Trigonometry Examples

$\cos (θ) = - 1$

Step 1

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 2

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 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

One to any power is one.

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

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 $1$ .

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

Step 4.3.1.2

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

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

Step 4.3.2

Add $1$ and $2$ .

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

Step 4.4

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

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

Step 4.5

Subtract $1$ from $1$ .

Opposite $= - \sqrt{0}$

Step 4.6

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

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

Step 4.7

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

Opposite $= - 0$

Step 4.8

Multiply $- 1$ by $0$ .

Opposite $= 0$

Step 5

Find the value of sine.

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Step 5.1

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

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (θ) = \frac{0}{1}$

Step 5.3

Divide $0$ by $1$ .

$\sin (θ) = 0$

Step 6

Find the value of tangent.

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Step 6.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (θ) = \frac{0}{- 1}$

Step 6.3

Divide $0$ by $- 1$ .

$\tan (θ) = 0$

Step 7

Find the value of cotangent.

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Step 7.1

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

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{- 1}{0}$

Step 7.3

Division by $0$ results in cotangent being undefined at $θ$ .

$\cot (θ) = Undefined$

Undefined

Step 8

Find the value of secant.

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Step 8.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{1}{- 1}$

Step 8.3

Divide $1$ by $- 1$ .

$\sec (θ) = - 1$

Step 9

Find the value of cosecant.

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Step 9.1

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

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{1}{0}$

Step 9.3

Division by $0$ results in cosecant being undefined at $θ$ .

$\csc (θ) = Undefined$

Undefined

Step 10

This is the solution to each trig value.

Undefined