Trigonometry Examples

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\begin{matrix} Side & Angle \begin{matrix} b = & 10 c = a = & 10 \end{matrix} & \begin{matrix} A = B = C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 10 \\ c = \\ a = & 10 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Assume that angle

C = 90

$C = 90$ .

C = 90

$C = 90$

Step 2

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

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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}

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

Step 2.2

Solve the equation for

c

$c$ .

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

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

Step 2.3

Substitute the actual values into the equation.

c = \sqrt{{(10)}^{2} + {(10)}^{2}}

$c = \sqrt{{(10)}^{2} + {(10)}^{2}}$

Step 2.4

Raise

10

$10$ to the power of

2

$2$ .

c = \sqrt{100 + {(10)}^{2}}

$c = \sqrt{100 + {(10)}^{2}}$

Step 2.5

Raise

10

$10$ to the power of

2

$2$ .

c = \sqrt{100 + 100}

$c = \sqrt{100 + 100}$

Step 2.6

Add

100

$100$ and

100

$100$ .

c = \sqrt{200}

$c = \sqrt{200}$

Step 2.7

Rewrite

200

$200$ as

10^{2} \cdot 2

$10^{2} \cdot 2$ .

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Step 2.7.1

Factor

100

$100$ out of

200

$200$ .

c = \sqrt{100 (2)}

$c = \sqrt{100 (2)}$

Step 2.7.2

Rewrite

100

$100$ as

10^{2}

$10^{2}$ .

c = \sqrt{10^{2} \cdot 2}

$c = \sqrt{10^{2} \cdot 2}$

c = \sqrt{10^{2} \cdot 2}

$c = \sqrt{10^{2} \cdot 2}$

Step 2.8

Pull terms out from under the radical.

c = 10 \sqrt{2}

$c = 10 \sqrt{2}$

c = 10 \sqrt{2}

$c = 10 \sqrt{2}$

Step 3

Find

B

$B$ .

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Step 3.1

The angle

B

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

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

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

Step 3.2

Substitute in the values of the opposite side to angle

B

$B$ and hypotenuse

10 \sqrt{2}

$10 \sqrt{2}$ of the triangle.

B = arcsin (\frac{10}{10 \sqrt{2}})

$B = \arcsin (\frac{10}{10 \sqrt{2}})$

Step 3.3

Cancel the common factor of

10

$10$ .

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Step 3.3.1

Cancel the common factor.

$B = \arcsin (\frac{10}{10 \sqrt{2}})$

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Multiply

$\frac{1}{\sqrt{2}}$ by

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

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

Multiply

$\frac{1}{\sqrt{2}}$ by

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

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

Step 3.5.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 3.5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 3.5.4

Use the power rule

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

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

Step 3.5.5

Add

$1$ and

$1$ .

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

Step 3.5.6

Rewrite

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

$2$ .

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Step 3.5.6.1

Use

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

$\sqrt{2}$ as

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

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

Step 3.5.6.2

Apply the power rule and multiply exponents,

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

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

Step 3.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 3.5.6.4

Cancel the common factor of

$2$ .

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Step 3.5.6.4.1

Cancel the common factor.

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

Step 3.5.6.4.2

Rewrite the expression.

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

Step 3.5.6.5

Evaluate the exponent.

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

Step 3.6

The exact value of

$\arcsin (\frac{\sqrt{2}}{2})$ is

$45$ .

$B = 45$

Step 4

Find the last angle of the triangle.

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Step 4.1

The sum of all the angles in a triangle is

$180$ degrees.

$A + 90 + 45 = 180$

Step 4.2

Solve the equation for

$A$ .

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Step 4.2.1

Add

$90$ and

$45$ .

$A + 135 = 180$

Step 4.2.2

Move all terms not containing

$A$ to the right side of the equation.

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Step 4.2.2.1

Subtract

$135$ from both sides of the equation.

$A = 180 - 135$

Step 4.2.2.2

Subtract

$135$ from

$180$ .

$A = 45$

Step 5

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

$A = 45$

$B = 45$

$C = 90$

$a = 10$

$b = 10$

$c = 10 \sqrt{2}$