Sines, cosines and triangles
The sine, cosine and tangent: in a right triangle, on the unit circle for any angle, the main identity and the law of sines.
How to solve it
In a right triangle the sine is the opposite leg over the hypotenuse and the cosine the adjacent leg over the hypotenuse. For any angle they come from the unit circle: the point rotated by $\alpha$ has the coordinates $(\cos\alpha,\ \sin\alpha)$.
Step by step
- An acute angle: remember the values for $30^\circ$, $45^\circ$, $60^\circ$ — fractions made of $1$, $\sqrt{2}$, $\sqrt{3}$ and $2$.
- Any angle: bring it to a turn between $0^\circ$ and $360^\circ$ (adding or removing full turns), find the quadrant and the sign, and take the value of the acute angle to the nearest $x$-axis.
- If one function is known, find the other from $\sin^2\alpha + \cos^2\alpha = 1$; the sign comes from the quadrant.
- In any triangle: the law of sines links the sides to the opposite angles, the law of cosines links three sides and an angle.
Common mistakes
- Mixing up the adjacent and the opposite leg: the adjacent one touches the angle, the opposite one faces it.
- Losing the sign: $\sin(-120^\circ) = -\frac{\sqrt{3}}{2}$ — the angle is in the third quadrant, where the sine is negative.
- Swapping the sine and cosine of $30^\circ$ and $60^\circ$: $\sin 30^\circ = \cos 60^\circ = \frac{1}{2}$.
- Getting the cosine from the identity and forgetting to choose the sign by the quadrant.