5.1 Unit Circle, Functions, and Graphs

Why This Topic Matters

The official AEPA Mathematics profile places Trigonometry and Calculus in a 19% domain, and its trigonometry competency expects candidates to use trig functions for distance, angles, the unit circle, graph analysis, and periodic modeling. That is a teacher-certification target, not just a formula target. A strong candidate can calculate sin(7pi/6), but can also explain why the value is negative and how the same reasoning appears in a graph or a word problem.

Start with the unit circle: a circle of radius 1 centered at the origin. When an angle theta is drawn in standard position, the terminal point has coordinates (cos theta, sin theta). Tangent is sin theta / cos theta when cos theta is not zero. This coordinate definition keeps special-angle values, quadrant signs, and graph behavior connected.

Unit-Circle Anchors

Use reference angles to extend those values. For 7pi/6, the reference angle is pi/6 and the angle lies in Quadrant III, so both coordinates are negative: cos(7pi/6) = -sqrt(3)/2 and sin(7pi/6) = -1/2. The mnemonic about quadrant signs can help, but the coordinate picture is more reliable for teaching because it explains the signs.

Graph Features

For y = a sin(bx - c) + d and y = a cos(bx - c) + d, identify four features before sketching. The amplitude is |a|, the period is 2pi/|b|, the midline is y = d, and the horizontal shift is c/b. A negative a reflects the graph across its midline. Sine normally starts at the midline and rises; cosine normally starts at a maximum.

Worked example: y = -3 cos(2x - pi) + 1 has amplitude 3, period pi, midline y = 1, and phase shift pi/2 to the right because 2x - pi = 2(x - pi/2). The minus sign reflects cosine, so at x = pi/2 the graph is 3 units below the midline, at y = -2. A common wrong sketch starts at y = 4 because it notices cosine at a maximum but ignores the reflection.

Periodic Modeling

AEPA-style modeling may describe daylight hours, height on a Ferris wheel, sound, temperature, or seasonal behavior. Choose a trig model when the situation repeats with a stable cycle. The midline is the average of the maximum and minimum values, the amplitude is half the difference, and the period is the time for one full cycle. If a Ferris wheel seat ranges from 4 feet to 44 feet above the ground, the midline is 24 and the amplitude is 20. If one rotation takes 60 seconds, b = 2pi/60 = pi/30 when time is in seconds.

Teacher-certification reasoning matters here. A student may say amplitude is maximum height, so 44. The fix is to ask, "44 measured from what line?" Amplitude measures distance from the midline, not distance from the ground. Another student may treat phase shift as the number after x, saying y = sin(2x - pi) shifts pi right. Factoring the coefficient of x shows the shift is pi/2 right.

Common Traps

• Degree-radian mix: pi/3 is 60 degrees, not 3.14/3 degrees.
• Calculator mode: a scientific calculator can evaluate values, but mode mistakes produce plausible-looking wrong decimals.
• Tangent undefined: tangent fails where cosine is zero, such as pi/2 and 3pi/2.
• Period vs frequency: larger |b| makes the graph complete cycles faster, so the period gets smaller.
• Visual guessing: a graph that appears shifted left or right should be confirmed from the transformed expression.

On the exam, write the base period and midline first, then transform. For exact values, locate the reference angle and quadrant before evaluating. For a model, translate max, min, period, and starting position into graph features before touching the calculator.

Distance And Angle Connections

The official formulas page for this test includes the laws of sines and cosines, so do not treat trigonometry as only right triangles. The law of sines is useful when you have an angle-side opposite pair and another angle or side. The law of cosines is useful for SAS or SSS information, and it reduces to the Pythagorean theorem when the included angle is 90 degrees.

Example: two sides of a triangle are 7 and 9 with included angle 60 degrees. The opposite side c satisfies c^2 = 7^2 + 9^2 - 2(7)(9)cos 60 degrees = 49 + 81 - 63 = 67, so c = sqrt(67). A distractor may add 49 and 81 only, as if every triangle were right. The teacher move is to ask which angle is included and whether the right-triangle condition has actually been given.

Graph and triangle questions also meet in angle units. If a model uses x in radians, period formulas use 2pi. If a triangle problem gives degrees, the calculator must match degrees. Before computing, label the unit on the angle.