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 for a point \(P(x, y)\) on the unit circle corresponding to an angle \(t\), the coordinates are given by \(x = \cos(t)\) and \(y = \sin(t)\).

Identify the values of \(x\) and \(y\) from the point \(P\) on the unit circle. These values represent \(\cos(t)\) and \(\sin(t)\) respectively.

Use the definitions of the six trigonometric functions in terms of \(\sin(t)\) and \(\cos(t)\): \(\sin(t) = y\) \(\cos(t) = x\) \(\tan(t) = \frac{y}{x}\) (provided \(x \neq 0\)) \(\csc(t) = \frac{1}{y}\) (provided \(y \neq 0\)) \(\sec(t) = \frac{1}{x}\) (provided \(x \neq 0\)) \(\cot(t) = \frac{x}{y}\) (provided \(y \neq 0\)).

Substitute the values of \(x\) and \(y\) into these formulas to express each trigonometric function in terms of the coordinates of point \(P\).

Check the quadrant of the angle \(t\) based on the signs of \(x\) and \(y\) to determine the signs of the trigonometric functions, ensuring the correct values for each function.

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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, where x = cos(t) and y = sin(t). This relationship allows us to find trigonometric function values directly from coordinates.

Trigonometric Functions on the Unit Circle

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—can be defined using the coordinates of point P on the unit circle. Specifically, sin(t) = y, cos(t) = x, and tan(t) = y/x, with reciprocal functions defined accordingly.

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent depends on the quadrant where point P lies. For example, sine is positive in quadrants I and II, cosine is positive in quadrants I and IV, and tangent is positive in quadrants I and III. This helps determine the correct sign of function values.

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.<IMAGE>