What are sine, cosine and tangent?
For a right triangle they are ratios of side lengths relative to an angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. The mnemonic SOH-CAH-TOA encodes exactly that. On the unit circle they become distances instead — cosine is the horizontal coordinate of the point where the radius meets the circle, sine the vertical one — which is why they still make sense for angles beyond 90 degrees where no right triangle exists.
Why does the unit circle matter?
Because the triangle definitions only cover angles between 0 and 90 degrees. The unit circle extends them to any angle, including obtuse, reflex and negative ones, and makes the sign pattern obvious: both positive in the first quadrant, sine only in the second, neither in the third, cosine only in the fourth. It also explains periodicity — go round again and the values repeat, which is why sine and cosine describe waves.
When is tangent undefined?
At 90 and 270 degrees, and every 180 degrees from there. Tangent equals sine divided by cosine, and cosine is zero at those angles, so the division is undefined — not infinite, undefined. Geometrically the radius is vertical and never meets the tangent line at x equals 1. Approaching from either side the value grows without limit in opposite directions, which is why the graph has vertical asymptotes rather than a peak.
What is the difference between degrees and radians?
Both measure angle; radians measure it as arc length on a unit circle. A full turn is 360 degrees or 2π radians, so 180 degrees equals π radians and one radian is about 57.3 degrees. Radians are not merely a convention — calculus results such as the derivative of sine being cosine only hold in radians. Mixing the two is the most frequent cause of a trigonometric answer that is wildly wrong rather than slightly off.
What are cosecant, secant and cotangent?
The reciprocals: cosecant is 1 over sine, secant is 1 over cosine, cotangent is 1 over tangent. The pairing is deliberately counter-intuitive — secant goes with cosine, not sine. They appear mainly in calculus and in some engineering conventions, where writing sec θ is tidier than 1/cos θ. Each is undefined wherever its base function is zero.
How do I find an angle from a ratio?
Use the inverse functions — arcsine, arccosine, arctangent, often shown as sin⁻¹ on a calculator. That notation means inverse function, not reciprocal: sin⁻¹(0.5) is 30 degrees, whereas 1/sin(0.5) is something else entirely. Inverse functions return only one angle from a restricted range, so in a real problem check whether another angle in a different quadrant also satisfies it.