De Moivre’s theorem says that to raise a complex number in trigonometric form to a power, raise the modulus to the power and multiply the angle by it: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ). A related formula finds all n of the nth roots of a complex number.
The trigonometric form of a complex number (also called polar form) writes a + bi as r(cos θ + i sin θ). Here r = the square root of a^2 + b^2 is the modulus, the distance from 0, and θ is the argument, the angle from the positive real axis. This form makes multiplying and dividing complex numbers easy.
Graphing polar equations means plotting every point (r, θ) that makes an equation like r = 2 + 2 cos θ true. Make a table of θ and r values, plot the points on a polar grid, and connect them in order. Learn the common shapes: circles, rose curves, cardioids and limaçons.
Polar coordinates name a point by its distance from the origin and its angle, written (r, θ), instead of by x and y. To convert from polar to rectangular, use x = r cos θ and y = r sin θ. To go back, use r = the square root of x^2 + y^2 and tan θ = y/x, checking the quadrant.
The area of an oblique triangle (a triangle with no right angle) can be found without knowing its height. If you know two sides and the angle between them, use Area = (1/2)bc sin A. If you know all three sides, use Heron’s formula: Area = the square root of s(s – a)(s – b)(s – c), where s is half the perimeter.
The law of cosines connects all three sides of a triangle with one angle: c^2 = a^2 + b^2 – 2ab cos C. Use it to find the third side when you know two sides and the angle between them (SAS), or to find an angle when you know all three sides (SSS). It works for every triangle.
The ambiguous case of the law of sines happens when you know two sides and an angle that is not between them (SSA). That information can make no triangle, one triangle or two different triangles. To decide, compare the side across from the angle with the height h = b sin A, or check whether sin B is more than 1.
The law of sines says that in any triangle, each side divided by the sine of its opposite angle gives the same number: a/sin A = b/sin B = c/sin C. Use it to find missing sides and angles of triangles that are not right triangles, when you know two angles and a side.
Solving trig equations with identities means rewriting an equation so it has only one trig function of one angle, then solving it. For sin 2x = cos x, replace sin 2x with 2 sin x cos x, factor, and set each factor equal to 0. Use Pythagorean, double-angle and reciprocal identities to make the swap.
Trig equations with multiple angles have the variable multiplied inside the function, like sin 2x = √3/2. To solve one, let u = 2x, stretch the interval to match, and find every value of u. Then divide each one by 2. A multiple angle usually gives more solutions in [0, 2π).