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A right triangle has sides of
6cm,8cm, and
10cm. Determine the measure of the smallest angle,
theta.

A right triangle has sides of $6 \mathrm{~cm}, 8 \mathrm{~cm}$ , and $10 \mathrm{~cm}$ . Determine the measure of the smallest angle, $\theta$ .

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Find missing angles in special triangles

Full solution

Q. A right triangle has sides of

6 \mathrm{~cm}, 8 \mathrm{~cm}

, and

10 \mathrm{~cm}

. Determine the measure of the smallest angle,

\theta

.

Identify Shortest Side: To find the smallest angle in a right triangle, we can use the trigonometric functions. The smallest angle will be opposite the shortest side. In this case, the shortest side is $6\,\text{cm}$ . We can use the cosine function since we know the lengths of the adjacent side ( $8\,\text{cm}$ ) and the hypotenuse ( $10\,\text{cm}$ ). $\newline$ $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$ $\newline$ $\cos(\theta) = \frac{8\,\text{cm}}{10\,\text{cm}}$
Calculate Cosine Value: Now we calculate the value of $\text{Cos}(\theta)$ . $\text{Cos}(\theta) = \frac{8}{10}$ $\text{Cos}(\theta) = 0.8$
Find Angle using Inverse Cosine: Next, we need to find the angle whose cosine is $0.8$ . We can do this by using the inverse cosine function, also known as arccos. $\newline$ $\theta = \arccos(0.8)$
Calculate Theta: Using a calculator, we find the measure of theta. $\theta \approx \arccos(0.8) \approx 36.87$ degrees

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