Trigonometry Examples

$\tan (0) = - (\frac{3}{4})$

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.

$\tan (0) = \frac{opposite}{adjacent}$

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.

$Hypotenuse = \sqrt{{(\frac{3}{4})}^{2} + {(- 1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Apply the product rule to $\frac{3}{4}$ .

Hypotenuse $= \sqrt{\frac{3^{2}}{4^{2}} + {(- 1)}^{2}}$

Step 4.2

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

Hypotenuse $= \sqrt{\frac{9}{4^{2}} + {(- 1)}^{2}}$

Step 4.3

Raise $4$ to the power of $2$ .

Hypotenuse $= \sqrt{\frac{9}{16} + {(- 1)}^{2}}$

Step 4.4

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{\frac{9}{16} + 1}$

Step 4.5

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

Hypotenuse $= \sqrt{\frac{9}{16} + \frac{16}{16}}$

Step 4.6

Combine the numerators over the common denominator.

Hypotenuse $= \sqrt{\frac{9 + 16}{16}}$

Step 4.7

Add $9$ and $16$ .

Hypotenuse $= \sqrt{\frac{25}{16}}$

Step 4.8

Rewrite $\sqrt{\frac{25}{16}}$ as $\frac{\sqrt{25}}{\sqrt{16}}$ .

Hypotenuse $= \frac{\sqrt{25}}{\sqrt{16}}$

Step 4.9

Simplify the numerator.

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Step 4.9.1

Rewrite $25$ as $5^{2}$ .

Hypotenuse $= \frac{\sqrt{5^{2}}}{\sqrt{16}}$

Step 4.9.2

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

Hypotenuse $= \frac{5}{\sqrt{16}}$

Step 4.10

Simplify the denominator.

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Step 4.10.1

Rewrite $16$ as $4^{2}$ .

Hypotenuse $= \frac{5}{\sqrt{4^{2}}}$

Step 4.10.2

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

Hypotenuse $= \frac{5}{4}$

Step 5

Find the value of sine.

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Step 5.1

Use the definition of sine to find the value of $\sin (0)$ .

$\sin (0) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (0) = \frac{\frac{3}{4}}{\frac{5}{4}}$

Step 5.3

Simplify the value of $\sin (0)$ .

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (0) = \frac{3}{4} \cdot \frac{4}{5}$

Step 5.3.2

Cancel the common factor of $4$ .

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Step 5.3.2.1

Cancel the common factor.

$\sin (0) = \frac{3}{4} \cdot \frac{4}{5}$

Step 5.3.2.2

Rewrite the expression.

$\sin (0) = 3 (\frac{1}{5})$

Step 5.3.3

Combine $3$ and $\frac{1}{5}$ .

$\sin (0) = \frac{3}{5}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of $\cos (0)$ .

$\cos (0) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (0) = \frac{- 1}{\frac{5}{4}}$

Step 6.3

Multiply the numerator by the reciprocal of the denominator.

$\cos (0) = - \frac{4}{5}$

Step 7

Multiply $- 1$ by $\frac{3}{4}$ .

$\tan (0) = - \frac{3}{4}$

Step 8

Find the value of cotangent.

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Step 8.1

Use the definition of cotangent to find the value of $\cot (0)$ .

$\cot (0) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (0) = \frac{- 1}{\frac{3}{4}}$

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

$\cot (0) = - \frac{4}{3}$

Step 9

Find the value of secant.

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Step 9.1

Use the definition of secant to find the value of $\sec (0)$ .

$\sec (0) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (0) = \frac{\frac{5}{4}}{- 1}$

Step 9.3

Simplify the value of $\sec (0)$ .

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Step 9.3.1

Move the negative one from the denominator of $\frac{\frac{5}{4}}{- 1}$ .

$\sec (0) = - 1 \cdot \frac{5}{4}$

Step 9.3.2

Rewrite $- 1 \cdot \frac{5}{4}$ as $- \frac{5}{4}$ .

$\sec (0) = - \frac{5}{4}$

Step 10

Find the value of cosecant.

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Step 10.1

Use the definition of cosecant to find the value of $\csc (0)$ .

$\csc (0) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (0) = \frac{\frac{5}{4}}{\frac{3}{4}}$

Step 10.3

Simplify the value of $\csc (0)$ .

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Step 10.3.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (0) = \frac{5}{4} \cdot \frac{4}{3}$

Step 10.3.2

Cancel the common factor of $4$ .

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Step 10.3.2.1

Cancel the common factor.

$\csc (0) = \frac{5}{4} \cdot \frac{4}{3}$

Step 10.3.2.2

Rewrite the expression.

$\csc (0) = 5 (\frac{1}{3})$

Step 10.3.3

Combine $5$ and $\frac{1}{3}$ .

$\csc (0) = \frac{5}{3}$

Step 11

This is the solution to each trig value.

$\sin (0) = \frac{3}{5}$

$\cos (0) = - \frac{4}{5}$

$\tan (0) = - \frac{3}{4}$

$\cot (0) = - \frac{4}{3}$

$\sec (0) = - \frac{5}{4}$

$\csc (0) = \frac{5}{3}$