The tangent function, its connection to slope, the reciprocal functions secant, cosecant and cotangent, and finding all six values from one.
Tangent
On the unit circle, \(\tan\theta = \frac{y}{x}\). Unlike sine and cosine, tangent can take any real value. It's positive in quadrants I and III (where \(x\) and \(y\) have the same sign) and negative in II and IV.
Tangent is slope
The terminal side of \(\theta\) is a ray from the origin through \((\cos\theta, \sin\theta)\). Its slope is \(\frac{\sin\theta}{\cos\theta}\).
So the line \(y = x\) makes a \(45°\) angle with the \(x\)-axis (\(\tan 45° = 1\)), and a line with slope \(-1\) makes a \(135°\) angle.
The reciprocal functions
Three more functions are the reciprocals of the first three.
Each reciprocal has the same sign as its partner.
More Pythagorean identities
Divide \(\sin^2\theta + \cos^2\theta = 1\) by \(\cos^2\theta\), and then by \(\sin^2\theta\):
Finding all six from one
Practice
Quick check
Where is \(\tan\theta\) undefined?
At \(\frac{\pi}{2}\) the cosine is \(0\), so \(\frac{\sin\theta}{\cos\theta}\) is undefined.
Quick check
If \(\cos\theta = \frac{2}{3}\), what is \(\sec\theta\)?
Secant is the reciprocal of cosine.
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