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.

Verified step by step guidance

1. Identify the coordinates of the point where the terminal side of the angle \( \theta \) intersects the unit circle. Since the circle has radius 1, these coordinates are \( (x, y) \) with \( x = \cos(\theta) \) and \( y = \sin(\theta) \).
2. Recall the definitions of the six circular (trigonometric) functions in terms of \( x \) and \( y \) on the unit circle: \[ \sin(\theta) = y \] \[ \cos(\theta) = x \] \[ \tan(\theta) = \frac{y}{x} \quad \text{(provided } x \neq 0) \] \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{y} \quad \text{(provided } y \neq 0) \] \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{x} \quad \text{(provided } x \neq 0) \] \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{x}{y} \quad \text{(provided } y \neq 0) \]
3. Substitute the \( x \) and \( y \) values from the point on the unit circle into each of the six function formulas to find their values.
4. Check the quadrant in which the terminal side lies to determine the signs (positive or negative) of each function value, since sine and cosine can be positive or negative depending on the quadrant.
5. Write down the six values: \( \sin(\theta) \), \( \cos(\theta) \), \( \tan(\theta) \), \( \csc(\theta) \), \( \sec(\theta) \), and \( \cot(\theta) \), ensuring to note any undefined values if division by zero occurs.

Key Concepts

Unit Circle and Standard Position

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. The terminal side intersects the unit circle at a point whose coordinates correspond to cosine and sine values of the angle.

Circular Functions (Trigonometric Functions)

The six circular functions are sine, cosine, tangent, cosecant, secant, and cotangent. For an angle θ on the unit circle, sine and cosine are the y- and x-coordinates of the intersection point, respectively. The other functions are ratios derived from sine and cosine, such as tangent = sine/cosine.

Evaluating Trigonometric Functions from Coordinates

To find the six circular function values, use the coordinates (x, y) of the terminal side's intersection with the unit circle: cosine(θ) = x, sine(θ) = y. Then, tangent(θ) = y/x, secant(θ) = 1/x, cosecant(θ) = 1/y, and cotangent(θ) = x/y, provided denominators are not zero.