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I'm very confused right now. If cos(x) = 7/25, then is the cosine of the angle the x component on the unit circle, or the adjacent side over the hypotenuse side of the triangle it forms? And how do I know what definition to use?

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  • $\begingroup$ In what context did you encounter the problem? Are you trying to determine the values of the other trigonometric functions? $\endgroup$ – N. F. Taussig Apr 3 '16 at 21:48
  • $\begingroup$ How long is the hypotenuse of the right triangle that it forms? $\endgroup$ – John Joy Apr 4 '16 at 11:26
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Use whichever definition is more convenient. They are equivalent and so will always lead to the same answer.

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$\cos \theta = \dfrac{7}{25}$

If you are talking about right triangles, then $\theta$ can be thought of as the larger acute angle of a $7:24:25$ right triangle. Or, if you want the hypotenuse to be $1$, a $\dfrac{7}{25}:\dfrac{24}{25}:1$ right triangle.

If you are talking about points on the unit circle, then $\theta$ is not yet well defined because there are two points on the unit circle which give us $\cos \theta = \dfrac{7}{25}$: the point $P(\theta_1) = \left( \dfrac{24}{25},\dfrac{7}{25} \right)$ and the point $P(\theta_2) = \left( \dfrac{24}{25},-\dfrac{7}{25} \right)$.

The right triangle definition is of very limited usefulness. The unit circle definition is much more useful.


There is a third definition that is a bit more practical and a bit more confusing. Let $P = (x,y)$ be any point in the plane except for the origin.Let $E_1 = (1,0)$ and let $O=(0,0)$.

Any point on the ray $\overrightarrow{OP}$ corresponds to the same family of angles $\theta + 2n\pi \; (n \in \mathbb Z)$ where $\theta$ is the measure of $\angle E_1OP$. We can't say that $P(\theta) = (x,y)$ now but we can still say that $(x,y)$ corresponds to the angle $\theta$. Define $r = \sqrt{x^2 + y^2}$. Then $\cos \theta = \dfrac xr$ and $\sin \theta = \dfrac yr$.

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