In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

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Recall that the unit circle is a circle with radius 1 centered at the origin (0,0) in the coordinate plane. Any point P(x, y) on the unit circle satisfies the equation \(x^2 + y^2 = 1\).

Understand that for a real number \(t\), the point \(P(x, y)\) on the unit circle corresponds to the angle \(t\) (measured in radians) from the positive x-axis. Here, \(x = \cos(t)\) and \(y = \sin(t)\).

Use the coordinates of point \(P(x, y)\) to find the primary trigonometric functions: \(\sin(t) = y\) and \(\cos(t) = x\).

Calculate the other trigonometric functions using the definitions: \(\tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{y}{x}\) (provided \(x \neq 0\)), \(\csc(t) = \frac{1}{\sin(t)} = \frac{1}{y}\) (provided \(y \neq 0\)), \(\sec(t) = \frac{1}{\cos(t)} = \frac{1}{x}\) (provided \(x \neq 0\)), and \(\cot(t) = \frac{1}{\tan(t)} = \frac{\cos(t)}{\sin(t)} = \frac{x}{y}\) (provided \(y \neq 0\)).

Summarize all the trigonometric function values at \(t\) using the coordinates of \(P(x, y)\) and ensure to check for any undefined values where denominators might be zero.

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주요 개념

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Unit Circle Definition

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point P(x, y) on the unit circle corresponds to an angle t measured from the positive x-axis, where x = cos(t) and y = sin(t). This relationship allows us to define trigonometric functions based on coordinates.

Trigonometric Functions on the Unit Circle

The primary trigonometric functions—sine, cosine, and tangent—can be derived from the coordinates of point P on the unit circle. Specifically, sin(t) = y, cos(t) = x, and tan(t) = y/x (where x ≠ 0). Other functions like secant, cosecant, and cotangent are reciprocals of these.

Evaluating Trigonometric Functions at a Given Angle

To find the values of trigonometric functions at a real number t, identify the corresponding point P(x, y) on the unit circle. Use the coordinates to compute sine, cosine, and tangent, and then find reciprocal functions if needed. This process links angle measures to function values.