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Unit 3
30–35% of examThis unit is about functions that repeat, like tides, daylight hours and Ferris wheels. You'll build sine, cosine and tangent from the unit circle, transform and model with them, solve trig equations, and then use angles to describe points and graphs in polar coordinates. It's the toolkit for anything periodic or circular.
Longer videos that cover the whole unit. Good for a first pass or a final review.
A periodic function repeats the same pattern of outputs over and over, and the length of one cycle is its period. You'll describe periodic graphs and data using their period, maximum and minimum values, and how they increase, decrease and change concavity in each cycle.
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For an angle in standard position, sine is the y-coordinate and cosine is the x-coordinate of the point where its terminal ray meets the unit circle, and tangent is sine divided by cosine (the slope of that ray). Angles are usually measured in radians: one radian cuts off an arc as long as the radius, so a full turn is 2π.
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Special right triangles (30-60-90 and 45-45-90) give exact values like sin(π/6) = 1/2 and cos(π/4) = √2/2. Reference angles and symmetry on the unit circle extend these to every quadrant.
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Plotting sin θ and cos θ against the angle gives repeating waves with period 2π, oscillating between −1 and 1. Cosine's graph is the sine graph shifted left by π/2.
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Any wave shaped like a sine curve is a sinusoidal function. Its amplitude is half the distance from max to min, its midline is the horizontal line halfway between, and its period is the length of one full cycle.
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In f(θ) = a·sin(b(θ + c)) + d, |a| is the amplitude, 2π/|b| is the period, c shifts the graph left by c (right if c is negative), and d moves the midline to y = d. You'll read these values off graphs and write equations from them.
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Real repeating data, like temperature over a year or height on a Ferris wheel, can be modeled with a sinusoidal function. You'll find the max, min, period and a starting point from the context or with sinusoidal regression, then make predictions.
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Tangent, tan θ = sin θ / cos θ, has period π and vertical asymptotes wherever cos θ = 0, such as θ = π/2. Its graph is always increasing between asymptotes, and you can transform it just like sine and cosine.
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arcsin, arccos and arctan take a ratio and give back an angle. Because trig functions repeat, each inverse only returns angles from a restricted range: arcsin gives −π/2 to π/2, arccos gives 0 to π, and arctan gives angles strictly between −π/2 and π/2.
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To solve an equation like 2 sin θ = 1, you find the solutions in one period using the unit circle or an inverse trig function, then add multiples of the period for all solutions. Inequalities ask for the intervals where one side is bigger.
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These are the reciprocals of the main three: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ. Their graphs have vertical asymptotes wherever the original function equals zero.
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Identities let you rewrite trig expressions. The Pythagorean identity sin²θ + cos²θ = 1 comes straight from the unit circle, and the angle-sum formulas for sine and cosine lead to the difference and double-angle formulas. A good rewrite can make an equation much easier to solve.
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Polar coordinates (r, θ) locate a point by its distance from the origin and its angle from the positive x-axis. You convert with x = r cos θ and y = r sin θ, and back with r = √(x² + y²) and tan θ = y/x (checking the quadrant). A complex number a + bi is the point (a, b), so it can also be written in polar (trigonometric) form as r cos θ + i·r sin θ.
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A polar function r = f(θ) gives a radius for each angle, making shapes like circles, roses and limaçons. Graphing r against θ on regular axes first helps you sketch the polar curve.
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The point moves away from the origin when r is positive and growing, or negative and getting more negative; in the other two cases it moves closer. Where r switches between increasing and decreasing, the point is locally farthest from or closest to the origin. The change in r divided by the change in θ tells you how fast the signed radius changes per radian, and you can use it to estimate values in between.
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