Trigonometric Functions of Angles of Any Magnitude

Trigonometric Functions of Angles of Any Magnitude: In some areas like Astronomy, Architecture, Physics, surveying, and Medicine, we come across the usage of trigonometric functions of non-standard angles, such as \(225^{\circ}, 315^{\circ}\), and \(\frac{2 \pi}{3}\). We can find the trigonometric functions of any angle of magnitude by observing the reference angle for the angle with respect to the quadrant it lies.

Each trigonometric function values vary from quadrant to quadrant. Some of the trigonometric functions are positive in the second quadrant, whereas others are positive in other quadrants. By observing the range of the trigonometric functions, we can depict the values of some angles of various functions.

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Trigonometric Functions of Angles of Any Magnitude

Trigonometric ratios are defined for acute angles as the ratio of the sides of a right-angled triangle. The extension of trigonometric ratios to any angle of radian measure (real numbers) are called trigonometric functions.

There is a total of six trigonometric functions sine ratio of the angle (sin), cosine ratio of the angle (cos), the tangent ratio of the angle (tan), cotangent ratio of the angle (cot), secant ratio of the angle (sec) and cosecant ratio of the angle (cosec) of the angles.

Trigonometric Functions of Standard Angles

Let us take a unit circle, centre at the origin. Let \(P(a, b)\) be any point on the circle. \(\angle A O P=x\) radians and the arc \(AP=x\).

From the figure, \(\cos x=a\) and \(\sin x=b\).

In right triangle \(O M P, O M^{2}+M P^{2}=O P^{2}\)

Thus, \(\cos ^{2} x+\sin ^{2} x=1\)

We know that, one complete revolution, the angle subtends at the center is \(2 \pi\) radians.

The quadrant angles \(\angle A O B=\frac{\pi}{2}, \angle A O C=\pi\) and \(\angle A O D=\frac{3 \pi}{2}\) are integral multiples of \(\frac{\pi}{2}\).

Consider the points \(A(1,0), B(0,1), C(-1,0)\) and \(D(0,-1)\).

Therefore, for quadrantal angles, we have

• \(\cos 0^{\circ}=1 ; \sin 0^{\circ}=0\)
• \(\cos \frac{\pi}{2}=0 ; \sin \frac{\pi}{2}=1\)
• \(\cos \pi=-1 ; \sin \pi=0\)
• \(\cos \frac{3 \pi}{2}=0 ; \sin \frac{3 \pi}{2}=-1\)
• \(\cos 2 \pi=1 ; \sin 2 \pi=0\)

Thus, we also observe that the values of sine and cosine functions do not change if \(x\) changes by any integral multiple of \(2 \pi\). Thus

• \(\sin (2 n \pi+x)=\sin x, n \in Z\)
• \(\cos (2 n \pi+x)=\cos x, n \in Z\)

Further,

\(\sin x=0\), if \(x=0, \pm \pi\), \(\pm 2 \pi\), \(\pm 3 \pi, \ldots\), i.e., when \(x\) is an integral multiple of \(\pi\). This implies that \(x=n \pi\), for the value of \(\sin x=0\).

\(\cos x=0\), if \(x=\pm \frac{\pi}{2}, \pm \frac{3 \pi}{2}, \pm \frac{5 \pi}{2}, \ldots\) i.e., \(\cos x\) vanishes when \(\mathrm{x}\) is an odd multiple of \(\frac{\pi}{2}\). Thus, \(\cos \mathrm{x}=0\) implies \(x=(2 n+1) \frac{\pi}{2}\), where \(\mathrm{n}\) is an integer.

The solutions to other trigonometric functions are listed below:

• \(\operatorname{cosec}=\frac{1}{\sin } x, x \neq n \pi\), where \(n\) is any integer.
• \(\sec x=\frac{1}{\cos x}, x \neq \frac{(2 n+1) \pi}{2}\), where \(n\) is any integer.
• \(\tan x=\frac{\sin x}{\cos x}, x \neq(2 n+1) \frac{\pi}{2}\), where \(n\) is any integer.
• \(\cot x=\frac{\cos x}{\sin x}, x \neq n \pi\), where \(n\) is any integer.

Signs of Trigonometric Functions

Since for every point \(P(a, b)\) on the unit circle, \(-1 \leq a \leq 1\) and \(-1 \leq b \leq 1\), we have \(-1 \leq \cos x \leq 1\) and \(-1 \leq \sin x \leq 1\) for all \(x\). We know that in the

• First quadrant \(\left(0<x<\frac{\pi}{2}\right)\) in which \(a\) and \(b\) are both positive
• Second quadrant \(\left(\frac{\pi}{2}<x<\pi\right)\), in which \(a\) is negative and \(b\) is positive.
• Third quadrant \((\pi<x<\frac{3 \pi}{2})\), in which \(a\) and \(b\) are both negative
• Fourth quadrant \(\left(\frac{3 \pi}{2}<x<2 \pi\right)\), in which \(a\) is positive and \(b\) is negative

Trigonometric Functions of Any Angle

To evaluate the trigonometric functions of any angle, we need to understand the reference angle. A reference angle is an angle made by the terminal side of an angle and the horizontal axis. The reference angle of \(\varnothing\) is called \(\emptyset\).

• It is positive.
• It is acute.

The trigonometric functions of any angle will be numerically the same as the respective functions of the reference angle with an accurate sign \((+/-)\) affixed. The sign is decided according to the quadrant where the terminal side is present. The reference angle will also be written in the same unit as the angle: degrees or radians.

Example:

• \(\frac{5 \pi}{3}\) can be written as \(2 \pi-\frac{\pi}{3}\)
• \(225^{\circ}\) can be written as \(\left(270^{\circ}-45^{\circ}\right)\)

A positive acute angle, \(\alpha\), produced by the terminal side of \(\alpha\), and the \(x\)-axis is the angle’s reference angle. The reference angles of various quadrants is listed below.

Values of Trigonometric Functions of Standard Angles

In trigonometry, a few common angles are employed more frequently. The values of trigonometric functions at such angles are listed in the table below.

Domain and Range of Trigonometric Functions

The input values of the trigonometric functions are known as the domain. The output values of the trigonometric functions are known as the range. The domain and range of trigonometric functions of any angle are listed below:

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Behaviour of Trigonometric Functions

Notice that, in the first quadrant, as \(x\) increases from \(0\) to \(\frac{\pi}{2}\), \(\sin x\) increases from \(0\) to \(1\), and as \(x\) increases from \(\frac{\pi}{2}\) to \(\pi\), \(\sin x\) decreases from \(1\) to \(0\). In the third quadrant, as \(x\) increases from \(\pi\) to \(\frac{3 \pi}{2}, \sin x\) decreases from \(0\) to \(-1\) and finally, in the fourth quadrant, \(x\) increases from \(\frac{3 \pi}{2}\) to \(2 \pi, \sin x\) increases from \(-1\) to \(0\). In the following table, we can discuss the behaviour of various trigonometric functions:

Solved Examples

Below are a few solved examples that can help in getting a better idea:

Q.1. If \(\cos x=-\frac{12}{13}\), and \(x\) lies in the third quadrant, determine the values of the other five trigonometric functions.

Ans: Given: \(\cos x=-\frac{12}{13}\)

We know that, \(\sec x=\frac{1}{\cos x}\)

\(\sec x=\frac{1}{-\frac{12}{13}}\)

\(\therefore \sec x=-\frac{13}{12}\)

We know that, \(\sin ^{2} x+\cos ^{2} x=1\), then \(\sin x=\sqrt{1-\cos ^{2} x}\)

\(\sin x=\sqrt{1-\left(-\frac{12}{13}\right)^{2}}\)

\(\sin x=\sqrt{1-\frac{144}{169}}\)

\(\sin x=\sqrt{\frac{169-144}{169}}\)

\(\sin x=\sqrt{\frac{25}{169}}\)

\(\therefore \sin x=\pm \frac{5}{13}\)

As \(x\) lies in third quadrant, \(\sin x\) is negative.

So, \(\sin x=-\frac{5}{13}\)

\(\operatorname{cosec} x=\frac{1}{\sin x}\)

\(\therefore \operatorname{cosec} x=\frac{-13}{5}\)

\(\tan x=\frac{\sin x}{\cos x}\)

\(\tan x=\frac{-\frac{5}{13}}{-\frac{12}{13}}\)

\(\therefore \tan x=\frac{5}{12}\)

\(\cot x=\frac{1}{\tan x}\)

\(\cot x=\frac{12}{5}\)

Q.2. Find the value of \(\sin 225^{\circ}\) and \(\cos 225^{\circ}\)

Ans: We know that \(225^{\circ}\) lies in the third quadrant.

Reference angle for this is \(270-\alpha\).

So, \(225=270^{\circ}-45^{\circ}\)

In the third quadrant, both sine and cosine of angles are negative. And, for multiples of \(\frac{\pi}{2}\) The sine of the angle becomes cosine and the angle’s cosine becomes sine.

\(\sin \left(\frac{n \pi}{2}-\theta\right)=\pm \cos \theta\) and \(\cos \left(\frac{n \pi}{2}-\theta\right)=\pm \sin \theta\)

So, \(\sin \left(225^{\circ}\right)=\sin \left(270^{\circ}-45^{\circ}\right)\)

\(\sin \left(225^{\circ}\right)=-\cos 45^{\circ}\)

\(\therefore \sin \left(225^{\circ}\right)=-\frac{1}{\sqrt{2}}\)

\(\cos \left(225^{\circ}\right)=\cos \left(270^{\circ}-45^{\circ}\right)\)

\(\cos \left(225^{\circ}\right)=-\sin 45^{\circ}\)

\(\therefore \cos \left(225^{\circ}\right)=-\frac{1}{\sqrt{2}}\)

Q.3. Find the values of other trigonometric functions. If \(x\) lies in the third quadrant and \(\cot x=-\frac{5}{12}\).

Ans: Given: \(\cot x=-\frac{5}{12}\)

\(\Rightarrow \tan x=\frac{1}{\cot x}\)

\(\tan x=\frac{1}{-\frac{5}{12}}\)

\(\therefore \tan x=-\frac{12}{5}\)

We know that, \(\sec ^{2} x=1+\tan ^{2} x\)

\(\Rightarrow \sec x=\sqrt{1+\left(-\frac{12}{5}\right)^{2}}\)

\(\sec x=\sqrt{1+\frac{144}{25}}\)

\(\sec x=\sqrt{\frac{169}{25}}\)

\(\therefore \sec x=\pm \frac{13}{5}\)

Since \(x\) lies in second quadrant, \(\sec x\) will be negative.

\(\therefore \sec x=-\frac{13}{5}\)

\(\cos x=\frac{1}{\sec x}\)

\(\cos x=\frac{1}{-\frac{13}{5}}\)

\(\therefore \cos x=-\frac{5}{13}\)

We know that, \(\sin ^{2} x+\cos ^{2} x=1\), then \(\sin x=\sqrt{1-\cos ^{2} x}\)

\(\sin x=\sqrt{1-\left(-\frac{5}{13}\right)^{2}}\)

\(\sin x=\sqrt{1-\frac{25}{169}}\)

\(\sin x=\sqrt{\frac{169-25}{169}}\)

\(\sin x=\sqrt{\frac{144}{169}}\)

\(\therefore \sin x=\pm \frac{12}{13}\)

As \(x\) lies in second quadrant, then \(\sin x\) is positive.

\(\therefore \sin x=\frac{12}{13}\)

\(\operatorname{cosec} x=\frac{1}{\sin x}\)

\(\therefore \operatorname{cosec} x=\frac{13}{12}\)

Q.4. Explain the behaviour of \(\text {cos}\) function in all quadrants.

Ans:

We can notice that, in the first quadrant, as \(x\) increases from \(0\) to \(\frac{\pi}{2}, \cos x\) decreases from \(1\) to \(0\), and as \(x\) increases from \(\frac{\pi}{2}\) to \(\pi, \cos x\) decreases from \(0\) to \(-1\). In the third quadrant, as \(x\) increases from \(\pi\) to \(\frac{3 \pi}{2}, \cos x\) increases from \(-1\) to \(0\) and finally, in the fourth quadrant, \(\cos x\) increases from \(0\) to \(1\) as \(x\) increases from \(\frac{3 \pi}{2}\) to \(2 \pi\).

Q.5. Find the trigonometric functions of angles of \(-x\).

Ans: The cosine and secant functions are positive for a negative angle, whereas other functions are negative only.

• \(\sin (-x)=-\sin x\)
• \(\cos (-x)=\cos x\)
• \(\tan (-x)=-\tan x\)
• \(\cot (-x)=-\cot x\)
• \(\sec (-x)=\sec x\)
• \(\operatorname{cosec}(-x)=-\operatorname{cosec} x\)

Summary

Trigonometric functions describe the relationship between sides and angles of a triangle. There are six trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant. They have their applications in the field of Marine, Engineering, and Medical. They are also used to find the height of objects and the depth of the oceans.

The trigonometric functions of any magnitude can be found by using the reference angle. The reference in the first quadrant is the angle itself, in the second quadrant, it is \(180^{\circ}-\alpha\), in the third quadrant, it is \(270^{\circ}-\alpha\), whereas, in the fourth quadrant, it is \(360^{\circ}-\alpha\). The reference angle is expressed in the same unit of the angle, radians or degrees.

FAQs on Trigonometric Functions of Angles of Any Magnitude

Students might be having many questions regarding the Trigonometric Functions of Angles of any Magnitude. Here are a few commonly asked questions and answers.

Q.3. What are the trigonometric functions of an angle?
Ans: In trigonometry, there exist six functions of an angle that are commonly used.

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Cotangent (cot)
• Secant (sec)
• Cosecant (cosec) 

Q.4. What are the reference angles for \(\alpha\) in various quadrants?
Ans: The reference angle for the given angle \(\alpha\) is listed below.

• First quadrant: \(\alpha\)
• Second quadrant: \(\left(180^{\circ}-\alpha\right)\)
• Third quadrant: \(\left(270^{\circ}-\alpha\right)\)
• Fourth quadrant: \(\left(360^{\circ}-\alpha\right)\)

Q.5. What are the uses of trigonometric functions?
Ans: In geometric figures, trigonometric functions calculate unknown angles and distances from known or measured angles. Trigonometry evolved due to the need to calculate angles and distances in Astronomy, mapmaking, surveying, and artillery range finding.

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