Exercises 105–107 will help you prepare for the material...

Exercises 105–107 will help you prepare for the material covered in the next section. In each exercise, let u be an acute angle in a right triangle, as shown in the figure. These exercises require the use of the Pythagorean Theorem. If $a=5$ and $b=12,$ find the ratio of the length of the side opposite $\theta$ to the length of the hypotenuse.

Exercises 105–107 will help you prepare for the material covered in the next section. In each exercise, let u be an acute angle in a right triangle, as shown in the figure. These exercises require the use of the Pythagorean Theorem.
If $a=5$ and $b=12,$ find the ratio of the length of the side opposite $\theta$ to the length of the hypotenuse.

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Key Concepts

Acute Angle

An acute angle is any angle less than 90 degrees. In a right triangle, the two non-right angles are acute, and understanding their properties is crucial for applying trigonometric ratios effectively.

Right Triangle

A right triangle is a triangle that has one angle measuring 90 degrees. This type of triangle forms the basis for many applications in geometry and trigonometry, where properties like the Pythagorean Theorem and trigonometric ratios are applied to solve for unknown side lengths and angles.

Pythagorean Theorem

The Pythagorean Theorem is a key geometric principle which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential for finding unknown side lengths in right triangles when two side lengths are known.

Trigonometric Ratios

Trigonometric ratios are relationships between the angles of a triangle and the lengths of its sides. In the context of right triangles, the sine, cosine, and tangent functions provide specific ratios that can be used to determine angles or side lengths, with sine being defined as the ratio of the length of the side opposite an angle to the hypotenuse.

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