The graph of tangent, y = tan x, is a repeating S-shaped curve with period π. It passes through (0, 0), rises through (π/4, 1) and has vertical asymptotes at x = ±π/2, where cosine is 0. For y = a tan(bx), the period is π/|b|. Its range is all real numbers, and it has no highest or lowest point.
A phase shift moves a sine or cosine graph left or right. In y = a sin(b(x − h)) + k, the graph shifts right h units (left if h is negative), and k shifts it up or down so the middle line is y = k. For y = a sin(bx − c) + d, the phase shift is c/b.
For y = a sin(bx) or y = a cos(bx), the amplitude and period describe the size and length of the wave. The amplitude is |a|, the distance from the middle of the wave to a peak. The period is 2π/|b|, the horizontal length of one full cycle. A negative a flips the graph upside down.
The graphs of sine and cosine are smooth waves that repeat every 2π radians (360°). The graph of y = sin x starts at 0, rises to 1 at π/2, returns to 0 at π, falls to −1 at 3π/2 and is back to 0 at 2π. The graph of y = cos x is the same wave shifted left π/2, so it starts at 1.
The unit circle is a circle with radius 1 centered at the origin. The terminal side of any angle θ in standard position crosses it at the point (cos θ, sin θ). So the x-coordinate is the cosine, the y-coordinate is the sine, and tan θ = y/x. For example, the point for 30° is (√3/2, 1/2).
A reference angle is the acute angle between an angle’s terminal side and the x-axis. Any trig value of an angle equals the trig value of its reference angle, with a plus or minus sign set by the quadrant. For example, 150° has a reference angle of 30°, so sin 150° = sin 30° = 1/2.
To find the trig functions of any angle, pick a point (x, y) on the angle’s terminal side and find its distance from the origin, r = √(x² + y²). Then sin θ = y/r, cos θ = x/r and tan θ = y/x. The signs of x and y give each value its sign in every quadrant.
The trig values of special angles are the exact values of the six trig functions at 30°, 45° and 60°, or π/6, π/4 and π/3 in radians. They come from two special right triangles. For example, sin 30° = 1/2, cos 45° = √2/2 and tan 60° = √3.
The six trigonometric functions are sine, cosine and tangent and their reciprocals: cosecant, secant and cotangent. In a right triangle, each one is a ratio of two sides for an acute angle θ. Cosecant flips sine, secant flips cosine, and cotangent flips tangent.
Linear speed is how fast a point moves along a circular path: v = s/t, the arc length per unit of time. Angular speed is how fast the angle changes: ω = θ/t, in radians per unit of time. The two are linked by v = rω, where r is the radius.