Trigonometry Examples

$\csc (x) = 2.3$

Step 1

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

$\csc (x) = \frac{hypotenuse}{opposite}$

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.

$Adjacent = - \sqrt{{(2.3)}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Negate $\sqrt{{(2.3)}^{2} - {(1)}^{2}}$ .

Adjacent $= - \sqrt{{(2.3)}^{2} - {(1)}^{2}}$

Step 4.2

Raise $2.3$ to the power of $2$ .

Adjacent $= - \sqrt{5.29 - {(1)}^{2}}$

Step 4.3

One to any power is one.

Adjacent $= - \sqrt{5.29 - 1 \cdot 1}$

Step 4.4

Multiply $- 1$ by $1$ .

Adjacent $= - \sqrt{5.29 - 1}$

Step 4.5

Subtract $1$ from $5.29$ .

Adjacent $= - \sqrt{4.29}$

Step 5

Find the value of sine.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{1}{2.3}$

Step 5.3

Divide $1$ by $2.3$ .

$\sin (x) = 0.4347826$

Step 6

Find the value of cosine.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{- \sqrt{4.29}}{2.3}$

Step 6.3

Simplify the value of $\cos (x)$ .

Tap for more steps...

Step 6.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{\sqrt{4.29}}{2.3}$

Step 6.3.2

Evaluate the root.

$\cos (x) = - \frac{2.07123151}{2.3}$

Step 6.3.3

Divide $2.07123151$ by $2.3$ .

$\cos (x) = - 1 \cdot 0.90053544$

Step 6.3.4

Multiply $- 1$ by $0.90053544$ .

$\cos (x) = - 0.90053544$

Step 7

Find the value of tangent.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Substitute in the known values.

$\tan (x) = \frac{1}{- \sqrt{4.29}}$

Step 7.3

Simplify the value of $\tan (x)$ .

Tap for more steps...

Step 7.3.1

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

Tap for more steps...

Step 7.3.1.1

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

$\tan (x) = \frac{- 1 \cdot - 1}{- \sqrt{4.29}}$

Step 7.3.1.2

Move the negative in front of the fraction.

$\tan (x) = - \frac{1}{\sqrt{4.29}}$

Step 7.3.2

Multiply $\frac{1}{\sqrt{4.29}}$ by $\frac{\sqrt{4.29}}{\sqrt{4.29}}$ .

$\tan (x) = - (\frac{1}{\sqrt{4.29}} \cdot \frac{\sqrt{4.29}}{\sqrt{4.29}})$

Step 7.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 7.3.3.1

Multiply $\frac{1}{\sqrt{4.29}}$ by $\frac{\sqrt{4.29}}{\sqrt{4.29}}$ .

$\tan (x) = - \frac{\sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 7.3.3.2

Raise $\sqrt{4.29}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 7.3.3.3

Raise $\sqrt{4.29}$ to the power of $1$ .

$\tan (x) = - \frac{\sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 7.3.3.4

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

$\tan (x) = - \frac{\sqrt{4.29}}{{\sqrt{4.29}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\tan (x) = - \frac{\sqrt{4.29}}{{\sqrt{4.29}}^{2}}$

Step 7.3.3.6

Rewrite ${\sqrt{4.29}}^{2}$ as $4.29$ .

Tap for more steps...

Step 7.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4.29}$ as ${4.29}^{\frac{1}{2}}$ .

$\tan (x) = - \frac{\sqrt{4.29}}{{({4.29}^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\tan (x) = - \frac{\sqrt{4.29}}{{4.29}^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\tan (x) = - \frac{\sqrt{4.29}}{{4.29}^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.3.3.6.4.1

Cancel the common factor.

$\tan (x) = - \frac{\sqrt{4.29}}{{4.29}^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\tan (x) = - \frac{\sqrt{4.29}}{4.29}$

Step 7.3.3.6.5

Evaluate the exponent.

$\tan (x) = - \frac{\sqrt{4.29}}{4.29}$

Step 7.3.4

Evaluate the root.

$\tan (x) = - \frac{2.07123151}{4.29}$

Step 7.3.5

Divide $2.07123151$ by $4.29$ .

$\tan (x) = - 1 \cdot 0.48280454$

Step 7.3.6

Multiply $- 1$ by $0.48280454$ .

$\tan (x) = - 0.48280454$

Step 8

Find the value of cotangent.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Substitute in the known values.

$\cot (x) = \frac{- \sqrt{4.29}}{1}$

Step 8.3

Divide $- \sqrt{4.29}$ by $1$ .

$\cot (x) = - \sqrt{4.29}$

Step 9

Find the value of secant.

Tap for more steps...

Step 9.1

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

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

Step 9.2

Substitute in the known values.

$\sec (x) = \frac{2.3}{- \sqrt{4.29}}$

Step 9.3

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

Tap for more steps...

Step 9.3.1

Move the negative in front of the fraction.

$\sec (x) = - \frac{2.3}{\sqrt{4.29}}$

Step 9.3.2

Multiply $\frac{2.3}{\sqrt{4.29}}$ by $\frac{\sqrt{4.29}}{\sqrt{4.29}}$ .

$\sec (x) = - (\frac{2.3}{\sqrt{4.29}} \cdot \frac{\sqrt{4.29}}{\sqrt{4.29}})$

Step 9.3.3

Combine and simplify the denominator.

Tap for more steps...

Step 9.3.3.1

Multiply $\frac{2.3}{\sqrt{4.29}}$ by $\frac{\sqrt{4.29}}{\sqrt{4.29}}$ .

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 9.3.3.2

Raise $\sqrt{4.29}$ to the power of $1$ .

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 9.3.3.3

Raise $\sqrt{4.29}$ to the power of $1$ .

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{\sqrt{4.29} \sqrt{4.29}}$

Step 9.3.3.4

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

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{\sqrt{4.29}}^{1 + 1}}$

Step 9.3.3.5

Add $1$ and $1$ .

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{\sqrt{4.29}}^{2}}$

Step 9.3.3.6

Rewrite ${\sqrt{4.29}}^{2}$ as $4.29$ .

Tap for more steps...

Step 9.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4.29}$ as ${4.29}^{\frac{1}{2}}$ .

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{({4.29}^{\frac{1}{2}})}^{2}}$

Step 9.3.3.6.2

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

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{4.29}^{\frac{1}{2} \cdot 2}}$

Step 9.3.3.6.3

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

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{4.29}^{\frac{2}{2}}}$

Step 9.3.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.3.3.6.4.1

Cancel the common factor.

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{{4.29}^{\frac{2}{2}}}$

Step 9.3.3.6.4.2

Rewrite the expression.

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{4.29}$

Step 9.3.3.6.5

Evaluate the exponent.

$\sec (x) = - \frac{2.3 \sqrt{4.29}}{4.29}$

Step 9.3.4

Multiply $2.3$ by $\sqrt{4.29}$ .

$\sec (x) = - \frac{4.76383249}{4.29}$

Step 9.3.5

Simplify the expression.

Tap for more steps...

Step 9.3.5.1

Divide $4.76383249$ by $4.29$ .

$\sec (x) = - 1 \cdot 1.11045046$

Step 9.3.5.2

Multiply $- 1$ by $1.11045046$ .

$\sec (x) = - 1.11045046$

Step 10

This is the solution to each trig value.

$\sin (x) = 0.4347826$

$\cos (x) = - 0.90053544$

$\tan (x) = - 0.48280454$

$\cot (x) = - \sqrt{4.29}$

$\sec (x) = - 1.11045046$

$\csc (x) = 2.3$