Trigonometry Examples

$A = 48.3$ , $B = 41.7$ , $C = 90$ , $c = 75$

Step 1

Find $b$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

$b = 75 \cdot \cos (48.3)$

Step 1.5

Multiply $75$ by $0.66523035$ .

$b = 49.89227659$

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).

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

Step 2.2

Solve the equation for $a$ .

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

Step 2.3

Substitute the actual values into the equation.

$a = \sqrt{{(75)}^{2} - {(49.89227659)}^{2}}$

Step 2.4

Raise $75$ to the power of $2$ .

$a = \sqrt{5625 - {(49.89227659)}^{2}}$

Step 2.5

Raise $49.89227659$ to the power of $2$ .

$a = \sqrt{5625 - 1 \cdot 2489.23926423}$

Step 2.6

Multiply $- 1$ by $2489.23926423$ .

$a = \sqrt{5625 - 2489.23926423}$

Step 2.7

Subtract $2489.23926423$ from $5625$ .

$a = \sqrt{3135.76073576}$

Step 3

Convert $\sqrt{3135.76073576}$ to a decimal.

$a = 55.99786367$

Step 4

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

$A = 48.3$

$B = 41.7$

$C = 90$

$a = 55.99786367$

$b = 49.89227659$

$c = 75$