Getting the tangent from the sine: The sine of a certain...

Getting the tangent from the sine: The sine of a certain acute angle $\theta$ is $\frac{2}{3}$. a. Make a right triangle with an acute angle $\theta$ that has a sine of $\frac{2}{3}$. Suggestion: Choose convenient lengths for the opposite side and hypotenuse. b. Find the length of the side adjacent to $\theta$ in the right triangle you made in part a. c. Find $\tan \theta$.

Getting the tangent from the sine: The sine of a certain acute angle $\theta$ is $\frac{2}{3}$.
a. Make a right triangle with an acute angle $\theta$ that has a sine of $\frac{2}{3}$. Suggestion: Choose convenient lengths for the opposite side and hypotenuse.
b. Find the length of the side adjacent to $\theta$ in the right triangle you made in part a.
c. Find $\tan \theta$.

Key Concepts

Tangent Function

The tangent function is one of the trigonometric functions and is defined as the ratio of the sine of an angle to its cosine, or equivalently, as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle. It is especially useful in problems where the relationship between the angle and the slope or incline is examined.

Pythagorean Theorem

The Pythagorean Theorem is a principle in geometry that relates the lengths of the sides in a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem is essential for finding a missing side length when two side lengths are known, and it is widely used in trigonometry and geometry.

Right Triangle

A right triangle is a triangle in which one of the angles is exactly 90 degrees. It is a fundamental concept in geometry and trigonometry because its structure allows for the definition of trigonometric functions, where the sides can be labeled as opposite, adjacent, and hypotenuse relative to a given acute angle.

Trigonometric Ratios

Trigonometric ratios, including sine, cosine, and tangent, are relationships between the angles of a triangle and the lengths of its sides. In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. These ratios are crucial for solving problems involving angles and distances.

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