Trigonometry Examples

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\begin{matrix} Side & Angle \begin{matrix} b = & 2 c = a = & 3 \end{matrix} & \begin{matrix} A = B = C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 2 \\ c = \\ a = & 3 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

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

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Step 1.1

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}

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

Step 1.2

Solve the equation for

c

$c$ .

c = \sqrt{b^{2} + a^{2}}

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

Step 1.3

Substitute the actual values into the equation.

c = \sqrt{{(2)}^{2} + {(3)}^{2}}

$c = \sqrt{{(2)}^{2} + {(3)}^{2}}$

Step 1.4

Raise

2

$2$ to the power of

2

$2$ .

c = \sqrt{4 + {(3)}^{2}}

$c = \sqrt{4 + {(3)}^{2}}$

Step 1.5

Raise

3

$3$ to the power of

2

$2$ .

c = \sqrt{4 + 9}

$c = \sqrt{4 + 9}$

Step 1.6

Add

4

$4$ and

9

$9$ .

c = \sqrt{13}

$c = \sqrt{13}$

c = \sqrt{13}

$c = \sqrt{13}$

Step 2

Find

B

$B$ .

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Step 2.1

The angle

B

$B$ can be found using the inverse sine function.

B = arcsin (\frac{opp}{hyp})

$B = \arcsin (\frac{opp}{hyp})$

Step 2.2

Substitute in the values of the opposite side to angle

B

$B$ and hypotenuse

\sqrt{13}

$\sqrt{13}$ of the triangle.

B = arcsin (\frac{2}{\sqrt{13}})

$B = \arcsin (\frac{2}{\sqrt{13}})$

Step 2.3

Multiply

\frac{2}{\sqrt{13}}

$\frac{2}{\sqrt{13}}$ by

\frac{\sqrt{13}}{\sqrt{13}}

$\frac{\sqrt{13}}{\sqrt{13}}$ .

B = arcsin (\frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})

$B = \arcsin (\frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply

\frac{2}{\sqrt{13}}

$\frac{2}{\sqrt{13}}$ by

\frac{\sqrt{13}}{\sqrt{13}}

$\frac{\sqrt{13}}{\sqrt{13}}$ .

B = arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})

$B = \arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.4.2

Raise

\sqrt{13}

$\sqrt{13}$ to the power of

1

$1$ .

B = arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})

$B = \arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.4.3

Raise

\sqrt{13}

$\sqrt{13}$ to the power of

1

$1$ .

B = arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})

$B = \arcsin (\frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}})$

Step 2.4.4

Use the power rule

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

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

B = arcsin (\frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})

$B = \arcsin (\frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}})$

Step 2.4.5

Add

1

$1$ and

1

$1$ .

B = arcsin (\frac{2 \sqrt{13}}{{\sqrt{13}}^{2}})

$B = \arcsin (\frac{2 \sqrt{13}}{{\sqrt{13}}^{2}})$

Step 2.4.6

Rewrite

{\sqrt{13}}^{2}

${\sqrt{13}}^{2}$ as

13

$13$ .

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Step 2.4.6.1

Use

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

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

\sqrt{13}

$\sqrt{13}$ as

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

$13^{\frac{1}{2}}$ .

B = arcsin ⎛ ⎜ ⎜ ⎝ \frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}} ⎞ ⎟ ⎟ ⎠

$B = \arcsin (\frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}})$

Step 2.4.6.2

Apply the power rule and multiply exponents,

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

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

B = arcsin (\frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})

$B = \arcsin (\frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}})$

Step 2.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

B = arcsin (\frac{2 \sqrt{13}}{13^{\frac{2}{2}}})

$B = \arcsin (\frac{2 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 2.4.6.4

Cancel the common factor of

2

$2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$B = \arcsin (\frac{2 \sqrt{13}}{13^{\frac{2}{2}}})$

Step 2.4.6.4.2

Rewrite the expression.

$B = \arcsin (\frac{2 \sqrt{13}}{13})$

Step 2.4.6.5

Evaluate the exponent.

$B = \arcsin (\frac{2 \sqrt{13}}{13})$

Step 2.5

Evaluate

$\arcsin (\frac{2 \sqrt{13}}{13})$ .

$B = 33.69006752$

Step 3

Find the last angle of the triangle.

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Step 3.1

The sum of all the angles in a triangle is

$180$ degrees.

$A + 90 + 33.69006752 = 180$

Step 3.2

Solve the equation for

$A$ .

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Step 3.2.1

Add

$90$ and

$33.69006752$ .

$A + 123.69006752 = 180$

Step 3.2.2

Move all terms not containing

$A$ to the right side of the equation.

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Step 3.2.2.1

Subtract

$123.69006752$ from both sides of the equation.

$A = 180 - 123.69006752$

Step 3.2.2.2

Subtract

$123.69006752$ from

$180$ .

$A = 56.30993247$

Step 4

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

$A = 56.30993247$

$B = 33.69006752$

$C = 90$

$a = 3$

$b = 2$

$c = \sqrt{13}$