What if the given angle is negative or greater than 2\pi?

Find the coterminal angle by adding or subtracting 2\pi until the angle is between 0 and 2\pi. Use this coterminal angle to find the quadrant and reference angle.

How do I evaluate quadrantal angles like \pi/2 or \pi?

Quadrantal angles lie on the axes. Use the axis intercepts: (1, 0) for 0 or 2\pi, (0, 1) for \pi/2, (-1, 0) for \pi, and (0, -1) for 3\pi/2. Evaluate the ratios normally.

Why is the tangent of \pi/2 undefined?

The coordinates for \pi/2 are (0, 1). The tangent function is y/x. Substituting these values gives 1/0, which is mathematically undefined.

How to find exact trig values on the unit circle

The unit circle method evaluates trigonometric functions for standard angles by mapping them to coordinates (x, y) on a circle of radius r=1. It applies to angles that are integer multiples of \pi/6 and \pi/4.

The setup

Identify the given angle heta in radians or degrees. Determine its reference angle, which is the acute angle formed with the x-axis. Standard reference angles are \pi/6 (30 degrees), \pi/4 (45 degrees), and \pi/3 (60 degrees).

The steps

Determine the quadrant containing heta. 2. Find the (x, y) coordinates for the reference angle in Quadrant I: (\sqrt{3}/2, 1/2) for \pi/6, (\sqrt{2}/2, \sqrt{2}/2) for \pi/4, and (1/2, \sqrt{3}/2) for \pi/3. 3. Apply the appropriate signs for the quadrant: Q1 is (+,+), Q2 is (-,+), Q3 is (-,-), Q4 is (+,-). 4. Evaluate the trigonometric function: \cos( heta) = x, \sin( heta) = y, an( heta) = y/x, \sec( heta) = 1/x, \csc( heta) = 1/y, \cot( heta) = x/y.

Checking the result

Verify the sign of your answer matches the ASTC rule (All, Sine, Tangent, Cosine are positive in quadrants I, II, III, IV respectively). Confirm that x^2 + y^2 = 1 for your designated coordinates.

Common errors

Mixing up the x and y coordinates, such as evaluating sine as the x-coordinate instead of the y-coordinate. Failing to adjust the sign of the ratio based on the target quadrant.

Worked example

Find the exact value of \sec(5\pi/6).

The angle heta = 5\pi/6 is in Quadrant II. The reference angle is \pi - 5\pi/6 = \pi/6. The Quadrant I coordinates for \pi/6 are (\sqrt{3}/2, 1/2). In Quadrant II, the x-coordinate is negative and the y-coordinate is positive, yielding the point (-\sqrt{3}/2, 1/2). The secant function is \sec( heta) = 1/x. Substituting the x-coordinate gives \sec(5\pi/6) = 1 / (-\sqrt{3}/2) = -2 / \sqrt{3}. Rationalizing the denominator yields -2\sqrt{3} / 3.

FAQ

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Answer form

References: OpenStax Precalculus, Chapter 5 · Stewart Calculus, Appendix D