Math / Geometry

QuestionSolve the right triangle. 40.

Studdy Solution

STEP 1

1. We are given a right triangle.
2. One angle is $75^\circ$ .
3. The hypotenuse is $100$ units.
4. We need to solve the triangle, meaning find all unknown sides and angles.

STEP 2

1. Determine the measure of the third angle.
2. Use trigonometric ratios to find the lengths of the unknown sides.
3. Verify the solution by checking the Pythagorean Theorem.

STEP 3

To find the third angle in the triangle, we use the fact that the sum of angles in a triangle is $180^\circ$ . Given that one angle is $75^\circ$ and the right angle is $90^\circ$ , we calculate the third angle as follows:
$180^\circ = 90^\circ + 75^\circ + (\text{third angle})$ $180^\circ - 90^\circ - 75^\circ = (\text{third angle})$ $15^\circ = (\text{third angle})$

STEP 4

To find the lengths of the unknown sides, we use trigonometric ratios. Let's denote the side opposite the $75^\circ$ angle as $a$ and the side opposite the $15^\circ$ angle as $b$ .

STEP 5

Use the sine function to find $a$ :
$\sin(75^\circ) = \frac{a}{100}$ $a = 100 \cdot \sin(75^\circ)$
Calculate $\sin(75^\circ)$ using a calculator or trigonometric tables:
$\sin(75^\circ) \approx 0.9659$ $a = 100 \cdot 0.9659$ $a \approx 96.59$

STEP 6

Use the cosine function to find $b$ :
$\cos(75^\circ) = \frac{b}{100}$ $b = 100 \cdot \cos(75^\circ)$
Calculate $\cos(75^\circ)$ :
$\cos(75^\circ) \approx 0.2588$ $b = 100 \cdot 0.2588$ $b \approx 25.88$

STEP 7

Verify the solution using the Pythagorean Theorem:
$a^2 + b^2 = 100^2$ $(96.59)^2 + (25.88)^2 \approx 100^2$ $9331.0281 + 669.7744 \approx 10000$ $10000.8025 \approx 10000$
The values are consistent with the Pythagorean Theorem.
The solved triangle has sides approximately $a = 96.59$ units, $b = 25.88$ units, and angles $75^\circ$ , $15^\circ$ , and $90^\circ$ .

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