Trigonometry Examples

$\sin (2 x) = 2 (- \frac{24}{25}) (- \frac{7}{25})$

Step 1

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

$\sin (2 x) = \frac{opposite}{hypotenuse}$

Step 2

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

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = - \sqrt{{(1)}^{2} - {(2 (- \frac{24}{25}))}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(1)}^{2} - {(2 (- \frac{24}{25}))}^{2}}$ .

Adjacent $= - \sqrt{{(1)}^{2} - {(2 (- \frac{24}{25}))}^{2}}$

Step 4.2

One to any power is one.

Adjacent $= - \sqrt{1 - {(2 (- \frac{24}{25}))}^{2}}$

Step 4.3

Multiply $2 (- \frac{24}{25})$ .

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Step 4.3.1

Multiply $- 1$ by $2$ .

Adjacent $= - \sqrt{1 - {(- 2 (\frac{24}{25}))}^{2}}$

Step 4.3.2

Combine $- 2$ and $\frac{24}{25}$ .

Adjacent $= - \sqrt{1 - {(\frac{- 2 \cdot 24}{25})}^{2}}$

Step 4.3.3

Multiply $- 2$ by $24$ .

Adjacent $= - \sqrt{1 - {(\frac{- 48}{25})}^{2}}$

Step 4.4

Move the negative in front of the fraction.

Adjacent $= - \sqrt{1 - {(- \frac{48}{25})}^{2}}$

Step 4.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.5.1

Apply the product rule to $- \frac{48}{25}$ .

Adjacent $= - \sqrt{1 - ({(- 1)}^{2} {(\frac{48}{25})}^{2})}$

Step 4.5.2

Apply the product rule to $\frac{48}{25}$ .

Adjacent $= - \sqrt{1 - ({(- 1)}^{2} (\frac{48^{2}}{25^{2}}))}$

Step 4.6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 4.6.1

Move ${(- 1)}^{2}$ .

Adjacent $= - \sqrt{1 + {(- 1)}^{2} \cdot (- 1 \frac{48^{2}}{25^{2}})}$

Step 4.6.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 4.6.2.1

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

Adjacent $= - \sqrt{1 + {(- 1)}^{2} \cdot ((- 1) (\frac{48^{2}}{25^{2}}))}$

Step 4.6.2.2

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

Adjacent $= - \sqrt{1 + {(- 1)}^{2 + 1} (\frac{48^{2}}{25^{2}})}$

Step 4.6.3

Add $2$ and $1$ .

Adjacent $= - \sqrt{1 + {(- 1)}^{3} (\frac{48^{2}}{25^{2}})}$

Step 4.7

Raise $- 1$ to the power of $3$ .

Adjacent $= - \sqrt{1 - \frac{48^{2}}{25^{2}}}$

Step 4.8

Raise $48$ to the power of $2$ .

Adjacent $= - \sqrt{1 - \frac{2304}{25^{2}}}$

Step 4.9

Raise $25$ to the power of $2$ .

Adjacent $= - \sqrt{1 - \frac{2304}{625}}$

Step 4.10

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

Adjacent $= - \sqrt{\frac{625}{625} - \frac{2304}{625}}$

Step 4.11

Combine the numerators over the common denominator.

Adjacent $= - \sqrt{\frac{625 - 2304}{625}}$

Step 4.12

Subtract $2304$ from $625$ .

Adjacent $= - \sqrt{\frac{- 1679}{625}}$

Step 4.13

Move the negative in front of the fraction.

Adjacent $= - \sqrt{- \frac{1679}{625}}$

Step 4.14

Rewrite $- \frac{1679}{625}$ as ${(\frac{i}{25})}^{2} \cdot 1679$ .

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Step 4.14.1

Rewrite $- 1$ as $i^{2}$ .

Adjacent $= - \sqrt{i^{2} (\frac{1679}{625})}$

Step 4.14.2

Factor the perfect power $1^{2}$ out of $1679$ .

Adjacent $= - \sqrt{i^{2} (\frac{1^{2} \cdot 1679}{625})}$

Step 4.14.3

Factor the perfect power $25^{2}$ out of $625$ .

Adjacent $= - \sqrt{i^{2} (\frac{1^{2} \cdot 1679}{25^{2} \cdot 1})}$

Step 4.14.4

Rearrange the fraction $\frac{1^{2} \cdot 1679}{25^{2} \cdot 1}$ .

Adjacent $= - \sqrt{i^{2} ({(\frac{1}{25})}^{2} \cdot 1679)}$

Step 4.14.5

Rewrite $i^{2} {(\frac{1}{25})}^{2}$ as ${(\frac{i}{25})}^{2}$ .

Adjacent $= - \sqrt{{(\frac{i}{25})}^{2} \cdot 1679}$

Step 4.15

Pull terms out from under the radical.

Adjacent $= - (\frac{i}{25} \cdot \sqrt{1679})$

Step 4.16

Combine $\frac{i}{25}$ and $\sqrt{1679}$ .

Adjacent $= - \frac{i \sqrt{1679}}{25}$

Step 5

Simplify the value of $s i n e$ .

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Step 5.1

Multiply $2 (- \frac{24}{25})$ .

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Step 5.1.1

Multiply $- 1$ by $2$ .

$\sin (2 x) = - 2 (\frac{24}{25}) \cdot (- \frac{7}{25})$

Step 5.1.2

Combine $- 2$ and $\frac{24}{25}$ .

$\sin (2 x) = \frac{- 2 \cdot 24}{25} \cdot (- \frac{7}{25})$

Step 5.1.3

Multiply $- 2$ by $24$ .

$\sin (2 x) = \frac{- 48}{25} \cdot (- \frac{7}{25})$

Step 5.2

Move the negative in front of the fraction.

$\sin (2 x) = - \frac{48}{25} \cdot (- \frac{7}{25})$

Step 5.3

Multiply $- \frac{48}{25} (- \frac{7}{25})$ .

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

$\sin (2 x) = 1 (\frac{48}{25}) \cdot \frac{7}{25}$

Step 5.3.2

Multiply $\frac{48}{25}$ by $1$ .

$\sin (2 x) = \frac{48}{25} \cdot \frac{7}{25}$

Step 5.3.3

Multiply $\frac{48}{25}$ by $\frac{7}{25}$ .

$\sin (2 x) = \frac{48 \cdot 7}{25 \cdot 25}$

Step 5.3.4

Multiply $48$ by $7$ .

$\sin (2 x) = \frac{336}{25 \cdot 25}$

Step 5.3.5

Multiply $25$ by $25$ .

$\sin (2 x) = \frac{336}{625}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of $\cos (2 x)$ .

$\cos (2 x) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (2 x) = \frac{- \frac{i \sqrt{1679}}{25}}{1}$

Step 6.3

Divide $- \frac{i \sqrt{1679}}{25}$ by $1$ .

$\cos (2 x) = - \frac{i \sqrt{1679}}{25}$

Step 7

Find the value of tangent.

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Step 7.1

Use the definition of tangent to find the value of $\tan (2 x)$ .

$\tan (2 x) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (2 x) = \frac{2 (- \frac{24}{25})}{- \frac{i \sqrt{1679}}{25}}$

Step 7.3

Simplify the value of $\tan (2 x)$ .

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Step 7.3.1

Dividing two negative values results in a positive value.

$\tan (2 x) = \frac{2 (\frac{24}{25})}{\frac{i \sqrt{1679}}{25}}$

Step 7.3.2

Multiply the numerator by the reciprocal of the denominator.

$\tan (2 x) = 2 (\frac{24}{25}) \cdot \frac{25}{i \sqrt{1679}}$

Step 7.3.3

Multiply $2 (\frac{24}{25})$ .

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Step 7.3.3.1

Combine $2$ and $\frac{24}{25}$ .

$\tan (2 x) = \frac{2 \cdot 24}{25} \cdot \frac{25}{i \sqrt{1679}}$

Step 7.3.3.2

Multiply $2$ by $24$ .

$\tan (2 x) = \frac{48}{25} \cdot \frac{25}{i \sqrt{1679}}$

Step 7.3.4

Cancel the common factor of $25$ .

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Step 7.3.4.1

Cancel the common factor.

$\tan (2 x) = \frac{48}{25} \cdot \frac{25}{i \sqrt{1679}}$

Step 7.3.4.2

Rewrite the expression.

$\tan (2 x) = 48 (\frac{1}{i \sqrt{1679}})$

Step 7.3.5

Combine $48$ and $\frac{1}{i \sqrt{1679}}$ .

$\tan (2 x) = \frac{48}{i \sqrt{1679}}$

Step 7.3.6

Multiply the numerator and denominator of $\frac{48}{40.97560249 i}$ by the conjugate of $40.97560249 i$ to make the denominator real.

$\tan (2 x) = \frac{48}{40.97560249 i} \cdot \frac{i}{i}$

Step 7.3.7

Multiply.

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Step 7.3.7.1

Combine.

$\tan (2 x) = \frac{48 i}{40.97560249 i i}$

Step 7.3.7.2

Simplify the denominator.

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Step 7.3.7.2.1

Add parentheses.

$\tan (2 x) = \frac{48 i}{40.97560249 (i i)}$

Step 7.3.7.2.2

Raise $i$ to the power of $1$ .

$\tan (2 x) = \frac{48 i}{40.97560249 (i i)}$

Step 7.3.7.2.3

Raise $i$ to the power of $1$ .

$\tan (2 x) = \frac{48 i}{40.97560249 (i i)}$

Step 7.3.7.2.4

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

$\tan (2 x) = \frac{48 i}{40.97560249 i^{1 + 1}}$

Step 7.3.7.2.5

Add $1$ and $1$ .

$\tan (2 x) = \frac{48 i}{40.97560249 i^{2}}$

Step 7.3.7.2.6

Rewrite $i^{2}$ as $- 1$ .

$\tan (2 x) = \frac{48 i}{40.97560249 \cdot - 1}$

Step 7.3.8

Simplify the expression.

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Step 7.3.8.1

Multiply $40.97560249$ by $- 1$ .

$\tan (2 x) = \frac{48 i}{- 40.97560249}$

Step 7.3.8.2

Move the negative in front of the fraction.

$\tan (2 x) = - \frac{48 i}{40.97560249}$

Step 7.3.9

Factor $48$ out of $48 i$ .

$\tan (2 x) = - \frac{48 (i)}{40.97560249}$

Step 7.3.10

Factor $40.97560249$ out of $40.97560249$ .

$\tan (2 x) = - \frac{48 (i)}{40.97560249 (1)}$

Step 7.3.11

Separate fractions.

$\tan (2 x) = - (\frac{48}{40.97560249} \cdot \frac{i}{1})$

Step 7.3.12

Divide $48$ by $40.97560249$ .

$\tan (2 x) = - (1.17142877 (\frac{i}{1}))$

Step 7.3.13

Divide $i$ by $1$ .

$\tan (2 x) = - (1.17142877 i)$

Step 7.3.14

Multiply $1.17142877$ by $- 1$ .

$\tan (2 x) = - 1.17142877 i$

Step 8

Find the value of cotangent.

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Step 8.1

Use the definition of cotangent to find the value of $\cot (2 x)$ .

$\cot (2 x) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (2 x) = \frac{- \frac{i \sqrt{1679}}{25}}{2 (- \frac{24}{25})}$

Step 8.3

Simplify the value of $\cot (2 x)$ .

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Step 8.3.1

Dividing two negative values results in a positive value.

$\cot (2 x) = \frac{\frac{i \sqrt{1679}}{25}}{2 (\frac{24}{25})}$

Step 8.3.2

Multiply the numerator by the reciprocal of the denominator.

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot \frac{1}{2 (\frac{24}{25})}$

Step 8.3.3

Combine $2$ and $\frac{24}{25}$ .

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot \frac{1}{\frac{2 \cdot 24}{25}}$

Step 8.3.4

Multiply $2$ by $24$ .

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot \frac{1}{\frac{48}{25}}$

Step 8.3.5

Multiply the numerator by the reciprocal of the denominator.

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot (1 (\frac{25}{48}))$

Step 8.3.6

Multiply $\frac{25}{48}$ by $1$ .

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot \frac{25}{48}$

Step 8.3.7

Cancel the common factor of $25$ .

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Step 8.3.7.1

Cancel the common factor.

$\cot (2 x) = \frac{i \sqrt{1679}}{25} \cdot \frac{25}{48}$

Step 8.3.7.2

Rewrite the expression.

$\cot (2 x) = i \sqrt{1679} (\frac{1}{48})$

Step 8.3.8

Combine $\frac{1}{48}$ and $i$ .

$\cot (2 x) = \frac{i}{48} \cdot \sqrt{1679}$

Step 8.3.9

Combine $\frac{i}{48}$ and $\sqrt{1679}$ .

$\cot (2 x) = \frac{i \sqrt{1679}}{48}$

Step 9

Find the value of secant.

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Step 9.1

Use the definition of secant to find the value of $\sec (2 x)$ .

$\sec (2 x) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (2 x) = \frac{1}{- \frac{i \sqrt{1679}}{25}}$

Step 9.3

Simplify the value of $\sec (2 x)$ .

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Step 9.3.1

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

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Step 9.3.1.1

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

$\sec (2 x) = \frac{- 1 \cdot - 1}{- \frac{i \sqrt{1679}}{25}}$

Step 9.3.1.2

Move the negative in front of the fraction.

$\sec (2 x) = - \frac{1}{\frac{i \sqrt{1679}}{25}}$

Step 9.3.2

Multiply the numerator by the reciprocal of the denominator.

$\sec (2 x) = - (1 (\frac{25}{i \sqrt{1679}}))$

Step 9.3.3

Multiply $\frac{25}{i \sqrt{1679}}$ by $1$ .

$\sec (2 x) = - \frac{25}{i \sqrt{1679}}$

Step 9.3.4

Multiply the numerator and denominator of $\frac{25}{40.97560249 i}$ by the conjugate of $40.97560249 i$ to make the denominator real.

$\sec (2 x) = - (\frac{25}{40.97560249 i} \cdot \frac{i}{i})$

Step 9.3.5

Multiply.

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Step 9.3.5.1

Combine.

$\sec (2 x) = - \frac{25 i}{40.97560249 i i}$

Step 9.3.5.2

Simplify the denominator.

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Step 9.3.5.2.1

Add parentheses.

$\sec (2 x) = - \frac{25 i}{40.97560249 (i i)}$

Step 9.3.5.2.2

Raise $i$ to the power of $1$ .

$\sec (2 x) = - \frac{25 i}{40.97560249 (i i)}$

Step 9.3.5.2.3

Raise $i$ to the power of $1$ .

$\sec (2 x) = - \frac{25 i}{40.97560249 (i i)}$

Step 9.3.5.2.4

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

$\sec (2 x) = - \frac{25 i}{40.97560249 i^{1 + 1}}$

Step 9.3.5.2.5

Add $1$ and $1$ .

$\sec (2 x) = - \frac{25 i}{40.97560249 i^{2}}$

Step 9.3.5.2.6

Rewrite $i^{2}$ as $- 1$ .

$\sec (2 x) = - \frac{25 i}{40.97560249 \cdot - 1}$

Step 9.3.6

Simplify the expression.

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Step 9.3.6.1

Multiply $40.97560249$ by $- 1$ .

$\sec (2 x) = - \frac{25 i}{- 40.97560249}$

Step 9.3.6.2

Move the negative in front of the fraction.

$\sec (2 x) = \frac{25 i}{40.97560249}$

Step 9.3.7

Factor $25$ out of $25 i$ .

$\sec (2 x) = \frac{25 (i)}{40.97560249}$

Step 9.3.8

Factor $40.97560249$ out of $40.97560249$ .

$\sec (2 x) = \frac{25 (i)}{40.97560249 (1)}$

Step 9.3.9

Separate fractions.

$\sec (2 x) = \frac{25}{40.97560249} \cdot \frac{i}{1}$

Step 9.3.10

Divide $25$ by $40.97560249$ .

$\sec (2 x) = 0.61011915 (\frac{i}{1})$

Step 9.3.11

Divide $i$ by $1$ .

$\sec (2 x) = 0.61011915 i$

Step 9.3.12

Multiply $0.61011915$ by $- 1$ .

$\sec (2 x) = - (- 0.61011915 i)$

Step 9.3.13

Multiply $- 0.61011915$ by $- 1$ .

$\sec (2 x) = 0.61011915 i$

Step 10

Find the value of cosecant.

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Step 10.1

Use the definition of cosecant to find the value of $\csc (2 x)$ .

$\csc (2 x) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (2 x) = \frac{1}{2 (- \frac{24}{25})}$

Step 10.3

Simplify the value of $\csc (2 x)$ .

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Step 10.3.1

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

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Step 10.3.1.1

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

$\csc (2 x) = \frac{- 1 \cdot - 1}{2 (- \frac{24}{25})}$

Step 10.3.1.2

Move the negative in front of the fraction.

$\csc (2 x) = - \frac{1}{2 (\frac{24}{25})}$

Step 10.3.2

Combine $2$ and $\frac{24}{25}$ .

$\csc (2 x) = - \frac{1}{\frac{2 \cdot 24}{25}}$

Step 10.3.3

Multiply $2$ by $24$ .

$\csc (2 x) = - \frac{1}{\frac{48}{25}}$

Step 10.3.4

Multiply the numerator by the reciprocal of the denominator.

$\csc (2 x) = - (1 (\frac{25}{48}))$

Step 10.3.5

Multiply $\frac{25}{48}$ by $1$ .

$\csc (2 x) = - \frac{25}{48}$

Step 11

This is the solution to each trig value.

$\sin (2 x) = \frac{336}{625}$

$\cos (2 x) = - \frac{i \sqrt{1679}}{25}$

$\tan (2 x) = - 1.17142877 i$

$\cot (2 x) = \frac{i \sqrt{1679}}{48}$

$\sec (2 x) = 0.61011915 i$

$\csc (2 x) = - \frac{25}{48}$