Trigonometry Examples



\begin{matrix} Side & Angle \begin{matrix} b = c = a = & 8 \end{matrix} & \begin{matrix} A = & 45 B = & 45 C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 8 \end{matrix} & \begin{matrix} A = & 45 \\ B = & 45 \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find

c

$c$ .

Tap for more steps...

Step 1.1

The cosine of an angle is equal to the ratio of the adjacent side to the hypotenuse.

cos (B) = \frac{adj}{hyp}

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

Step 1.2

Substitute the name of each side into the definition of the cosine function.

cos (B) = \frac{a}{c}

$\cos (B) = \frac{a}{c}$

Step 1.3

Set up the equation to solve for the hypotenuse, in this case

c

$c$ .

c = \frac{a}{cos (B)}

$c = \frac{a}{\cos (B)}$

Step 1.4

Substitute the values of each variable into the formula for cosine.

c = \frac{8}{cos (45)}

$c = \frac{8}{\cos (45)}$

Step 1.5

The value of

cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

c = \frac{8}{\frac{\sqrt{2}}{2}}

$c = \frac{8}{\frac{\sqrt{2}}{2}}$

Step 1.6

Multiply the numerator by the reciprocal of the denominator.

c = 8 (\frac{2}{\sqrt{2}})

$c = 8 (\frac{2}{\sqrt{2}})$

Step 1.7

Multiply

\frac{2}{\sqrt{2}}

$\frac{2}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

c = 8 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})

$c = 8 (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.8

Combine and simplify the denominator.

Tap for more steps...

Step 1.8.1

Multiply

\frac{2}{\sqrt{2}}

$\frac{2}{\sqrt{2}}$ by

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ .

c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})

$c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})

$c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})

$c = 8 (\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})$

Step 1.8.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

c = 8 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})

$c = 8 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Step 1.8.5

Add

1

$1$ and

1

$1$ .

c = 8 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})

$c = 8 (\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})$

Step 1.8.6

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

Tap for more steps...

Step 1.8.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

c = 8 ⎛ ⎜ ⎜ ⎝ \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} ⎞ ⎟ ⎟ ⎠

$c = 8 (\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Step 1.8.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

c = 8 (\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})

$c = 8 (\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Step 1.8.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

c = 8 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})

$c = 8 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.8.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 1.8.6.4.1

Cancel the common factor.

$c = 8 (\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})$

Step 1.8.6.4.2

Rewrite the expression.

$c = 8 (\frac{2 \sqrt{2}}{2})$

Step 1.8.6.5

Evaluate the exponent.

$c = 8 (\frac{2 \sqrt{2}}{2})$

Step 1.9

Cancel the common factor of

$2$ .

Tap for more steps...

Step 1.9.1

Factor

$2$ out of

$8$ .

$c = 2 (4) (\frac{2 \sqrt{2}}{2})$

Step 1.9.2

Cancel the common factor.

$c = 2 \cdot (4 (\frac{2 \sqrt{2}}{2}))$

Step 1.9.3

Rewrite the expression.

$c = 4 (2 \sqrt{2})$

Step 1.10

Multiply

$2$ by

$4$ .

$c = 8 \sqrt{2}$

Step 2

Find the last side of the triangle using the Pythagorean theorem.

Tap for more steps...

Step 2.1

Use the Pythagorean theorem to find the unknown side. In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (the two sides other than the hypotenuse).

$a^{2} + b^{2} = c^{2}$

Step 2.2

Solve the equation for

$b$ .

$b = \sqrt{c^{2} - a^{2}}$

Step 2.3

Substitute the actual values into the equation.

$b = \sqrt{{(8 \sqrt{2})}^{2} - {(8)}^{2}}$

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Apply the product rule to

$8 \sqrt{2}$ .

$b = \sqrt{8^{2} {\sqrt{2}}^{2} - {(8)}^{2}}$

Step 2.4.2

Raise

$8$ to the power of

$2$ .

$b = \sqrt{64 {\sqrt{2}}^{2} - {(8)}^{2}}$

Step 2.5

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

Tap for more steps...

Step 2.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$b = \sqrt{64 {(2^{\frac{1}{2}})}^{2} - {(8)}^{2}}$

Step 2.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$b = \sqrt{64 \cdot 2^{\frac{1}{2} \cdot 2} - {(8)}^{2}}$

Step 2.5.3

Combine

$\frac{1}{2}$ and

$2$ .

$b = \sqrt{64 \cdot 2^{\frac{2}{2}} - {(8)}^{2}}$

Step 2.5.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 2.5.4.1

Cancel the common factor.

$b = \sqrt{64 \cdot 2^{\frac{2}{2}} - {(8)}^{2}}$

Step 2.5.4.2

Rewrite the expression.

$b = \sqrt{64 \cdot 2 - {(8)}^{2}}$

Step 2.5.5

Evaluate the exponent.

$b = \sqrt{64 \cdot 2 - {(8)}^{2}}$

Step 2.6

Simplify the expression.

Tap for more steps...

Step 2.6.1

Multiply

$64$ by

$2$ .

$b = \sqrt{128 - {(8)}^{2}}$

Step 2.6.2

Raise

$8$ to the power of

$2$ .

$b = \sqrt{128 - 1 \cdot 64}$

Step 2.6.3

Multiply

$- 1$ by

$64$ .

$b = \sqrt{128 - 64}$

Step 2.6.4

Subtract

$64$ from

$128$ .

$b = \sqrt{64}$

Step 2.6.5

Rewrite

$64$ as

$8^{2}$ .

$b = \sqrt{8^{2}}$

Step 2.6.6

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

$b = 8$

Step 3

These are the results for all angles and sides for the given triangle.

$A = 45$

$B = 45$

$C = 90$

$a = 8$

$b = 8$

$c = 8 \sqrt{2}$