3.12 Trigonometric values, equations and identities

In this module, we discuss trigonometric values and angles. In particular, we shall learn about two very useful algorithms which help us to find (i) value of trigonometric function when angle is given and (ii) angles when value of trigonometric function is given. In addition, we shall go through various trigonometric equations and identities. We are expected to be already familiar with them. For this reason, solutions of equations and identities are presented here without deduction and are included for reference purpose.

Values of trigonometric function

It is sufficient to know values of trigonometric functions for angles in first quarter. These angles are called acute angles (angle value less than π/2). Here, we develop algorithm, which converts angles in other quadrants in terms of acute angles. Basic idea is that angles can be expressed in terms of combination of acute angle and reference angles like 0, π/2, π and 2π. These angles demark quadrants. Using certain procedure, we can find value of trigonometric function of any angle provided we know the trigonometric value of corresponding acute angle. For the sake of convenience, we shall concentrate on acute angles π/6, π/4 and π/3, whose trigonometric function values are known to us. We follow an algorithm to determine trigonometric values as given here :

1 : Express given angle as sum or difference of acute angle and reference angles 0, π/2, π and 2π.

2 : Write trigonometric function of sum or difference as trigonometric function of acute angle. A trigonometric sum/difference combination of angles involving angles of 0, π and 2π does not change the function. However, a combination involving π/2 changes function from sine to cosine and vice-versa, tangent to cotangent and vice-versa and cosecant to secant and vice-versa.

3 : Apply sign before trigonometric function determined as above in accordance with the sign rule of trigonometric function.

$$f\left(r+a\right)=\left(+or-\right)g\left(a\right)$$

where “f” and “g” denote trigonometric functions, “r” denotes reference angles like 0, π/2, π and 2π and “a” denotes acute angle.

Let us consider an angle 7π/6. We are required to find sine and cotangent values of this angle. Here, we see that 7π/6 is greater than π. Hence, it is equal to π plus some acute angle, say, x.

$$\pi +x=\frac{7\pi }{6}$$ $$\Rightarrow x=\frac{7\pi }{6}-x=\frac{\pi }{6}$$ $$\Rightarrow \sin \frac{7\pi }{6}=\sin \left(\pi +\frac{\pi }{6}\right)$$

Since combination involves angle π, the sine of given angle retains the trigonometric function form. However, angle 7π/6 falls in third quadrant, in which sine is negative. Thus,

$$\Rightarrow \sin \frac{7\pi }{6}=\sin \left(\pi +\frac{\pi }{6}\right)=-\sin \frac{\pi }{6}=-\frac{\sqrt{3}}{2}$$

Similarly,

$$\Rightarrow \cot \frac{7\pi }{6}=\cot \left(\pi +\frac{\pi }{6}\right)=\cot \frac{\pi }{6}=\frac{1}{\sqrt{3}}$$

This method is very helpful to determine value of trigonometric function provided we know the value of trigonometric function of corresponding acute angle resulting from combination involving angles 0, π/2, π and 2π. Here, we shall work out few standard identities involving combination of angles with reference angles. We need not remember these identities. Rather, we should rely on the procedure discussed here as all of these can be derived on spot easily.