Precalculus Examples

$\sin (2 x) = - 1$

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.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

One to any power is one.

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

Step 4.3

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

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

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

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

Step 4.3.2

Add $1$ and $2$ .

Adjacent $= - \sqrt{1 + {(- 1)}^{3}}$

Step 4.4

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

Adjacent $= - \sqrt{1 - 1}$

Step 4.5

Subtract $1$ from $1$ .

Adjacent $= - \sqrt{0}$

Step 4.6

Rewrite $0$ as $0^{2}$ .

Adjacent $= - \sqrt{0^{2}}$

Step 4.7

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

Adjacent $= - 0$

Step 4.8

Multiply $- 1$ by $0$ .

Adjacent $= 0$

Step 5

Find the value of cosine.

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Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\cos (2 x) = \frac{0}{1}$

Step 5.3

Divide $0$ by $1$ .

$\cos (2 x) = 0$

Step 6

Find the value of tangent.

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Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\tan (2 x) = \frac{- 1}{0}$

Step 6.3

Division by $0$ results in tangent being undefined at $2 x$ .

$\tan (2 x) = Undefined$

Undefined

Step 7

Find the value of cotangent.

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Step 7.1

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

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

Step 7.2

Substitute in the known values.

$\cot (2 x) = \frac{0}{- 1}$

Step 7.3

Divide $0$ by $- 1$ .

$\cot (2 x) = 0$

Step 8

Find the value of secant.

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Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\sec (2 x) = \frac{1}{0}$

Step 8.3

Division by $0$ results in secant being undefined at $2 x$ .

$\sec (2 x) = Undefined$

Undefined

Step 9

Find the value of cosecant.

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Step 9.1

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

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

Step 9.2

Substitute in the known values.

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

Step 9.3

Divide $1$ by $- 1$ .

$\csc (2 x) = - 1$

Step 10

This is the solution to each trig value.

Undefined