Question

using trigonometry to find lengths
find the missing side. round to the nearest tenth.
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Explanation:

Problem 2:

Step 1: Identify trigonometric ratio

We have a right triangle with angle \(25^\circ\), adjacent side \(10\), and opposite side \(x\). So we use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
\(\tan(25^\circ)=\frac{x}{10}\)

Step 2: Solve for \(x\)

Multiply both sides by \(10\): \(x = 10\times\tan(25^\circ)\)
Calculate \(\tan(25^\circ)\approx0.4663\), so \(x\approx10\times0.4663 = 4.7\) (rounded to nearest tenth)

Step 1: Identify trigonometric ratio

We have hypotenuse \(7\), angle \(39^\circ\), and adjacent side \(x\). So we use \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).
\(\cos(39^\circ)=\frac{x}{7}\)

Step 2: Solve for \(x\)

Multiply both sides by \(7\): \(x = 7\times\cos(39^\circ)\)
Calculate \(\cos(39^\circ)\approx0.7771\), so \(x\approx7\times0.7771 = 5.4\) (rounded to nearest tenth)

Step 1: Identify trigonometric ratio

We have angle \(23^\circ\), adjacent side \(6\), and opposite side \(x\). So we use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
\(\tan(23^\circ)=\frac{x}{6}\)

Step 2: Solve for \(x\)

Multiply both sides by \(6\): \(x = 6\times\tan(23^\circ)\)
Calculate \(\tan(23^\circ)\approx0.4245\), so \(x\approx6\times0.4245 = 2.5\) (rounded to nearest tenth)

Answer:

\(4.7\)

Problem 3: