Unit 4 of Pre-Calculus: Trigonometric Functions.
This unit covers unit circle, graphing trig functions and amplitude period phase shift — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
Pick a mode. Play.
Answer questions as fast as you can. 2 minutes on the clock. Build streaks for bonus points!
Don't want to play?
Review the questions traditionally. Click to expand.
Questions loading...
Focus on understanding.
Focus on understanding core concepts before memorizing details. Use the game modes to test yourself repeatedly — spaced repetition is proven to boost long-term retention.
This unit covers unit circle, graphing trig functions and amplitude period phase shift — essential concepts for Pre-Calculus. 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 defines the sine and cosine of any angle using coordinates (cos θ, sin θ) on a circle of radius 1. Students must memorize exact values at the 16 standard angles (multiples of 30° and 45°). Knowing the signs of trig functions in each quadrant is essential for solving equations and evaluating expressions.
Key Points
- Coordinates on the unit circle are (cos θ, sin θ); tan θ = sin θ / cos θ
- Key angles and exact values: 0°, 30°, 45°, 60°, 90° and their reflections into all four quadrants
- ASTC rule (All Students Take Calculus): quadrants I–IV tell you which functions are positive
- Reference angle = the acute angle formed with the x-axis; used to find exact values in any quadrant
Find the exact value of sin(240°).
240° is in Quadrant III (between 180° and 270°), so sine is negative. The reference angle is 240° − 180° = 60°. Since sin(60°) = √3/2, we get sin(240°) = −√3/2.
2 Graphing Trig Functions
Students must be able to sketch y = sin x and y = cos x from memory, including key points, period, domain, and range. Recognizing how the basic graph changes when the equation is transformed is the core skill tested. Tangent graphs have asymptotes where cosine equals zero and must be handled separately.
Key Points
- y = sin x: starts at (0, 0), period = 2π, range [−1, 1]; y = cos x: starts at (0, 1), same period and range
- Five key points per cycle: start, peak (or trough), midpoint, trough (or peak), end
- y = tan x has vertical asymptotes at x = π/2 + nπ and period = π
- Identifying the equation from a graph: read amplitude (peak value), period (one full cycle), and any vertical or horizontal shift
Identify the amplitude, period, and key features of y = −3 sin(2x).
The amplitude is |−3| = 3, meaning the graph reaches a maximum of 3 and a minimum of −3. The period is 2π / 2 = π, so one full cycle completes in π units. The negative sign reflects the graph over the x-axis, so the first notable movement is downward from (0, 0).
3 Amplitude Period Phase Shift
The standard form y = A sin(Bx − C) + D packages every transformation into four parameters that students must extract and apply in order. Amplitude, period, phase shift, and vertical shift each affect the graph independently and are all commonly tested. Students should be able to both analyze a given equation and write an equation from a described or pictured graph.
Key Points
- Amplitude = |A|; Period = 2π / |B| (or π / |B| for tangent)
- Phase shift = C / B (shift right if positive, left if negative); do NOT confuse C with the shift itself
- Vertical shift = D; moves the midline of the graph up or down from y = 0
- To graph: plot the midline first, then apply amplitude, then shift the five key points by the phase shift
State the amplitude, period, phase shift, and vertical shift of y = 2 cos(3x + π) − 1.
Rewrite as y = 2 cos(3(x + π/3)) − 1 to match the form A cos(B(x − C/B)) + D. Amplitude = 2, period = 2π/3, phase shift = −π/3 (shift left π/3 units), and vertical shift = −1 (midline at y = −1).
Questions, answered.
What is Trigonometric Functions?
Trigonometric Functions is Unit 4 of Pre-Calculus, covering unit circle, graphing trig functions and amplitude period phase shift.
How to study for Pre-Calculus Unit 4?
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.