Basic Trigonometry

Here we will cover the basic trigonometry definitions, holding back most of the trig identities for a later section. A large part (but certainly not all) of this subject is about right triangles and some definitions which follow are all defined in terms of right triangles and an angle named $\theta$. In these definitions, $\theta$ is never the $90^{\circ}$ angle. It will be one of the other two angles, and it must be designated before any of this makes sense. If we look Figure 1, and the angle $\theta$, we define the horizontal side next to angle $\theta$ to be the “adjacent length” side. The vertical side is called the “opposite length” side. The adjacent side is always the side that goes from angle $\theta$ to the right angle.

Figure 1: Trig Triangles. Trigonometry Triangle with sides identified relative to the named angle, $\theta$. In the panel on the left, The adjacent side is the side that is adjacent to angle $\theta$ and goes to the right angle. Likewise, the opposite side is the one that is opposite the angle identified as $\theta$. The bottom left triangle in the figure is rotated in the plane, but since $\theta$ is identified and the right angle is obvious, we can correctly label the adjacent and opposite legs. This labeling is critical to applying the trig definitions. However: You can designate either acute angle as $\theta$.

$\begin{array}{c} \cos\theta=\frac{adjacent}{hypotenuse}=\frac{adj}{r}\\ \sin\theta=\frac{opposite}{hypotenuse}=\frac{opp}{r}\\ \tan\theta=\frac{opposite}{adjacent}=\frac{\sin\theta}{\cos\theta} \end{array}$ $\begin{array}{c} \sec\theta=\frac{hypotenuse}{adjacent}\\ \csc\theta=\frac{hypotenuse}{opposite}\\ \cot\theta=\frac{adjacent}{opposite} \end{array}$

Figure 2: The left figure shows a unit circle with a right triangle and sides labeled $\sin\theta$ and $\cos\theta$. These lengths come directly from the definitions of sine and cosine, since $\sin(\theta)=opposite/hypotenuse=\sin\theta/1$. In the right figure, the vertical leg is also labeled as $y=\sin\theta$, but the horizontal leg is found by the Pythagorean theorem expressed for $y$. Both figures will lead directly to the trig identity,$\cos^{2}\theta+\sin^{2}\theta=1$ by application of the Pythagorean theorem.