QuestionFind the legs of a right triangle with an angle of $65^{\circ}$ and a hypotenuse of 12. Round to two decimals.

Studdy Solution

STEP 1

Assumptions1. The triangle is a right triangle. . One acute angle measures $65^{\circ}$ .
3. The hypotenuse measures12 units.
4. We are asked to find the lengths of the legs of the triangle.
5. We will use the trigonometric functions sine and cosine to find the lengths of the legs.

STEP 2

We know that in a right triangle, the sine of an angle is equal to the opposite side divided by the hypotenuse, and the cosine of an angle is equal to the adjacent side divided by the hypotenuse. We can use these facts to find the lengths of the legs.
Let's denote the side opposite to the $65^{\circ}$ angle as $a$ and the side adjacent to the $65^{\circ}$ angle as $b$ .
Then we can write $\sin(65^{\circ}) = \frac{a}{12}$ $\cos(65^{\circ}) = \frac{b}{12}$

STEP 3

Now, we can solve these equations for $a$ and $b$ .
For $a$ $a =12 \cdot \sin(65^{\circ})$ For $b$ $b =12 \cdot \cos(65^{\circ})$

STEP 4

Now, we can calculate the values of $a$ and $b$ using a calculator. Remember to set the calculator to degree mode because the angle is given in degrees.
$a =12 \cdot \sin(65^{\circ}) \approx10.88$ $b =12 \cdot \cos(65^{\circ}) \approx.07$ So, the lengths of the legs of the triangle are approximately10.88 units and.07 units. The answer is A. $.07,10.88$ .

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