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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)”?