Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

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Verified step by step guidance

1. Identify the coordinates of the point where the terminal side of the angle \( \theta \) intersects the unit circle. Let's denote this point as \((x, y)\).
2. Recall that for any angle \( \theta \) in standard position intersecting the unit circle, the coordinates \((x, y)\) correspond to \(\cos(\theta)\) and \(\sin(\theta)\) respectively.
3. Use the coordinates \((x, y)\) to find the six trigonometric functions: \(\sin(\theta) = y\), \(\cos(\theta) = x\), \(\tan(\theta) = \frac{y}{x}\), \(\csc(\theta) = \frac{1}{y}\), \(\sec(\theta) = \frac{1}{x}\), and \(\cot(\theta) = \frac{x}{y}\).
4. Check the quadrant in which the angle \( \theta \) lies to determine the signs of the trigonometric functions. Remember: In Quadrant I, all are positive; in Quadrant II, sine and cosecant are positive; in Quadrant III, tangent and cotangent are positive; in Quadrant IV, cosine and secant are positive.
5. Substitute the values of \(x\) and \(y\) into the expressions for the six trigonometric functions to evaluate them.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of any point on the unit circle correspond to the cosine and sine values of the angle formed with the positive x-axis, allowing for easy calculation of trigonometric functions.

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are derived from the relationships between the angles and sides of right triangles. In the context of the unit circle, these functions can be defined as ratios of the coordinates of points on the circle. For example, sine is the y-coordinate, and cosine is the x-coordinate of a point on the unit circle corresponding to angle θ.

Standard Position of an Angle

An angle is said to be in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. The terminal side of the angle is formed by rotating the initial side counterclockwise for positive angles and clockwise for negative angles. Understanding this concept is crucial for accurately determining the position of angles and their corresponding trigonometric function values on the unit circle.