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.

검증된 단계별 안내

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 coordinates of the point \(P(x, y)\) on the unit circle correspond to \(x = \cos(t)\) 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 in terms of sine and cosine: \(\tan(t) = \frac{\sin(t)}{\cos(t)}\), \(\csc(t) = \frac{1}{\sin(t)}\), \(\sec(t) = \frac{1}{\cos(t)}\), and \(\cot(t) = \frac{\cos(t)}{\sin(t)}\).

Make sure to consider the signs of \(x\) and \(y\) based on the quadrant where point \(P\) lies to determine the correct signs of the trigonometric functions.

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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 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 for 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 values directly. This method simplifies evaluating trig functions without a calculator.