CONCEPT PREVIEW Match each equation in Column I with the appropriate right triangle in Column II. In each case, the goal is to find the value of x.

x = 5 tan 38°

Verified step by step guidance

1. Step 1: Understand that each equation involving a trigonometric function (like tangent) relates an angle in a right triangle to the ratio of two sides. For example, \(x = 5 \tan 38^\circ\) means that \(x\) is the length of the side opposite the 38° angle, and 5 is the length of the adjacent side.
2. Step 2: Recall the definition of the tangent function in a right triangle: \(\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\). This helps identify which sides correspond to the given values in the equation.
3. Step 3: For the equation \(x = 5 \tan 38^\circ\), recognize that the side with length 5 is adjacent to the 38° angle, and \(x\) is opposite that angle. So, look for the triangle in Column II where the side adjacent to 38° is 5 and the side opposite is labeled \(x\).
4. Step 4: Repeat this reasoning for each equation in Column I, matching the trigonometric function and given values to the sides and angles in the triangles in Column II.
5. Step 5: Confirm your matches by checking that the ratios from the equations correspond exactly to the side lengths and angles in the triangles, ensuring the correct pairing.

Key Concepts

Right Triangle Trigonometry

Right triangle trigonometry involves relationships between the angles and sides of a right triangle. The primary trigonometric ratios—sine, cosine, and tangent—relate an angle to the ratios of two sides, enabling the calculation of unknown side lengths or angles.

Tangent Function

The tangent of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. It is often used to find a missing side when one side and an angle are known, as in the equation x = 5 tan 38°.

Solving for Unknown Sides Using Trigonometric Equations

To find an unknown side in a right triangle, set up an equation using the appropriate trigonometric ratio based on the given angle and known side. Solving this equation involves substituting known values and using inverse operations or calculator functions to isolate the variable.