Solving trig equations means finding the angles that make an equation like 2 sin x − 1 = 0 true. Isolate the trig function, find the reference angle from the unit circle or an inverse trig function, and list every solution in [0, 2π). For all solutions, add multiples of the period: x = π/6 + 2πn.
The half-angle formulas find the sine, cosine or tangent of θ/2 from cos θ: sin(θ/2) = ±√((1 − cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2). The quadrant of θ/2 decides the sign. They give exact values for angles like 15° and 22.5°, and they come from the double-angle formulas for cosine.
The double-angle formulas rewrite trig functions of 2θ in terms of θ: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ and tan 2θ = 2 tan θ/(1 − tan²θ). Cosine also has two other forms, 2cos²θ − 1 and 1 − 2sin²θ. These formulas all come from the sum formulas.
The sum and difference formulas give the sine, cosine or tangent of A + B or A − B from the values for A and B. For example, sin(A + B) = sin A cos B + cos A sin B. They let you find exact values like sin 75° by writing 75° as 45° + 30°. The cosine and tangent formulas work the same way.
Verifying trig identities means proving that two trig expressions are equal for every angle where both are defined. Start with the more complicated side and rewrite it, one step at a time, until it matches the other side. Useful moves are rewriting in sine and cosine, using Pythagorean identities, combining fractions and factoring.
The fundamental trig identities are equations that are true for every angle where both sides are defined. They include the reciprocal identities, such as csc θ = 1/sin θ, the quotient identities, such as tan θ = sin θ/cos θ, and the Pythagorean identities, such as sin²θ + cos²θ = 1. They let you rewrite and simplify trig expressions.
Compositions of inverse trig functions combine a trig function with an inverse trig function, like sin(arcsin x) or cos(arctan 3/4). A function undoes its own inverse when the input is allowed: sin(arcsin x) = x. For a mixed pair, draw a right triangle for the inner angle and read the outer ratio from it.
Inverse trig functions undo sine, cosine and tangent: they take a ratio and give back an angle. Because trig functions repeat, each inverse uses a restricted range. Arcsine and arctangent give angles from −π/2 to π/2, and arccosine gives angles from 0 to π. For example, arcsin(1/2) = π/6.
Sinusoidal models use sine or cosine functions to describe things that repeat, like tides, Ferris wheels and seasonal temperatures. From the highest value, the lowest value and the time for one cycle, you can write y = A cos(B(t − h)) + k: the amplitude is A = (max − min)/2, the midline is k = (max + min)/2 and B = 2π/period.
The graphs of secant, cosecant and cotangent come from flipping the graphs of cosine, sine and tangent. Wherever the original function is 0, the reciprocal has a vertical asymptote. Secant and cosecant are U-shaped curves with period 2π and never take values between −1 and 1. Cotangent has period π and falls from left to right.