Trigonometry Examples

$\sec (0) = 1.499$

Step 1

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

$\sec (0) = \frac{hypotenuse}{adjacent}$

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.

$Opposite = - \sqrt{{(1.499)}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(1.499)}^{2} - {(1)}^{2}}$ .

Opposite $= - \sqrt{{(1.499)}^{2} - {(1)}^{2}}$

Step 4.2

Raise $1.499$ to the power of $2$ .

Opposite $= - \sqrt{2.247001 - {(1)}^{2}}$

Step 4.3

One to any power is one.

Opposite $= - \sqrt{2.247001 - 1 \cdot 1}$

Step 4.4

Multiply $- 1$ by $1$ .

Opposite $= - \sqrt{2.247001 - 1}$

Step 4.5

Subtract $1$ from $2.247001$ .

Opposite $= - \sqrt{1.247001}$

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.

$\sin (0) = \frac{- \sqrt{1.247001}}{1.499}$

Step 5.3

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

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Step 5.3.1

Move the negative in front of the fraction.

$\sin (0) = - \frac{\sqrt{1.247001}}{1.499}$

Step 5.3.2

Evaluate the root.

$\sin (0) = - \frac{1.11669198}{1.499}$

Step 5.3.3

Divide $1.11669198$ by $1.499$ .

$\sin (0) = - 1 \cdot 0.74495796$

Step 5.3.4

Multiply $- 1$ by $0.74495796$ .

$\sin (0) = - 0.74495796$

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.

$\cos (0) = \frac{1}{1.499}$

Step 6.3

Divide $1$ by $1.499$ .

$\cos (0) = 0.6671114$

Step 7

Find the value of tangent.

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Step 7.1

Use the definition of tangent to find the value of $\tan (0)$ .

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

Step 7.2

Substitute in the known values.

$\tan (0) = \frac{- \sqrt{1.247001}}{1}$

Step 7.3

Divide $- \sqrt{1.247001}$ by $1$ .

$\tan (0) = - \sqrt{1.247001}$

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.

$\cot (0) = \frac{1}{- \sqrt{1.247001}}$

Step 8.3

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

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Step 8.3.1

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

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Step 8.3.1.1

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

$\cot (0) = \frac{- 1 \cdot - 1}{- \sqrt{1.247001}}$

Step 8.3.1.2

Move the negative in front of the fraction.

$\cot (0) = - \frac{1}{\sqrt{1.247001}}$

Step 8.3.2

Multiply $\frac{1}{\sqrt{1.247001}}$ by $\frac{\sqrt{1.247001}}{\sqrt{1.247001}}$ .

$\cot (0) = - (\frac{1}{\sqrt{1.247001}} \cdot \frac{\sqrt{1.247001}}{\sqrt{1.247001}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{1}{\sqrt{1.247001}}$ by $\frac{\sqrt{1.247001}}{\sqrt{1.247001}}$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 8.3.3.2

Raise $\sqrt{1.247001}$ to the power of $1$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 8.3.3.3

Raise $\sqrt{1.247001}$ to the power of $1$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 8.3.3.4

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

$\cot (0) = - \frac{\sqrt{1.247001}}{{\sqrt{1.247001}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{{\sqrt{1.247001}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{1.247001}}^{2}$ as $1.247001$ .

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Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1.247001}$ as ${1.247001}^{\frac{1}{2}}$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{{({1.247001}^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{{1.247001}^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\cot (0) = - \frac{\sqrt{1.247001}}{{1.247001}^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\cot (0) = - \frac{\sqrt{1.247001}}{{1.247001}^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\cot (0) = - \frac{\sqrt{1.247001}}{1.247001}$

Step 8.3.3.6.5

Evaluate the exponent.

$\cot (0) = - \frac{\sqrt{1.247001}}{1.247001}$

Step 8.3.4

Evaluate the root.

$\cot (0) = - \frac{1.11669198}{1.247001}$

Step 8.3.5

Divide $1.11669198$ by $1.247001$ .

$\cot (0) = - 1 \cdot 0.89550208$

Step 8.3.6

Multiply $- 1$ by $0.89550208$ .

$\cot (0) = - 0.89550208$

Step 9

Find the value of cosecant.

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Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\csc (0) = \frac{1.499}{- \sqrt{1.247001}}$

Step 9.3

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

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Step 9.3.1

Move the negative in front of the fraction.

$\csc (0) = - \frac{1.499}{\sqrt{1.247001}}$

Step 9.3.2

Multiply $\frac{1.499}{\sqrt{1.247001}}$ by $\frac{\sqrt{1.247001}}{\sqrt{1.247001}}$ .

$\csc (0) = - (\frac{1.499}{\sqrt{1.247001}} \cdot \frac{\sqrt{1.247001}}{\sqrt{1.247001}})$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{1.499}{\sqrt{1.247001}}$ by $\frac{\sqrt{1.247001}}{\sqrt{1.247001}}$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 9.3.3.2

Raise $\sqrt{1.247001}$ to the power of $1$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 9.3.3.3

Raise $\sqrt{1.247001}$ to the power of $1$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{\sqrt{1.247001} \sqrt{1.247001}}$

Step 9.3.3.4

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

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{\sqrt{1.247001}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{\sqrt{1.247001}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{1.247001}}^{2}$ as $1.247001$ .

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Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1.247001}$ as ${1.247001}^{\frac{1}{2}}$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{({1.247001}^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{1.247001}^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{1.247001}^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{{1.247001}^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{1.247001}$

Step 9.3.3.6.5

Evaluate the exponent.

$\csc (0) = - \frac{1.499 \sqrt{1.247001}}{1.247001}$

Step 9.3.4

Multiply $1.499$ by $\sqrt{1.247001}$ .

$\csc (0) = - \frac{1.67392129}{1.247001}$

Step 9.3.5

Simplify the expression.

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Step 9.3.5.1

Divide $1.67392129$ by $1.247001$ .

$\csc (0) = - 1 \cdot 1.34235761$

Step 9.3.5.2

Multiply $- 1$ by $1.34235761$ .

$\csc (0) = - 1.34235761$

Step 10

This is the solution to each trig value.

$\sin (0) = - 0.74495796$

$\cos (0) = 0.6671114$

$\tan (0) = - \sqrt{1.247001}$

$\cot (0) = - 0.89550208$

$\sec (0) = 1.499$

$\csc (0) = - 1.34235761$