To find \(\tan \theta\) using a triangle on the unit circle, we can use the basic definitions of the sine and cosine functions, since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
-
Identify the coordinates on the unit circle: If the angle \(\theta\) corresponds to a point \((x, y)\) on the unit circle, then:
- \(\cos \theta = x\)
- \(\sin \theta = y\)
-
Using a right triangle: The tangent is defined as the ratio of the opposite side over the adjacent side. Given that for any angle \(\theta\) on the unit circle, the hypotenuse is always 1, the opposite side is \(\sin \theta\) (which is \(y\)) and the adjacent side is \(\cos \theta\) (which is \(x\)).
Thus, we can express tangent as: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x} \]
- Finding \(\tan \theta\): If we have values for \(\sin \theta\) and \(\cos \theta\), we can plug them in to calculate \(\tan \theta\).
Please provide the specific coordinates or angles if you wish to calculate \(\tan \theta\) for a specific triangle on the unit circle or clarify your question further so I can assist you better!
1 answer