Trigonometry Examples



\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 26.4 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 35 \\ C = & 90 \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 26.4 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 35 \\ C = & 90 \end{matrix} \end{matrix}$

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

$180$ degrees.

A + 90 + 35 = 180

$A + 90 + 35 = 180$

Step 1.2

Solve the equation for

A

$A$ .

Tap for more steps...

Step 1.2.1

Add

90

$90$ and

35

$35$ .

A + 125 = 180

$A + 125 = 180$

Step 1.2.2

Move all terms not containing

A

$A$ to the right side of the equation.

Tap for more steps...

Step 1.2.2.1

Subtract

125

$125$ from both sides of the equation.

A = 180 - 125

$A = 180 - 125$

Step 1.2.2.2

Subtract

125

$125$ from

180

$180$ .

A = 55

$A = 55$

A = 55

$A = 55$

A = 55

$A = 55$

A = 55

$A = 55$

Step 2

Find

b

$b$ .

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}

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

Step 2.2

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

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

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

Step 2.3

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

b

$b$ .

b = c \cdot \cos (A)

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

Step 2.4

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

b = 26.4 \cdot \cos (55)

$b = 26.4 \cdot \cos (55)$

Step 2.5

Multiply

26.4

$26.4$ by

0.57357643

$0.57357643$ .

b = 15.14241791

$b = 15.14241791$

b = 15.14241791

$b = 15.14241791$

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}

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

Step 3.2

Solve the equation for

a

$a$ .

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

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

Step 3.3

Substitute the actual values into the equation.

a = \sqrt{{(26.4)}^{2} - {(15.14241791)}^{2}}

$a = \sqrt{{(26.4)}^{2} - {(15.14241791)}^{2}}$

Step 3.4

Raise

26.4

$26.4$ to the power of

2

$2$ .

a = \sqrt{696.96 - {(15.14241791)}^{2}}

$a = \sqrt{696.96 - {(15.14241791)}^{2}}$

Step 3.5

Raise

15.14241791

$15.14241791$ to the power of

2

$2$ .

a = \sqrt{696.96 - 1 \cdot 229.29282045}

$a = \sqrt{696.96 - 1 \cdot 229.29282045}$

Step 3.6

Multiply

- 1

$- 1$ by

229.29282045

$229.29282045$ .

a = \sqrt{696.96 - 229.29282045}

$a = \sqrt{696.96 - 229.29282045}$

Step 3.7

Subtract

229.29282045

$229.29282045$ from

696.96

$696.96$ .

a = \sqrt{467.66717954}

$a = \sqrt{467.66717954}$

a = \sqrt{467.66717954}

$a = \sqrt{467.66717954}$

Step 4

Convert

\sqrt{467.66717954}

$\sqrt{467.66717954}$ to a decimal.

a = 21.62561396

$a = 21.62561396$

Step 5

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

A = 55

$A = 55$

B = 35

$B = 35$

C = 90

$C = 90$

a = 21.62561396

$a = 21.62561396$

b = 15.14241791

$b = 15.14241791$

c = 26.4

$c = 26.4$