Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―2√3 , 2)

검증된 단계별 안내

Step 1: Identify the coordinates of the point given on the terminal side of the angle \( \theta \). Here, the point is \( (-2\sqrt{3}, 2) \).

Step 2: Calculate the distance \( r \) from the origin to the point using the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-2\sqrt{3})^2 + 2^2} \]. This will be used to find the trigonometric functions.

Step 3: Determine the quadrant where the point lies. Since \( x = -2\sqrt{3} < 0 \) and \( y = 2 > 0 \), the point is in the second quadrant. This helps in determining the signs of the trigonometric functions.

Step 4: Find the six trigonometric functions using the definitions: \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}, \quad \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]. Substitute the values of \( x \), \( y \), and \( r \) into these formulas.

Step 5: Rationalize the denominators of the trigonometric function values where necessary to express the answers in simplest form.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is determined by rotating the initial side counterclockwise by the angle measure θ. Understanding this helps in visualizing and sketching the angle based on a given point on its terminal side.

Finding the Reference Angle and Quadrant

The given point (―2√3, 2) lies in the second quadrant because x is negative and y is positive. The reference angle is the acute angle formed between the terminal side and the x-axis. Identifying the quadrant and reference angle is essential to determine the least positive measure of θ and the signs of trigonometric functions.

Six Trigonometric Functions from Coordinates

The six trigonometric functions (sin, cos, tan, csc, sec, cot) can be found using the coordinates of the point on the terminal side. First, calculate the radius r = √(x² + y²). Then, sin θ = y/r, cos θ = x/r, and tan θ = y/x. The reciprocal functions are csc θ = r/y, sec θ = r/x, and cot θ = x/y. Rationalizing denominators ensures simplified exact values.