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Section 2.6 The Tangent Function in Trigonometry


It is unfortunate that the word “tangent” in trigonometry has a different meaning than in geometry. In geometry, and speaking loosely, a tangent is a line that touches a curve in just one point. However, the tangent function in trigonometry is a function that gives the slope of a line. Figure 2.6.1 shows that the slope of a line that passes through the origin and makes an angle of θ radians is $\dfrac{\sin(\theta)}{\cos(\theta)}$. This is why, in trigonometry, tan(θ) is defined as $$\tan(θ)=\dfrac{\sin(\theta)}{\cos(\theta)},$$where θ is the measure in radians of the arc that the line subtends, measured from 0 radians. It gives the slope of a line.

Sin, cos, and tan on a unit circle

Figure 2.6.1. Slope of the ray from $(0,0)$ that subtends an arc of $\theta$ radii is $\dfrac{\sin(\theta)}{\cos(\theta)}$.

The quotient $\left( \dfrac{\sin(\theta)}{\cos(\theta)} \right)$ gives the size of $\sin(\theta)$ in units of $\cos(\theta)$. A value of the quotient does not have the radius as a unit of measure, as does sine and cosine. But to graph $y = \tan(x)$ in the same coordinate system as sine and cosine, the quotient must have the circle’s radius as a unit of measure. We can do that by using similar triangles.

The animation in Figure 2.6.2 uses the relationship $$\dfrac{\sin(\theta)}{\cos(\theta)}=\dfrac{\tan(\theta)}{1}.$$By displaying the value of $\tan(\theta)$ at $x =1$, the directed segment that represents the value of $\tan(\theta)$ is measured in units of the circle’s radius. We can thus graph $y = \tan(x)$ in the same coordinate system as cosine and sine because “1” stands for the same thing for all of them — the length of the circle’s radius.



Figure 2.6.2. Value of tan(θ) in units of the circle’s radius.

Reflection 2.4. Convince yourself that the value labeled $\tan(\theta)$ gives the slope of the line, and that this value is measured in units of the circle’s radius.

Reflection 2.5. Play the animation in Figure 2.15, stopping it when the value of $\cos(\theta)$ is close to 0. What happens to the quotient sin($\theta$)/cos($\theta$) as $\cos(\theta)$ is near 0 but less than 0? What happens to the quotient sin($\theta$)/cos($\theta$) as $\cos(\theta)$ is near 0 but greater than 0?

Reflection 2.6. Why does $\tan(\theta)$ make sudden jumps from positive infinity to negative infinity?

Reflection 2.7. Is $\tan(\theta)$ always increasing over every interval contained in the domain of its independent variable? That is, does it ever decrease as $\theta$ increases continuously?

Reflection 2.8. Why is $\theta$ given as the input to sine, cosine, and tangent functions in Figure 2.6.2 instead of x? When is it appropriate to write “sin(x)”, “cos(x)”, and “tan(x)”?