Trigonometry — common mistakes

When solving trigonometric equations like $\sin x = k$, your calculator will only give you one solution (the principal value). You must use the symmetry of the graph or a CAST diagram to find all other solutions within the given range. For $\sin x = k$, the second solution is typically $\pi - x$ (in radians) or $180° - x$ (in degrees).

While degrees are practical for geometry, radians are the 'natural' unit for angles in mathematics. Using radians simplifies formulas in calculus, particularly for differentiation and integration of trigonometric functions. For example, the derivative of $\sin x$ is $\cos x$ only when $x$ is in radians.

Draw an equilateral triangle with side length 2. If you cut it in half, you get a right-angled triangle with angles $\pi/6$ (30°), $\pi/3$ (60°), and $\pi/2$ (90°). The sides will be 1, $\sqrt{3}$, and 2. You can then use SOH CAH TOA on this triangle to find the sin, cos, and tan of $\pi/3$ and $\pi/6$.

They are very different. $\sin^2 x$ is shorthand for $(\sin x)^2$, which means you first find the sine of the angle $x$, and then you square the result. In contrast, $\sin(x^2)$ means you first square the angle $x$, and then you find the sine of that new angle.

Yes, you can use the graphs. For example, to solve $\cos x = 0.5$, sketch the $y=\cos x$ graph and the line $y=0.5$. The x-coordinates of the intersection points are your solutions. The symmetry of the graph helps you find all solutions in the required range from the principal value your calculator gives you.