In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined.
(0, -1)

검증된 단계별 안내

Identify the coordinates of the point on the terminal side of angle \( \theta \). Here, the point is \( (0, -1) \), so \( x = 0 \) and \( y = -1 \).

Calculate the radius \( r \), which is the distance from the origin to the point, using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the values to get \( r = \sqrt{0^2 + (-1)^2} \).

Recall the definitions of the six trigonometric functions in terms of \( x \), \( y \), and \( r \): \[ \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x} \] \[ \csc \theta = \frac{r}{y}, \quad \sec \theta = \frac{r}{x}, \quad \cot \theta = \frac{x}{y} \]

Substitute the values of \( x \), \( y \), and \( r \) into each function. Note that if the denominator in any function is zero, that function is undefined.

Simplify each expression to find the exact values of the six trigonometric functions or state which are undefined based on the point \( (0, -1) \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Coordinates and the Terminal Side of an Angle

The terminal side of an angle θ in standard position passes through a point (x, y) on the coordinate plane. This point helps determine the angle's trigonometric values by relating x and y to the radius r, which is the distance from the origin to the point.

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are defined using the coordinates (x, y) and radius r as follows: sin(θ) = y/r, cos(θ) = x/r, tan(θ) = y/x, and their reciprocals cosec(θ) = r/y, sec(θ) = r/x, cot(θ) = x/y. These definitions allow calculation of exact values.

Handling Undefined Trigonometric Functions

Some trigonometric functions become undefined when their denominators are zero, such as tan(θ) = y/x when x = 0. Recognizing when a function is undefined is crucial, especially when the point lies on an axis, as in (0, -1), where division by zero may occur.

In Exercises 37–38, a point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of θ, or state that the function is undefined. (0, -1)