The cotangent of a certain acute angle θis (2)/(3). a. Make a right triangle with an acute angle

step 1 We are given that cot theta equal to 3 and theta is in quadrant third. We can write cot theta eq

The cotangent of a certain acute angle θis (2)/(3). a. Make...

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

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

Key Concepts

Pythagorean Theorem

A key theorem in geometry that states the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. This theorem is essential for finding an unknown side length when two side lengths of a right triangle are known.

Sine Function

A trigonometric function that gives the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right triangle. It is widely used for determining angle measures and side lengths in various applications of trigonometry.

Trigonometric Ratios

Definitions that relate the angles of a triangle to the ratios of its sides. In a right triangle, these include sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), tangent (opposite/adjacent), and cotangent (adjacent/opposite), forming the basis for solving various geometric problems.

Right Triangle

A triangle with one 90-degree angle, which serves as the foundational structure in trigonometry for relating angles to side lengths. It allows the application of trigonometric ratios and the Pythagorean theorem to solve for unknown sides or angles.

Cotangent Ratio

A specific trigonometric ratio defined as the length of the side adjacent to an angle divided by the length of the side opposite that angle in a right triangle. It is the reciprocal of the tangent, and it guides the construction of triangles based on known relationships.

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