Trigonometry Examples

$\tan (x) = 0$

Step 1

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

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

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.

$Hypotenuse = \sqrt{{(0)}^{2} + {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Raising $0$ to any positive power yields $0$ .

Hypotenuse $= \sqrt{0 + {(1)}^{2}}$

One to any power is one.

Hypotenuse $= \sqrt{0 + 1}$

Add $0$ and $1$ .

Hypotenuse $= \sqrt{1}$

Any root of $1$ is $1$ .

Hypotenuse $= 1$

Step 5

Find the value of sine.

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

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

Substitute in the known values.

$\sin (x) = \frac{0}{1}$

Divide $0$ by $1$ .

$\sin (x) = 0$

Step 6

Find the value of cosine.

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

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

Substitute in the known values.

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

Divide $1$ by $1$ .

$\cos (x) = 1$

Step 7

Find the value of cotangent.

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

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

Substitute in the known values.

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

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

$\cot (x) = Undefined$

Undefined

Step 8

Find the value of secant.

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

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

Substitute in the known values.

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

Divide $1$ by $1$ .

$\sec (x) = 1$

Step 9

Find the value of cosecant.

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

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

Substitute in the known values.

$\csc (x) = \frac{1}{0}$

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

$\csc (x) = Undefined$

Undefined

Step 10

This is the solution to each trig value.

Undefined