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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Verified step by step guidance

1. 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 \(y = \sin(t)\).
2. Identify the values of \(x\) and \(y\) from the point \(P\) on the unit circle. These values represent \(\cos(t)\) and \(\sin(t)\) respectively.
3. 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 \(y \neq 0\)).
4. Substitute the values of \(x\) and \(y\) into these formulas to express each trigonometric function in terms of the coordinates of point \(P\).
5. Check the domain restrictions for each function to ensure the denominators are not zero, and interpret the results accordingly.

Key Concepts

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 (in radians) measured from the positive x-axis. The coordinates (x, y) represent the cosine and sine of t, respectively, which are fundamental for defining trigonometric functions.

Trigonometric Functions on the Unit Circle

On the unit circle, the primary trigonometric functions sine and cosine are given by the y- and x-coordinates of point P, respectively. Other functions like tangent, cotangent, secant, and cosecant can be derived from these values using their definitions as ratios, such as tangent = sine/cosine.

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 x = cos(t) and y = sin(t) to compute sine and cosine, then calculate other functions like tangent (y/x), secant (1/x), cosecant (1/y), and cotangent (x/y), ensuring denominators are not zero.