot (θ) = - \frac{\sqrt{2}}{4}$

$\sec (θ) = 3$

$\csc (θ) = - \frac{3 \sqrt{2}}{4}$