Understanding sine, cosine, and tangent in the unit circle

Question

In the following diagram I understand how to use angle $\theta$ to find cosine and sine. However, I'm having a hard time visualizing how to arrive at tangent. Furthermore, is it true that in all right triangle trig ratios we always need to use one of the non-right angles?

Answer 1

• In the first and third quadrants, $\tan(\theta)$ is the length from $(\cos(\theta),\sin(\theta))$ to the $x$ axis along the line tangent to the circle at $(\cos(\theta),\sin(\theta))$. In the second and fourth quadrants the situation is essentially the same but we use the opposite sign.
• One can think of $\cos(\pi/2)$ and $\sin(\pi/2)$ from the triangle point of view as the ratios of a degenerate triangle with angles $(\pi/2,\pi/2,0)$, which is really just a line segment. Now you identify one of the $\pi/2$'s as the "right" angle and the other as the "acute" angle and measure ratios relative to that "acute" angle.

Answer 2

The angle between the red and blue lines is $\theta$. So in that triangle, $$\cos\theta=\frac{\rm red}{\rm blue}$$ Or$${\rm blue}=\frac{\rm red}{\cos\theta}=\frac{\sin\theta}{\cos\theta}=\tan\theta$$

Answer 3

If you pay attention, the smaller left right triangle is similar to the larger right triangle adjacent to it as they are both right triangles with angles of measure $\frac{\pi}{2}$, $\theta$, and $(\frac{\pi}{2}-\theta)$.

When two triangles are similar, the ratio of their corresponding sides will be equivalent.

$$\frac{\cos\theta}{1}$$ In the smaller right triangle, $\cos\theta$ is opposite the $(\frac{\pi}{2}-\theta)$ angle and $1$ is the hypotenuse.

In the larger right triangle, $\sin\theta$ is opposite to the $(\frac{\pi}{2}-\theta)$ angle and $x$ (unknown variable) is the hypotenuse.

$$\frac{\sin\theta}{x}$$

Using the rules of similarity, we can say the two ratios are equivalent.

$$\frac{\cos\theta}{1} = \frac{\sin\theta}{x}$$ $$x\cos\theta = \sin\theta \implies x = \frac{\sin\theta}{\cos\theta} =\implies \boxed{x = \tan\theta}$$

You can check this page out for $\csc\theta$, $\sec\theta$, and $\cot\theta$: graphical representation of trig functions.

Answer 4

Commit this SohCahToa Table to memory so that you can always write it out:

For the $sin$ and $cos$ in the unit circle you can think of putting the values in the table over a big $1$ for the hypotenuse.

Example: Think of

$sin(\frac{\pi}{6})= \frac{{\frac{\sqrt1}{2}}}{1}$

$cos(\frac{\pi}{6})= \frac{{\frac{\sqrt3}{2}}}{1}$

and the unit circle placement of $\left(cos(\frac{\pi}{6}),sin(\frac{\pi}{6})\right)$.