Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 10 \\ a = \end{matrix} & \begin{matrix} A = & 30 \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

Step 1

Find the last angle of the triangle.

Tap for more steps...

Step 1.1

The sum of all the angles in a triangle is $180$ degrees.

$30 + 90 + B = 180$

Step 1.2

Solve the equation for $B$ .

Tap for more steps...

Step 1.2.1

Add $30$ and $90$ .

$120 + B = 180$

Step 1.2.2

Move all terms not containing $B$ to the right side of the equation.

Tap for more steps...

Step 1.2.2.1

Subtract $120$ from both sides of the equation.

$B = 180 - 120$

Step 1.2.2.2

Subtract $120$ from $180$ .

$B = 60$

Step 2

Find $b$ .

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

$\cos (A) = \frac{b}{c}$

Step 2.3

Set up the equation to solve for the adjacent side, in this case $b$ .

$b = c \cdot \cos (A)$

Step 2.4

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

$b = 10 \cdot \cos (30)$

Step 2.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.1

Factor $2$ out of $10$ .

$b = 2 (5) \cdot \frac{\sqrt{3}}{2}$

Step 2.5.2

Cancel the common factor.

$b = 2 \cdot 5 \cdot \frac{\sqrt{3}}{2}$

Step 2.5.3

Rewrite the expression.

$b = 5 \cdot \sqrt{3}$

$b = 5 \sqrt{3}$

Step 3

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

Tap for more steps...

Step 3.1

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 3.2

Solve the equation for $a$ .

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

Step 3.3

Substitute the actual values into the equation.

$a = \sqrt{{(10)}^{2} - {(5 \sqrt{3})}^{2}}$

Step 3.4

Simplify the expression.

Tap for more steps...

Step 3.4.1

Raise $10$ to the power of $2$ .

$a = \sqrt{100 - {(5 \sqrt{3})}^{2}}$

Step 3.4.2

Apply the product rule to $5 \sqrt{3}$ .

$a = \sqrt{100 - (5^{2} {\sqrt{3}}^{2})}$

Step 3.4.3

Raise $5$ to the power of $2$ .

$a = \sqrt{100 - (25 {\sqrt{3}}^{2})}$

Step 3.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$a = \sqrt{100 - (25 {(3^{\frac{1}{2}})}^{2})}$

Step 3.5.2

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

$a = \sqrt{100 - (25 \cdot 3^{\frac{1}{2} \cdot 2})}$

Step 3.5.3

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

$a = \sqrt{100 - (25 \cdot 3^{\frac{2}{2}})}$

Step 3.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.4.1

Cancel the common factor.

$a = \sqrt{100 - (25 \cdot 3^{\frac{2}{2}})}$

Step 3.5.4.2

Rewrite the expression.

$a = \sqrt{100 - (25 \cdot 3)}$

Step 3.5.5

Evaluate the exponent.

$a = \sqrt{100 - (25 \cdot 3)}$

Step 3.6

Multiply $- (25 \cdot 3)$ .

Tap for more steps...

Step 3.6.1

Multiply $25$ by $3$ .

$a = \sqrt{100 - 1 \cdot 75}$

Step 3.6.2

Multiply $- 1$ by $75$ .

$a = \sqrt{100 - 75}$

Step 3.7

Subtract $75$ from $100$ .

$a = \sqrt{25}$

Step 3.8

Rewrite $25$ as $5^{2}$ .

$a = \sqrt{5^{2}}$

Step 3.9

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

$a = 5$

Step 4

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

$A = 30$

$B = 60$

$C = 90$

$a = 5$

$b = 5 \sqrt{3}$

$c = 10$