Algebra 2 Unit 9 — Trigonometric Functions.
This unit covers unit circle, graphing sine and cosine and amplitude and period — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers unit circle, graphing sine and cosine and amplitude and period — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Unit Circle
The unit circle is a circle with radius 1 centered at the origin. Students must know the coordinates (cos θ, sin θ) at the standard angles 0°, 30°, 45°, 60°, 90°, and their equivalents in all four quadrants. Recognizing reference angles and the signs of sine and cosine in each quadrant is essential for evaluating trig functions without a calculator.
Key Points
- Coordinates on the unit circle are (cos θ, sin θ)
- Key angles in radians: 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π
- ASTC rule (All Students Take Calculus): tells which trig functions are positive in each quadrant
- Reference angle is the acute angle formed with the x-axis; use it to find trig values in any quadrant
Find the exact value of sin(5π/6).
The angle 5π/6 is in Quadrant II, so sine is positive. The reference angle is π - 5π/6 = π/6. Since sin(π/6) = 1/2, we get sin(5π/6) = 1/2.
2 Graphing Sine And Cosine
Students must be able to sketch one full period of y = sin x and y = cos x, labeling key points (maxima, minima, and zeros). Exams commonly ask students to identify or graph transformations of these parent functions in the form y = a·sin(bx + c) + d. Understanding how each parameter shifts or scales the graph is critical.
Key Points
- y = sin x starts at (0, 0); y = cos x starts at (0, 1)
- Both parent functions have amplitude 1, period 2π, midline y = 0
- Vertical shift d moves the midline up or down; phase shift = −c/b shifts left or right
- Key points to plot: start, quarter-period, half-period, three-quarter-period, end
Describe the transformation of y = sin x represented by y = sin(x − π/2).
The graph is shifted horizontally to the right by π/2 units (phase shift = +π/2). The amplitude and period are unchanged at 1 and 2π respectively. This transformation makes y = sin(x − π/2) identical to y = cos x.
3 Amplitude And Period
Amplitude is the distance from the midline to the maximum (or minimum) of the graph and equals |a| in y = a·sin(bx) + d. Period is the length of one complete cycle and equals 2π/|b|. Students must be able to read these values from an equation or from a graph.
Key Points
- Amplitude = |a|; a negative value of a reflects the graph over the midline
- Period = 2π/|b|; larger |b| compresses the graph horizontally (shorter period)
- Midline is y = d; it is not the same as amplitude
- Given a graph, period = (x-value of end of cycle) − (x-value of start of cycle)
State the amplitude and period of y = −3·sin(4x).
The amplitude is |−3| = 3; the negative sign reflects the graph but does not change the amplitude. The period is 2π/|4| = π/2. So the graph reaches a maximum of 3 and a minimum of −3 and completes one full cycle every π/2 units.
Questions, answered.
What is Trigonometric Functions?
Trigonometric Functions is Unit 9 of Algebra 2, covering unit circle, graphing sine and cosine and amplitude and period.
How to study for Algebra 2 Unit 9?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.