True or false: cos(180 degrees) = -1.

Step-by-step solution

1. 1

   Recall what cosine measures on the unit circle. For an angle $\theta$ measured counterclockwise from the positive $x$-axis, the terminal point on the unit circle is $(\cos\theta,\sin\theta)$. So cosine is simply the $x$-coordinate of that point - not a ratio you need a triangle for.

2. 2

   Locate $180^\circ$ on the circle. A rotation of $180^\circ$ is half a turn, which carries $(1,0)$ to the diametrically opposite point $(-1,0).$ This is a quadrantal angle, sitting exactly on the negative $x$-axis rather than inside any quadrant.

3. 3

   Read off the coordinates. Since the terminal point is $(-1,0)$, $\cos(180^\circ)=-1,\qquad \sin(180^\circ)=0.$ Both values are exact - no decimal approximation is involved.

4. 4

   Cross-check with an identity. The double-angle formula gives $\cos(180^\circ)=\cos(2\cdot 90^\circ)=1-2\sin^{2}(90^\circ)=1-2(1)^{2}=-1$, and the supplement identity gives $\cos(180^\circ)=-\cos(0^\circ)=-1$. Both routes agree.

5. 5

   Conclude, and note the common trap. The statement is true. The usual slip is confusing $180^\circ$ with $\pi$ radians used as a raw number: a calculator left in radian mode returns $\cos(180)=-0.5985$, because it is evaluating $180$ radians, not $180$ degrees. In radians the correct input is $\pi$, and $\cos\pi=-1$.